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Member of American Society of Ciiil Engineers 
Professor cj Civil Engineering in Tujls College 




&^^_P^^y^ NEW YORK 


London: CHAPMAN & HALL, Limited 



Copyright, 1902, 1906, 




There is an opinion among engineers that too 
often students are not well grounded in the practical 
problems of Mechanics ; that they know more of 
theory and mathematical deductions than of practical 
applications. A prominent educator has recently 
said to me, in regard to the teaching of Mechanics, 
" I am convinced that it is to be done more thoroughly 
in the future than in the past ; " and it will be done, 
he believes, by sticking close to elementary principles 
as developed by well-chosen practical problems. Fur- 
thermore, he adds, " it will have to be recognized that 
all an engineering baccalaureate course can worthily 
accomplish is to give the raw recruit the ' setting-up ' 
exercises in Mechanics." 

It is now generally recognized, I think, that this 
subject should cover first of all the elements and 
fundamental principles that form the basis of every 
engineer's knowledge ; that these necessary elements 
and principles are best understood and best remem- 
bered by actually solving numerous problems that 
present important facts illustrative of every-day engi- 
neering practice, and arouse the student's interest 
far better than abstract examples which can be easily 
formulated from imaginary conditions. 

Therefore, for the reasons indicated above, an effort 



has been made in preparing this book to present, from 
actual conditions, many practical problems together 
with brief definitions and solutions of typical prob- 
lems which should help the student in Mechanics 
to follow the advice once given by George Stephenson 
to his son Robert : 

" Learn for yourself, think for yourself, 
make yourself master of principles." 

Photographs or electroplates have been furnished 
for certain of the illustrations as follows : 

Page 17 by Otto Gas Engine Works; pages 20 and 32, Pelton 
Water Wheel Company ; page 24, Wellington- Wild Coal Company ; 
page 25, Harrisburg Foundry and Machine Company; page 29, 
Fall River Iron Works Company ; page 35, Associated Factory 
Mutual Fire Insurance Companies; page d^,, Maryland Steel Com- 
pany; page 64, Bucyrus Company; page 120, A. J. Lloyd & Co.; 
page 146, CUnton Wire Cloth Company; page 148, The Detroit 
Graphite Manufacturing Company ; page 149, The Engineering 
Record; pages 151 and 169, Brown Hoisting and Conveying Machin- 
ery Company; page 153, Cement Age; page 157, Fig. 84, American 
Locomotive Company ; page 161, Carson Trench Machinery Com- 
pany ; page 164, Chicago Bridge and Iron Works ; page 165, Chap- 
man Valve Manufacturing Company. 


Tufts College, Mass., 
June, 1906. 



Problems i to 172. 

Raising weights, overcoming resistances of railroad 
trains, macliine punch, construction of wells and 
chimneys, operation of pumping engines. Force and 
distance or foot-pounds required in cases of pile- 
driver, horse, differential pulley, tackle, tram car . . 7 


Required by windmills, planing machines, gas engine, 
locomotive, steam engines — simple, compound, triple, 
slow speed, high speed engines. Horse-power from 
indicator cards, required by electric lamps, driving- 
belts, steam crane, coal towers, pumping engine, 
canals, streams, turbines, water-wheels. Efficiency, 
force or distance required in cases of fire pumps, 
mines, bicycles, shafts, railroad trains, air brakes, the 
tide, electric motors, freight cars, ships 16 


Foot-pounds, horse-power, velocity: — Ram, hoisting- 
engine, blacksmith, electric car, bullet, cannon, nail, 
pendulum. Energy resulting from motion of fly-wheel 
and energy required by jack-screw 44 



Problems 172 to 414. 


Canal boat being towed, rods, struts, beams, derrick, 
cranes set as in action ; balloon held by rope, ham- 
mock supported ; wagon, trucks, picture supported ; 
forces in frames of car dumper, tripod, shear legs, 
dipper dredge ; also in triangle, square, sailing vessel, 
rudder, foot-bridge, roof-truss .... 51 


Beam balanced, pressure on supports, propelling force 
of oars, raising anchor force at capstan, bridge loaded 
pressure on abutments, lifting one end of shaft, boat 
hoisted on davit, forces acting on triangle, square, 
supports of loaded table and floor 72 


Brake wheel, forces acting on square 84 


Beam leaning against wall, post in truss, rope pull on 
chimney, connecting rod of engines, trap-door held up 
by chain 86 


Rods with loads, metal square and triangle, circular 
disk with circular hole punched out, box with cover 
open, rectangular plane with weight on one end, 
irregular shapes, solid cylinder in hollow cylinder, 
cone on top of hemisphere 90 


Weight moved on level table, stone on ground, 
block on inclined plane, gun dragged up hill, cone 
sliding on inclined plane ; friction of planing machine. 



locomotives, trains, ladder against wall, bolt thread, 
rope around a post ; belts, pulleys and water-wheels 
in action ; heat generated in axles and bearings. . . 96 


Problems 414 to 527. 


Railroad train, ice boat, stone falling and depth of 
well, balloon ascending, cable car running wild. . . .119 


Aim in front of deer, rowing across river, bullet hit- 
ting balloon ascending, rain on passenger train, wind 
on steamer, two passing railroad trains 126 

Train stopped, steamer approaching dock, cannon 
recoil, locomotive increasing speed, body moved on 
table, box-machine, motion of table, barrel of flour on 
elevator, man's weight on elevator, cage drawn up 
coal shaft. 121 


Inclination for bullet to strike given point, motion 
down plane, stone dropped from train, thrown from 
tower, projectile from hill, from bay over fortification 
wall 133 


Simple, conical, ball in passenger car 141 


Water suddenly shut off, cricket ball struck, hammer 
falling on pile, shot from gun, bullet from rifle, freight 
and passenger trains collide 142 


Problems 528 to 600. 


Water turbine test, suspension bridge, Niagara tower, 
launching data, coal-wharf incline, typical American 
bridge, modern locomotive tests, wood in compression, 
actual cableway, St. Elmo water-tower, outside-screw- 
and-yoke valve, cast-iron pipe, retaining walls, geared 
drum, gas-engine test 145 


Yale, Tufts, Harvard 174 


600 problems, besides 43 under Examinations. About 
one-half have answers given 1S4 


Work, force, and motion and their sub-divisions .... 2 


FalHng Bodies, Functions of Angles, Unit Values — 
heights and velocities • 190 

INDEX 193 



The problems and solutions that follow have been 
arranged in the order of Work, Force, and Motion. 
At the beginning of each important section one 
problem has been solved so as to explain the method 
of solving similar problems and to serve as a guide for 
solutions to be put in note-books. An effort has been 
made throughout the book to simplify. Few methods 
have been presented ; the calculus has been used only 
where necessary ; no discussion has been offered of 
the term mass — many such subjects have been left 
for more advanced courses or extended treatises. 

The " gravitation system " of units — the foot- 
pound-second system, or meter-kilogram-second sys- 
tem — known as the engineers' system has /been 
used exclusively. 

In engineering practice one is often puzzled to tell 
just what data to collect and afterward how much of 
it to use ; because of this, I have left more data in 
some of the problems, and especially those under 
Review, than is absolutely necessary for solving the 
problem, and the student will have opportunity '* to 
pick and choose " just as he would do in actual cases. 



Mechanics is the science that treats of the action 
of forces at rest and in motion. 

Work, Force, and Motion are the three sub-divisions 
of Mechanics considered in this book. 


Work is done by the action of force through some 

Work is measured by the product of force times the 
distance through which it acts. 

Work = force x distance, — a formula fundamental 
for all Work problems. 

Energy is the amount of work that a body possesses. 

Potential energy is the work that a body possesses 
by virtue of its position above the earth's surface. 

Kinetic energy is the work that a body possesses by 
virtue of its velocity. 

Horse-power is the rate of doing work. One horse- 
power is the equivalent of 33 000 foot-pounds of 
work done per minute. 


Force in Mechanics has both magnitude and direc- 
tion, and in this treatise 


Force Magnitude is usually expressed in pounds. 
It may act as pressure, a push, or as tension, a pull. 

Concurrent forces acting on a body arc those that 
have the same point of application. 

Non-concurrent have different points of application. 

Moment of a force about a point or axis is the 
product obtained by multiplying the magnitude of 
the force by the shortest distance from the point or 
axis to the line of action of the force. 

Moment = force x perpendicular. Clockwise ten- 
dency of rotation is usually taken positive. 

Resultant of a system of concurrent forces is a 
single force that might be substituted for them with- 
out changing the effect. 

Equilibriant of a system of forces is a single force 
that balances them. The equilibriant is equal and 
opposite to the resultant. 

Components of a single force are the forces that 
might be substituted for it without changing the 

Parallelogram of forces. When three forces that 
are in equilibrium meet in a point they can be repre- 
sented in magnitude and direction by the sides of a 
parallelogram. This parallelogram is called the 
parallelogram of forces. 


1. S Vertical components = o. When the forces 
acting in one plane upon a body are in equihbrium, 
the forces can be resolved into components in any 
one direction, and the algebraic sum of the compo- 
nents will equal o. Likewise, 

2. 2 Horizontal components = o. The algebraic 
sum of the components in a direction perpendicular 
to that of I will equal zero ; and 

3. S Moments = o. The algebraic sum of the 
moments of the forces taken about any point or axis 
in the plane will equal zero. 

These three axioms can frequently be used to 
formulate three equations that contain unknown 
quantities which can then be determined. 

A Couple consists of two equal, opposite, parallel 
forces not acting in the same straight line. 

Moment of a couple is the product of one of the 
equal forces by the perpendicular distance between 

Center of gravity of a body or a system of bodies is a 
point about which the body or system can be imagined 
to balance and the forces of gravity will cause no 

Centroid and Center of Mass are terms that are 
sometimes used in preferenc? to center of gravity. 

Centroid is the point of application of a system of 
parallel forces. 



Motion (uniform) is that in wliich a body moves 
through equal distances in equal times. 

Motion (accelerated) is that in which a body moves 
through unequal distances in equal times. 

Motion (uniform-accelerated) is that in which the 
velocity increases the same amount in each unit of 
time, which is generally taken as the second. 

Acceleration is the gain or loss in velocity per unit 
of time. 

Centrifugal force. When a body is compelled to 
move in a curved path it exerts a force directed out- 
wards from the center ; its amount is the centrifugal 

W If" 
force = 

g r 

Impact is said to take place when one body strikes 
against another. 

A period of compression thus occurs, and the forces 
acting are Impulsive forces of compression. Then 
follows a period of restitution. 

Coefficient of restitution e for any pair of substances 
is the ratio of the impulsive force of restitution to the 
impulsive force of compression. 

/■ >'' OK THE 





1. A 20th-century express of 100 tons weight goes 
up a grade of i vertical in 1 20 horizontal ; the re- 
sistances are 1 5 pounds per ton. Find the amount 
of work that locomotive expends per mile of travel. 


of locomotive 

= work + work 

of friction of lifting train 


of friction 

= force X distance 


= 15 X 100 

= I 500 pounds 


= I mile 

= 5 280 feet 


of friction 

= I 500 X 5 280 

= 792 000 foot-pounds 
Work = force X distance 

of lifting train 

Force = 100 x 2 000 

= 200 000 pounds 
Distance = 5 2 8dNX y^^ 
= 44 feet 
.-. Work = 200 000 X 44 

of lifting train 

= 8 800 000 foot-pound 
Work = 792 000 + 8 800 000 

of locomotive 

= 9 592 000 foot-pounds 

2. Find the work done by an cni;ine in drawing a 
train i mile along a level railroad, w hen the constant 
resistances of friction, air, and so on, are i ton. 



3. A punch exerts a uniform pressure of 36 tons 
in punching a hole through an iron plate of one-half 
inch thickness. Find the foot-pounds of work done. 

4. Find what work is being done per minute by 
an engine that is raising 2 000 gallons of water an 
hour from a mine 300 feet deep. 

5. If a weight of i 130 povmds be lifted up 20 
feet by 20 men twice in a minute, how much work 
does each man do per hour } 

6. A number of men can each do, on the average, 
495 000 foot-pounds of work per day of 8 hours. 
How many such men are required to do 33 000 x 10 
foot-pounds of work per minute.? 

7. A centrifugal pump delivers water 10 feet 
above the level of a lake of half a square mile area. 
At the end of a day's pumping, the water has been 
lowered i|^ feet. How much work has been done } 

" Distance " will be lo feet plus | of 1 1 feet. 

8. Water in a well is 20 feet below the surface of 
the ground, and when 500 gallons have been pumped 
out it is 26 feet below. Find the work done. 

9. Brick and mortar for a chimney 100 feet high 
are raised to an average height of 35 feet. Total 
amount of material used 40 000 cubic feet or about 
5 600 000 pounds. What work was done } 

10. What work is done in winding up a chain that 
hangs vertically, is 130 feet long, and weighs 20 
pounds per foot 1 


11. A chain of weight 300 pounds and length i 50 
feet, with a weight of 500 pounds at the end of it, is 
wound up by a capstan. What work is done ? 

12. A stream of width 20 feet, average depth 3 
feet, and mean velocity of 3 miles per hour has an 
available fall of 80 feet. What work is stored in the 
quantity of water flowing each minute ? 

Find tlie pounds of water flowing by observing that 
Quantity = area X velocity. 

13. A horse draws 150 pounds of earth out of a 
well, by means of a rope going over a fixed pulley, 
which moves at the rate of 2\ miles an hour. Neg- 
lecting friction, how many units of work does this 
horse perform a minute } 

14. A cylindrical shaft 14 feet in diameter must be 
sunk to a depth of 10 fathoms through chalk, the 
weight of which is 144 pounds per cubic foot. Find 
the work done. 

15. A well is to be dug 20 feet deep and 4 feet in 
diameter. Find the work in raising the material, sup- 
posing that a cubic foot of it weighs 140 pounds. 

16. A horse draws earth from a trench by means 
of a rope going over a pulley. He pulls up, twice 
every 5 minutes, a man weighing 130 pounds, and a 
barrowful of earth weighing 260 pounds. Each time 
the horse goes forward 40 feet. Find the useful 
work done per hour. 

17. A body weighing 50 pounds slides a distance 


of 8 feet down a plane inclined 20° to the horizontal, 
against a constant retarding force of 4 pounds. 
Compute the total work done upon the body by 
(gravity) its weight and the friction. 

18. What work is stored in a cross-bow whose 
cord has been pulled 1 5 inches with a maximum force 
of 224 pounds } 

19. If 25 cubic feet of water are pumped every 
5 minutes from a mine 140 fathoms deep, what 
amount of work is expended per minute ? 

20. In pumping i 000 gallons from a water-cistern 
with vertical sides the surface of the water is lowered 
5 feet. Find the work done, the discharge being 10 
feet above the original surface. 

21. A uniform beam weighs i 000 pounds, and is 
20 feet long, it hangs by one end, round which it can 
turn freely. How many foot-pounds of work must be 
done to raise it from its lowest to its highest 
position ? 

22. A body is suspended by an elastic string of 
unstretched length 4 feet. Under a pull of 10 
pounds the string stretches to a length of 5 feet. 
Required the work done on the body by the tension 
of the string while its length changes from 6 feet to 
4 feet. 

23. A weight of 200 pounds is to be raised to a 
height of 40 feet by a chord passing over a fixed 
smooth pulley ; it is found that a constant force P, 
pulling the chord at its other end for three-fourths of 


the ascent, communicates sufficient velocity to the 
weight to enable it to reach the required height. 
Find P. 

Work = force x distance 

Work =200 X 40 

on weight 

Work = P X 2 of 40 

by pull 

Work = Work 

on weight by pull 

200 X 40 = P X 30 

P = 2665 pounds 

24. A horse drawing a cart along a level road at 
the rate of 2 miles per hour performs 29 216 foot- 
pounds of work in 3 minutes. What pull in pounds 
does the horse exert in drawing the cart > 

25. It is said that a horse can do about 1 3 200 000 
foot-pounds of work in a day of 8 hours, walking at 
the rate of 2^ miles per hour. What pull in pounds 
could such a horse exert continuously during the 
working-day } 

26. If a horse walking once round a circle i o yards 
across raises a ton weight 18 inches, what force does 
he exert over and above that necessary to overcome 
friction 1 

27. A building of weight 50 000 pounds is being 
moved on rollers by a horse that is pulling on a pole 
with a distance of 10 feet from the center of a capstan 
that is 1 8 inches in diameter. If the total friction is 
200 pounds per ton, what force must the horse exert ? 



28. The 500-pound hammer of a pile-driver is 
raised to a height of 20 feet and then allowed to 
fall upon the head of a pile, which is driven into the 
ground i inch by the blow. Find the average force 
which the hammer exerts upon the head of the pile. 

Work = force X distance 
= 500 X 20 
= 10 000 foot-pounds 
Distance =tV ^^^^ 
/. 10 000 foot-pounds = force x iV ^^^t 
.-. force =10 000 X 12 

= 120 000 pounds 

29. A hammer weighing i ton falls from a height 
of 24 feet on the end of a vertical pile, and drives it 
half an inch deeper into the ground. Assume the 
driving force of the hammer on the pile to be con- 
stant while it lasts, and find its amount expressed in 
tons weight. 

30. Determine by the principle of work, 
neglecting friction, the relation between the 
pull P and the load W in case of the differ- 
ential wheel-and-axle of Fig. i. 
For one revolution, 

Work = P X 2 TT^ 

of P 

Work = I W X 2 -nr' - ^ W X 2 irr 

on weight 

P X 2 TT^? = I \\V X 2 (r' — r) 
P X 2 ^ = W {?■' - r) 
W _ 7' - r 
"P ~ 2a * 


31. A barrel of Portland cement that weighs 396 
pounds is to be hoisted by a wheel and axle as in Fig. 
I ; the radii are 6, 12 and 18 inches. What force 
will be required ? 

32. If, neglecting frictions, a power of 10 pounds, 
acting on an arm 2 feet long, produces in a screw- 
press a pressure of half a ton, what would be the 
pitch of the screw .? 

33. What is the ratio of the weight to the power, 
in a screw-press working without friction, when the 
screw makes 4 turns in the inch, and the arm to 
which the power is applied is 2 feet long ? 

34. What force applied at the end of an arm 
18 inches long will produce a pressure of i 000 
pounds upon the head of a smooth screw when 1 1 
turns cause the head to advance two-thirds of an 
inch } 

35. Find the mechanical advantage in a differential 
screw, if the length of the power arm is 2 feet, and 
there are 4 threads to the inch in the large screw, 
and 5 threads to the inch in the small screw. 

36. In a differential pulley, if the radii of the 
pulleys in the fixed block are as 3 to 2 ; and if the 
weight of the lower block is \\ pounds, what weight 
can be raised by a force of 5 pounds } 

37. In a wheel and axle the diameter of the wheel 
is 7 feet, of the axle 7 inches. What weight can be 


raised by a force of lo pounds acting at the circum- 
ference of the wheel ? 

38. A weight of 448 pounds is raised by a cord 

which passes round a drum 3 feet in diameter, having 

on its shaft a toothed wheel also 3 feet in diameter ; 

a pinion 8 inches in diameter, and driven by a winch 

handle 16 inches long, gears with the 

wheel. Find the power to be applied to 

the winch handle in order to raise the 


39. A tackle is formed of two blocks, 
each weighing 1 5 pounds, the lower one 
being a single movable pulley, and the 
upper or fixed block having two sheaves ; 
the parts of the cord are vertical, and 
the standing end is fixed to the movable 
block. What pull on the cord will sup- 
port 200 pounds hung from the movable 

block .!^ and what will then be the pull on the staple 

at the upper block 1 

40. A weight of 400 pounds is being raised by a 
pair of pulley blocks, each having two sheaves ; the 
standing part of the rope is fixed to the upper block, 
and the parts of the rope, whose weight may be dis- 
regarded, are considered to be vertical ; each block 
weighs 10 pounds. What is the pressure on the 
point from which the upper block hangs } 

41. Two equal weights, each 1 12 pounds, are joined 
by a rope which runs over two pulleys A and B 12 


feet apart and in the same horizontal Hne. If a 
weight of ten pounds is lowered on to the rope half- 
way between A and B how far will the rope deflect ? 

Work = Work 

of lo-pound weight of two 112-pound weights. 

42. A weight of 500 pounds, by falling through 
36 feet, lifts, by means of a machine, a weight of 60 
pounds to a height of 200 feet. How many units of 
work has been expended on friction, and what ratio 
does it bear to the whole amount of work done t 

43. The pull on a tram-car was registered when 
the car was at the following distances along the track ; 
0,200 pounds; 10 feet, 150 pounds; 25 feet, 160 
pounds ; 32 feet, 156 pounds; 41 feet, 163 pounds; 
56 feet, 170 pounds; 60 feet, 165 pounds; 73 feet, 
160 pounds. What effective work was done in pulling 
the car through the distance of 73 feet, and what 
constant pull would have produced the same work ? 

44. In lifting an anchor of i J tons from a depth 
of 1 5 fathoms in 6 minutes, what is the useful man- 
power, if a man-power is defined as 3 500 foot-pounds 
per minute } 

45. Four hundred weight of material are drawn 
from a depth of 80 fathoms by a rope weighing i . 1 5 
pounds per linear foot. How much work is done 
altogether, and how much per cent is done in lifting 
the rope.-* How many units of 33000 foot-pounds 
per minute would be required to raise the material in 
4| minutes } 


H O R S E-P O W E R 

46. A gas engine must hoist 3 tons of grain through 
a vertical height of 50 feet every minute. What 
horse-power must be provided ? 

Work = force X distance 

of engine [minUtC 

= (3 X 2 000) pounds X 50 feet per 
Now I horse-power = t^t^ 000 foot-pounds per minute 

Axr 1 3X2 000 X £50 
- .-.Work = ^— ^ 

of engine ZZ ^^^ 

— 9tt horse-power 

47. A hod-carrier who weighs 1 5 5 pounds carries 
65 pounds of brick to the third story, a vertical height 
of 20 feet. How many foot-pounds of work has he 
done ? If he makes 10 such trips in an hour, at what 
rate in horse-power does he work ? 

48. A windmill raises by means of a pump 22 tons 
of water per hour to a height of 60 feet. Supposing 
it to work uniformly, calculate its horse-power. 

49. The travel of the table of a planing-machine 
which cuts both ways is 9 feet. If the resistance 
while cutting be taken at 400 pounds, and the 
number of revolutions or double strokes per hour be 
80, find the horse-power absorbed in cutting. 

50. A forge hammer weighing 300 pounds makes 
100 lifts a minute ; the perpendicular height of each 
lift is 2 feet. What is the horse-power of the engine 
that operates 20 such hammers ? 



51. An Otto gas engine is shown in the above 
illustration. It has a belt pulley that is 36 inches in 
diameter, and makes 150 revolutions per minute. 


What force for driving, shafting, and machinery, 
therefore, can the belt transmit when the engine 
is developing its rated horse-power of twenty- 

52. How many horse-power would it take to raise 
3 hundred weight of coal a minute from a pit whose 
depth is 66o feet ? 

53. Find the horse-power of an engine which is to 
raise 30 cubic feet of water per minute from a depth 
of 440 feet. 

54. Find the horse-power required to draw a train 
of 100 tons, at the rate of 30 miles an hour, along a 
level railroad, the resistance from friction being 16 
pounds per ton. 

55. Each of the two cylinders in a locomotive 
engine is 16 inches in diameter and the length of 
crank is i foot. If the driving-wheels make 105 
revolutions per minute, and the mean effective steam- 
pressure is 85 pounds per square inch, what is the 
horse-power } 

56. The weight of a train is 95.5 tons, and the 
drawbar pull is 6 pounds per ton. Find the horse- 
power required to keep the train running at 25 miles 
per hour. 

57. A train, whose weight including the engine is 
100 tons, is drawn by an engine of 150 horse-power ; 
friction is 14 pounds per ton — all other resistances 
neglected. Find the maximum speed which the 
engine is capable of maintaining on a level track. 


In the electrical problems that follow observe that 
I kilowatt = 1 ,340 horse-power 
I horse-power = 746 watts 

Watts = volts X amperes 

58. A dynamo is driven by an engine that develops 
230 horse-power. If the efficiency of dynamo is 0.81 » 
what ** activity " in known kilowatts is represented by 
the current generated ? 

59. Electric lamps giving i candle-power for 4 
watts (a) how many 10- and (d) how many i6-candle 
lamps may be worked per electric horse-power ? The 
combined efficiency of engine, dynamo, and gearing 
being 70 per cent, what is the candle-power available 
for every indicated horse-power ? 

60. What electrical current expressed in amperes 
will be used by a 250-volt electric hoist when raising 
2 500 pounds of coal per minute from a ship's hold 
1 50 feet below dump cars on trestle work, the effi- 
ciency of the whole arrangement being 50 per cent ? 

61. A prospective electric company can find a 
market for 900 electrical horse-power at a city 20 
miles from a suitable water-power. Engineers esti- 
mate losses in generating machinery 10% ; in line 
y% ; in transformers at load end 10% ; and the 
efficiency of turbines 85%. The average velocity of 
the riv^er is 2 feet per second ; width available near 
dam 40 feet ; depth 5 feet. Eind (a) the water- 
power that would be required (b) the net fall that 
proposed dam must afford. 



Fig. 3. 

62. A water-motor is driven by two jets i inch in 
diameter, flowing with velocity of 8o feet per second. 
Theoretic horse-power would be 9.9 ; and if efficiency 
of wheel is 85 per cent, and the generator which 
the wheel drives also 85 per cent, what power in 
kilowatts does the current represent t 

63. What is the difference in tensions of the two 
sides of a 30-inch driving belt that is running 4 200 
feet a minute, and transmitting 300 horse-power .? 

In belt problems the difference in tensions represents " force." 

64. Find the speed of a driving-pulley 3.5 feet in 
diameter to transmit 6 horse-power, the driving-force 
of the belt being 150 pounds. 


65. A belt is designed to stand a difference in 
tension of lOO pounds only. Find the least speed at 
which it can be driven to transmit 20 horse-power. 

66. A pulley 3 feet 6 inches in diameter, and mak- 
ing 150 revolutions a minute, drives by means of a 
belt, a machine which absorbs 7 horse-power. What 
must be the width of the belt so that its greatest ten- 
sion may be 70 pounds per inch of width, it being 
assumed that the tension in the driving-side is twice 
that on the slack side .-* 

67. An endless cord stretched and running over 
grooved pulleys with a linear velocity of 3 000 feet 
per minute, transmits 5 horse-power. Find the dif- 
ference in tensions of the cord in pounds. 

68. A rope drive has a grooved pulley 14 feet in 
diameter that makes 30 revolutions per minute. The 
difference in tensions being 100 pounds, find the 
horse-power transmitted. 

69. A locomotive that can develop 500 horse- 
power is drawing a train of total weight 100 tons up 
a 2 per cent grade ; resistances are 10 pounds per 
ton. Find the highest speed that can be attained. 

Work = Work -f Work 

of locomotive of resistances of lifting train 

500 X 2>2> 000 = 10 X iooX^/-f- 100 X 2 000 X Too X '^ 
5 X 33 000 = 10 X ^-h 20 X 2 X d 

50 d = 5 X 33 000 

d = 3 300 feet per minute 
= 37^ miles per hour. 


70. A train of lOO tons weight runs at 42 miles 
an hour on a level track ; resistances are 8 pounds per 
ton. Find the speed of train up a i per cent grade 
(i foot rise in 100 feet horizontal) if the engine-power 
is kept constant. 

71. In 1895 a passenger engine on the Lake Shore 
Railroad made a run of Z6 miles at the rate of 73 
miles an hour. Weight of train, 250 tons ; resistance 
on level track, 1 5 pounds per ton. The engine was a 
10- wheeler, having drivers 5 feet 8 inches in diameter 
and cylinders 17 X 24 inches. When 730 horse- 
power was developed up a i per cent grade what was 
the average draw-bar pull t 

72. A 98 -horse-power automobile has by test in 
Colorado drawn a special 36-ton locomotive up a 12 
per cent highway grade at the rate of four miles an 
hour. What were the frictional resistances per ton 1 

73. A modern farming machine equipped with a 
loo-horse-power automobile will plow, sow, and harrow, 
all at the same time, a strip 30 feet wide at the rate 
of 3|- miles an hour, or 80 acres a day. What force 
is developed for each foot width of ground } 

74. Find the total horse-power of two engines 
which are taking a train of 250 tons down a grade of 
I in 200 at 60 miles an hour, supposing the resistance 
on the level at this speed to be 35 pounds a ton. 


75. An automobile that weighs 5 tons goes up a 
rough road of grade i vertical to 10 horizontal ; air 
and frictional resistances are 16 pounds per ton. 
What horse-power must the motor develop to main- 
tain a speed of 20 miles an hour } 

76. Find the horse-power of a locomotive which is 
to move at the rate of 20 miles an hour up an incline 
which rises i foot in 100, the weight of the locomo- 
tive and load being 60 tons, and the resistance from 
friction 12 pounds per ton. 

77. A steam-crane, working at 3 horse-power, is 
able to raise a weight of 10 tons to a height of 50 feet 
in 20 minutes. What part of the work is done against 
friction t If the crane is kept at similar work for 8 
hours, how many foot-pounds of work are wasted on 
friction .'* 

78. The six-master shown on the next page carries 
5 500 tons of coal. It is unloaded by small engines 
which take up i ton at each hoist ; average lift from 
hold of ship to top of chutes which lead to cars, 
35 feet; weight of bucket, i ton; 2 trips are made 
per minute, and 25 per cent of power of engine is 
lost in friction and transmission. When two towers 
are working how long will it take to unload the 
vessel ? 



The illustration of six-master on opposite page accompanies 
Problem 78. 

79. An average size coal barge will carry i 600 
tons. If it is unloaded by two simple direct engines, 
the coal being hoisted 65 feet to an elevated hopper 
on the wharf, weight of bucket i ton, and carrying i 
ton of coal, what horse-power of engines would be re- 
quired to unload the i 600 tons in 20 hours? 

80. The locomotive of problem 71 made 360.7 
revolutions per minute. What was the mean effective 
cylinder pressure } 

Work = force X distance 
Force = ] tt 1 7- x /^ X 2 
Distance = 2 x 360.7 x \% 
730 X ZZ 000 = (} It if X /'X 2) X (24 X 360.7 X fl) 

Fig. 4. 

81. The engine shown in Fig. 4 has steam cylin- 
der 1 5 inches in diameter ; length of stroke, 1 5 inches ; 



revolutions per minute, 275 ; mean effective pressure, 
^6 pounds per square inch. Find the horse-power. 

82. The indicator cards illustrated herewith were 
taken from an engine of the type shown in problem 
81, diameter of steam cylinder being 14 inches, 
length of stroke 12 inches, revolutions per minute 
300. Scale on cut the mean ordinates, which were 
produced by indicator springs of stiffness 40 pounds 
to an inch, and compute the indicated horse-power of 
the engine. 

Fig. 5. Full Load Indication. 

83. The indicator cards shown below were taken 
from one of the triple-expansion pumping-engines at 
the East Boston Station of the Metropolitan Sewerage. 
The cards were from two ends of a high-pressure 
cylinder. Refer to the cards and compute the indi- 
cated horse-power. (A twenty-four hours' duty trial 
of this pumping-engine was made January 17-18, 
1 90 1, by engineering students of Tufts College.) 



Fig. 6. Head end. Card shown, one-half size ; area of original, 4.69 square 
inches ; stiffness of spring, 50 pounds per square inch ; length of stroke, 
30 inches ; revolutions per minute, 84 ; piston diameter, 13^ inches. 

Fig. 7. Crank end. Card shown, one-half size ; area of original, 4.62 square 
inches : stiffness of spring, 50 pounds per square inch ; length of stroke 
30 inches ; revolutions per minute, 84 ; piston diameter, 13^ inches. 

84. The average breadth of an indicator diagram 
for one end of a piston is 1.58 inches, and for the 
other end it is 1.42 inches, and i inch represents 32 
pounds per square inch. Piston, 1 2 inches diameter ; 
crank, i foot long; revokitions per minute, no. 
What is the indicated horse-power } 

85. The cyHnder of a steam-engine has an internal 
diameter of 3 feet ; length of stroke, 6 feet ; and it 
makes 10 strokes per minute. Under what effective 
pressure per square inch would it have to work in 
order that the piston may develop 1 2 5 horse-power ? 


The illustration of triple-expansion engines on opposite p»age 
accompanies Problem 86. 

86. Four pairs of triple-expansion steam-engines 
are used to drive the cotton machinery of the largest 
Fall River corporation. One of these engines shown 
in illustration has cylinders 26|- inches diameter, 36J, 
and 54. The steam pressures are : In main pipe, i 50 
pounds per square inch ; in receiver between high and 
intermediate cylinders, 40 pounds ; in receiver between 
intermediate and low, 5 pounds. Vacuum is 27 
inches. The mean effective pressures in the cylin- 
ders are respectively 54 pounds per square inch, 23^- 
and I2|. Length of stroke is 5 feet; piston speed, 
660 feet per minute. Calculate the horse-power. 

87. An engine is required to drive an overhead 
traveling crane for lifting a load of 30 tons at 4 feet 
per minute. The power is transmitted by means of 
2j-inch shafting, making 160 revolutions per minute. 
The length of the shafting is 250 feet ; the power is 
transmitted from the shaft through two pairs of bevel 
gears (efficiency 90^;^ each, including bearings), and 
one worm and wheel (efficiency 85<%, including bear- 
ings). Taking the mechanical efficiency of the steam- 
engine at 80 '}^, calculate the required horse-power of 
the engine. 

88. An engine 'working at 50 horse-power is 
driven by steam at 75 pounds pressure acting on pis- 
tons in two cylinders. If the area of each piston is 72 
square inches, and the length of stroke 2 feet, how 
many revolutions does the fly-wheel make per minute ? 


89. The steam-engine in use at the Worsted 
Weaving Mill of the Pacific Mills at Lawrence, 
Mass., is a Corliss type cross-compound with steam 
cylinders 19 and 36 inches diameter; stroke, 42 
inches; revolutions, 100 per minute ; mean effective 
pressures, 60 pounds and 1 3 pounds. Find how many 
looms weaving worsted dress-goods said engine will 
drive, each loom requiring \ horse-power. 

90. A ship laden with coal must be unloaded at 
the rate of 22 tons of coal in 10 minutes. If the 
height of lift is 150 feet, what horse-power of engines 
will be required } 

91. The fuel used in running a steam-engine is 
coal of such composition that the combustion of i 
pound produces heat sufficient to raise the tempera- 
ture of 12 000 pounds of water 1° Fahr. It is found- 
that 3^^ pounds of fuel are consumed per horse-power 
per hour. What is the efficiency of the entire appa- 
ratus } 

92. A steam-engine uses coal of such composition 
that the combustion of i pound generates 10 000 
British thermal units. If 40 pounds of coal are used 
per hour, and if the efficiency is 0.08, what horse- 
power is realized t 

93. The cyhnder of a Corliss-type steam-engine is 
30 inches in diameter, stroke 48 inches, and it makes 
85 revolutions per minute. The steam pressure be- 
ing 90 pounds per square inch, what is the horse- 
power of the engine ? 


94. The piston of a steam-engine is 15 inches in 
diameter ; its stroke is 2 J feet, and it makes 20 revo- 
lutions per minute ; the mean pressure of the steam 
on it is 15 pounds per square inch. How many foot- 
pounds of work are done by the steam per minute, 
and what is the horse-power of the engine t 

95. An engine has a 6-foot stroke, the shaft makes 
30 revolutions per minute, the average steam pres- 
sure is 25 pounds per square inch. Required the 
horse-power when the area of the piston is i 800 
square inches, the modulus of the engine being i J. 

96. The diameter of a steam-engine cylinder is 9 
inches ; the length of crank, 9 inches ; the number of 
revolutions per minute, no; the mean effective pres- 
sure of the steam 35 pounds per square inch. Find 
the indicated horse-power. 

97. The 21 horse-power gas engine of problem 
51 has a 12-inch piston and 8-inch crank. When 
making 150 revolutions per minute with an explosion 
every 2 revolutions what will be the mean effective 
pressure during a cycle } 

98. The area of a cross-section of the Charles River 
at Riverside, Massachusetts, is 408 square feet. The 
velocity of current as found by rod floats and current 
meter, April 17 and 22, 1902, was 1.12 feet per sec- 
ond. What would be the theoretic horse-power of 
this quantity of water at the Waltham dam, which 
gives a fall of 12.58 feet 1 



In water-power problems use 

Work = force X distance 

(area X velocity X 62J) 

Force = pounds of water flowing 
Distance = height of dam or available fall 

99. Find the 
useful horse-power 
of a water-wheel, 
supposing the 
stream to be loo 
feet wide and 5 feet 
deep, and to flow 
with a velocity of 
\ foot per second ; 
the height of the 
fall is 24 feet, and 
the efficiency of 
tlic wheel 70 per 

100. A small 
Fig. 8.--water-Power. stream has mean 

velocity of 35 feet per minute, fall of 13 feet and a 
mean section of 5 feet by 2. On this stream is 
erected a water-wheel whose modulus is 0.65. Find 
the horse-power of the wheel. 

101. On page 32 is shown the canal at Manchester, 
N.H., as it passes the mills of the Amoskeag Manufac- 
turing Company. Width is 5 i feet, depth of water 8.9 
feet, velocity 1.13 feet per second. What quantity of 
water is flowing .? The height of fall for the turbines 
being 27.3 feet, what is the theoretic horse-power ? 


102. The reaction turbines of problem loi have 
an efficiency of 80 per cent ; the electric generators, 
90 per cent. What kilowatts are available .-* 

103T In winter, if 2 feet of ice forms on this 
canal, and the velocity drops to 0.75 feet per second, 
and the available fall becomes 25.0 feet, what will 
be the kilowatts available .!* 

104. The mean section of the Merrimac Canal just 
before it enters the mills of the Merrimac Manufac- 
turing Company at Lowell, Mass., is 48.2 feet by 10.6 
feet; mean velocity on Nov. 23, 1901, was 2.37 feet 
per second ; the water-wheels had a net fall of 35.67 
feet, and gave an efficiency of about J J per cent. Find 
the number of broad looms weaving cotton sheetings 
that may be driven 2^ looms requiring one horse- 

105. The estimated discharge of the nine turbines 
at Niagara Falls in 1898 was 430 cubic feet per sec- 
ond for each turbine. The average pressure head on 
the wheels was that due to a fall of about 136 feet. 
Compute the actual horse-power available from all tur- 
bines, allowing an efficiency of 82 per cent. 

106. The average flow over Niagara Falls is 270 oco 
cubic feet per second. The height of fall is 1 6 1 feet. 
In round numbers what horse-power is developed } 

107. Calculate the horse-power that can be obtained 
for one minute from an accumulator which makes 



one stroke in a minute and has a ram of 20 inches 
diameter, 23 feet stroke, loaded to a pressure of 750 
pounds per square inch. 

Fig. 9. An Underwriter Fire-Pump with Standard Fittings. 

108. A fire-pump for protection of a 50 000-spindle 
cotton-mill will deliver i 000 gallons of water per 
minute at 100 pounds pressure. Large boiler capa- 


city is required for such a fire-pump and for the above 
size 150 horse-power would be used. What portion 
of this boiler capacity would be required in actual 
work of delivering water ? 

Work = force x distance 

of pumping 

Force = i 000 x ^\ pounds per minute 

Distance = 100 x 2.304 feet head (i pound = 2.304 ft.) 

.-.Work = I 000 X 81- X 100 X 2.304 

of pumping 

= I 920 000 foot-pounds per minute 
= 58.3 horse-power 

Portion of boiler used = 

= 0.39, or about one-third 

109. An Underwriter fire-pump to protect an av- 
erage-sized factory will deliver four streams of water 
through i|-inch smooth nozzles with pressure at base 
of play pipes of 50 pounds per square inch. This 
would correspond to a discharge of i 060 gallons per 
minute. Loss of pressure through nozzle can be neg- 
lected ; and loss in quantity of discharge by slippage, 
short strokage, and so on, will be about 10 per cent. 
Find the work done by the pump. 

110. A pump of medium size used for fire pro- 
tection of a factory will deliver three i|-inch fire 
streams, or 750 gallons per minute at 80 pounds 
pressure. A boiler should be provided large enough 
to allow 70 per cent of its capacity to remain as extra. 
What should be the nominal horse-power of boiler } 


HI. A Silsby steam fire-engine delivers water 
through a Siamese nozzle that is 2 inches in diameter, 
with a pressure of 80 pounds per square inch and a 
mean velocity of 106 feet per second. Find (i) the 
number of cubic feet discharged per second ; (2) 
the weight of water discharged per minute ; (3) the 
work possessed by each pound of water due to 80 
pounds pressure ; (4) the horse-power of the engine 
required to drive the pump, assuming the efficiency 
to be 70 per cent. 

112. At the Chestnut Hill High-Service Pumping 
Station (Boston) for the month of October, 1904, 
Kngine No. 4 pumped 950 780 000 gallons of water ; 
average lift was 130.63 feet ; total time of pumping, 
744 hours. What average horse-power was de- 
veloped } 

113. The amount of coal burned during the month 
was 783 148 pounds. How many foot-pounds of 
work were done for every 100 pounds of coal burned, 
that i.s, what was the Duty of the pumping-engine } 

114. The ordinary fire-engine when in full opera- 
tion burns soft coal, and will consume in an hour about 
60 pounds per fire-stream of 250 gallons per minute. 
Therefore at the 70-million dollar fire in Baltimore, 
February, 1904, a 500-gallon engine that was running 
30 hours, before the fire was under control, consumed 
how many pounds of coal } 

115. Find the useful work done each second by a 
fire-engine which discharges water at the rate of 500 


gallons per minute against a pressure of lOO pounds 
per square inch. 

116. There were 6 ooo cubic feet of water in a 
mine of 6o-fathom depth when a 50-horse-power 
pump began to pump it out. It took 5 hours to 
empty it. Find the number of cubic feet of water 
that ran into the mine during the 5 hours, supposing 
one-fourth of the work of the pump to have been 

117. Find the horse-power necessary to pump out 
the Saint Mary's Falls Canal Lock, Sault Ste. Marie, 
in 24 hours, the length of the lock being 500 feet, 
width 80 feet, and depth of water 1 8 feet, the water 
being delivered at a height of 42 feet above the bot- 
tom of the lock. 

118. The mean section of the branch of the First 
Level Canal at the headgates of No. i Mill, Whiting 
Paper Co., Holyoke, Mass., is 78 feet wide by 14 
deep ; from this canal to the Second Level there is a 
fall of 20 feet, but about 2 feet is lost in penstock and 
tail-race ; velocity of flow in canal during the daytime 
is 0.20 feet per second, and the turbines that are 
driven have an efficiency of 77%. Find how many 
96-inch Fourdrinier Paper Machines can be driven, 
each machine requiring 1 00 horse-power. 

119. What is the horse-power of a stream that 
passes through a section of 6 square feet at the rate 
of 2^ miles an hour, and has a water-fall of 1 8 feet ? 


120. What horse-power is involved in lowering by 
2 feet the level of the surface of a lake 2 square miles 
in area in 300 hours, the water being lifted to an 
average height of 5 feet ? 

121. Taking the average power of a man as ^^^th 
of a horse-power, and the efficiency of the pump used 
as 0.4, in what time will 10 men empty a tank of 
50 feet X 30 feet x 6 feet filled with water, the lift 
being an average height of 30 feet ? 

122. A shaft 560 feet deep and 5 feet in diameter 
is full of water. How many foot-pounds of work are 
required to empty it, and how long would it take an 
engine of 3^^ horse-power to do the work } 

123. Required the number of horse-power to raise 
2 200 cubic feet of water an hour from a mine whose 
depth is 61 fathoms. 

124. What weight of coal will an engine of 4 horse- 
power raise in one hour from a pit whose depth is 200 
feet ? 

125. A cut is being made on a 4-inch wrought-iron 
shaft revolving at 10 revolutions per minute; the 
traverse feed is 0.3 inch per revolution; the pressure 
on the tool is found to be 435 pounds. What is the 
horse-power expended at the tool } How much metal 
is removed per hour per horse-power when the depth 
of cut is .06 inch, the breadth .06 inch (triangular 
section) ? 

ff OF THE ' 



126. A man rides a bicycle up a hill whose slope is 
I in 20 at the rate of 4 miles an hour. The weight 
of man and machine is iS/^ pounds. What work 
per minute is he doing ? 

127. At the top of the hill the bicyclist referred to 
in example 126 is met by a strong head-wind, and he 
finds that he has to work twice as hard to keep the 
same rate of 4 miles an hour on the level. What 
force is the wind exerting against him } 

128. A bicyclist works at the rate of one-tenth of a 
horse-power, and goes 1 2 miles an hour on the level. 
Prove that the constant resistance of the road is 3.125 

Prove that up an incline of i vertical to 50 horizon- 
tal the speed will be reduced to about 5.8 miles per 
hour, supposing that the man and machine together 
weigh 168 pounds. 

129. A man rows a miles per hour uniformly. If R 
pounds be the resistance of the water, and P foot- 
pounds of useful work are done at each stroke, find 
the number of strokes made per minute. 

130. The resistance offered by still water to the 
passage of a certain steamer at 10 knots an hour is 
15 000 pounds. If \2''io of the engine power is lost 
by '' slip " — in pushing aside and backward the 
water acted on by the screw or paddle — and 8% is 
lost in friction of machinery, what must be the 
total horse-power of the engines t 


131. The United States warship Cokimbia has a 
speed of 23 knots, with an indicated horse-power of 
22 000. Find the resistance offered to her passage. 

132. The rise and fall of the tide at Boston, Mass., 
is about 9 feet. If the in-coming water for one 
square mile of ocean surface could be stored and its 
potential energy used during the next 6 hours with 
an average fall of 3 feet, what horse-power would be 
available ? 

133. A nail 2 inches long was driven into a block 
by successive blows from a hammer weighing 5.01 
pounds ; after one blow it was found that the head of 
the nail projected 0.8 inches above the surface of the 
block ; the hammer was then raised to a height of 
1.5 feet and allowed to fall upon the head of the nail, 
which, after the blow, was found to be 0.46 inches 
above the surface. Find the force which the hammer 
exerted upon the nail at this blow. 

134. A 5 00- volt motor drives a lo-ton car up a 
5 per cent grade at a speed of 12 miles per hour : 
75 per cent of the work of the motor is usefully ex- 
pended. What electric current, expressed in am- 
peres, will be required t 

Work X 0.75 = work 

of motor of lifting car 

135. The speed of the "■ Exposition Flyer " on the 
Lake Shore and Michigan Southern Railroad, when 
running at its maximum, is 100 miles per hour. At 


that speed what pull by the engine would represent 
one horse-power? What pull when running at 50 
miles an hour ? 

136. An express train of weight 250 tons covers 
40 miles in 40 minutes. Taking the train resistances 
on a level track to be 20 pounds per ton at this speed 
find the horse-power that engine must develop. 

137. A train goes down a grade of i % for a dis- 
tance of I mile, steam throttle being kept shut ; it 
then runs up an equal grade with its acquired velocity 
for a distance of 500 yards before stopping. Find 
the total resistances, frictional or other, in pounds per 
ton, which are stopping it. 

138. A caboose and three cars break away from a 
freight train and *' coast " down a grade of 2 in 100 
for a distance of i mile ; then brakes are applied and 
the cars stopped in 200 feet. Frictional resistances 
over whole distance being 15 pounds per ton, what 
are the brake resistances per ton t 

Work = Work + Work 

down grade of friction' of brakes 

139. The Baltimore and Ohio Railroad has now in its 
service (1906) six electric locomotives. Two of recent 
construction are used in handling eastbound freight 
trains with steam locomotives through the city of 
Baltimore, which includes a distance of about two 
miles of tunnel. These locomotives can start and 
accelerate on a level track a train of 3 000 tons weight 
with a current consumption of 2 200 amperes, which 


is supplied from a power station at 560 volts, but 
reaches the locomotives through booster stations and 
a storage battery at 625 volts. What horse-power do 
they thus develop ? 

140. These electric locomotives will draw on a i % 
grade a freight train of i 400 tons weight at 10 miles 
an hour. Frictional resistances being 20 pounds per 
ton what amperes are necessary with voltage of 625 ? 



141. A train of 150 tons is running at 50 miles an 
hour. What brake force is required to stop it in a 
quarter of a mile on a down grade of 2*%, frictional 
resistance being 1 5 pounds per ton ? 

Work + Work = Work + Work 

possessed by train gained in \ mile of brakes of resistances 

Work = force X distance 

possessed by train 

Force =150 tons 

Distance = ? 

The distance is found by determining the vertical height that a 
body would have to fall in order to acquire a velocity of 50 miles an 
hour. 30 miles an hour = 44 feet per second. Therefore the 
velocity is f ^ x 44 = -f^ feet per second. Now to acquire a 
velocity of ^|^ feet per second a body would fall a vertical distance 
that can be found from a fundamental formula of falling bodies, 

V = N/a gJh in which ^* varies for different localities 

h = 84.2 feet. 

That is, if the train had fallen by the action of gravity from a 

vertical height of 84.2 feet it would have a velocity of ^|^ feet per 

second, or 50 miles an hour. Work can now be analyzed as in 

previous problems. 

Work =^150 X 84.^ foot-tons 

possessed by train 

Work = 150 X (t-Io X^^-V^) 

gained in \ mile 

* The value of g for London is 32.19 feet per second per second ; for San Fran- 
isco, 32.15 ; for Chicago, 32.16 ; for Boston, 32.16. Practical limiting values for the 
United States are 32.186 at sea level for latitude 49"^ ; and 32.089, latitude 25° and 
ID 000 feet above sea level. In this book the value 32 is used for^ so that com- 
putations may be shortened. In many cases the table on page 190 will be of 
assistance. The values of \^2gh there given are based on g as 32.16. 


142. In the Westinghouse brake tests (Jan., 1887), 
at Weehawken, a passenger-train moving 22 miles an 
hour on a clown grade of 1% was stopped in 91 feet. 
There was 94% of the train braked. Taking the 
frictional resistance as 8 pounds per ton, find the net 
brake resistance per ton on the part of the train that 
was braked, and the grade to which this is equivalent. 

143. A freight-car weighing 20 000 pounds requires 
a net pull of 10 pounds per ton to overcome frictional 
resistance. If '* kicked " to a level side track with 
velocity of 10 miles per hour, how far will it run 
before stopping } 

144. A cake of ice weighing 150 pounds slides 
down a chute the height of which is 25 feet; it 
reaches the foot of the shute with a velocity of 30 
feet per second. During the motion how many foot- 
pounds of energy must have been lost t 

145. A ship and its cradle that weigh 5 000 tons 
slides down ways that slope i foot in 20 to the hori- 
zontal ; frictional resistances amount to a constant 
retarding force of 100 tons. What v^ould be the 
equivalent height of fall that would produce the same 
velocity as the ship possesses when she takes the 
water 150 feet down the ways } 

146. If the resistance of the water, anchors, and 
stop ropes amount to a constant force of 50 tons, 
how far will the ship of the preceding problem run 
after she takes the water ? 


147. A six-inch rapid-fire gun discharges 5 projec- 
tiles per minute, each of weight 100 pounds, with 
a velocity of 2 800 feet per second. What is the 
horse-power expended } 

Consider from what vertical height a body would fall to have a 
velocity of 2 Soo feet per second, {v ^-= s/ig/i). 

148. A railway car of 4 tons, moving at the rate 
of 5 miles an hour, strikes a pair of buffers which 
yield to the extent of 6 inches. Find the average 
force exerted upon them. 

Work = work 

possessed by train by buffers 

149. What is the kinetic energy of a 2^ -ton cable 
car moving at 6 miles per hour, loaded with 36 pas- 
sengers, each of average weight 154 pounds.-* If 
stopped in 2 seconds, what is the average force .-^ 

150. What is the kinetic energy of an electric car 
weighing 2|- tons, moving at 10 miles an hour, and 
loaded with 50 passengers, of average weight 150 
pouilds } 

151. The weight of a ram is 600 pounds, and at 
the end of a blow it has a velocity of 40 feet per 
second. What work is done in raising it } 

152. Find the horse-power of a man who strikes 
25 blows per minute on an anvil with a hammer of 
weight 14 pounds, the velocity of the hammer on 
striking being 32 feet per second. 

153. A blacksmith's helper using a 16-pound 
sledge strikes 20 times a minute and with a velocity 
of 30 feet per second. Find his rate of work. 


154. A ball weighing 5 ounces, and moving at 
I 000 feet per second, pierces a shield, and moves on 
with a velocity of 400 feet per second. What energy 
is lost in piercing the shield ? 

155. A shot of I 000 pounds moving at the rate of 
I 600 feet per second strikes a fixed target. How far 
will the shot penetrate the target, exerting upon it an 
average pressure equal to a weight of 1 2 000 tons ? 

156. A bullet weighing i ounce leaves the mouth 
of a r'He with a velocity of i 500 feet per second. If 
the barrel be 4 feet long, calculate the mean pressure 
of the powder, neglecting all friction. 

157. The bullet referred to in the preceding 
problem penetrates a sand bank to the depth of 3 
feet. What is the mean pressure exerted by the sand ? 

158o An 8-hundred weight shot leaves a 40-ton gun 
with velocity of 2 000 feet per second : the length of 
the gun is 20 feet. What is the average force of 
the powder ? 

159. A 2-ounce bullet leaves the barrel of a gun 
with a velocity of i 000 feet per second. Find the 
work stored up in the bullet, and the height from 
which it must fall to acquire that velocity. 

160. What is the kinetic energy of a 5 -hundred 
weight projectile fired with a velocity of 2 000 feet 
per second .? 

161. An 8-inch projectile, weight 250 pounds, 
strikes a sand butt going 2 000 feet per second, and 


is stopped in 25 feet. If the resistance is uniform, 
what is its value in pounds ? 

162. A hammer weighing i pound has a velocity 
of 20 feet per second at the instant it strikes the head 
of a nail. Find the force which the hammer exerts on 
the nail if it is driven into the wood ^^ of an inch. 

163. A fly-wheel weighs 10 000 pounds, and is of 
such a size that its mass may be treated as if concen- 
trated on the circumference of a circle 1 2 feet in radius. 
What is its kinetic energy when moving at the rate of 
15 revolutions a minute .-* 

164. How many turns would the above fly-wheel 
make before coming to rest, if the steam were cut off, 
and it moved against a friction of 400 pounds exerted 
on the circumference of an axle i foot in diameter 1 

165. A fly-wheel on a 21 -horse-power gas engine 
of nominal speed 1 50 revolutions per minute, must 
store what energy to provide for an increase or de- 
crease in speed of 3 revolutions per minute .'* 

166. The fly-wheel of a 4-horse-power engine 
running at 75 revolutions per minute is equivalent to 
a heavy rim of mean diameter 2 feet 9 inches, and 
weight 500 pounds. What is the ratio of the work 
stored in the fly-wheel to the work developed in a 
revolution .-* 

167. A 3 horse- power stamping machine presses 
down once in every 2 seconds ; its speed fluctuates 
from 80 to 1 20 revolutions per minute ; and to pro- 
vide for this fluctuation the fly wheel stores |ths of 


all the energy supply for 2 seconds. What energy is 
thus stored per revolution ? 

168. A nozzle discharges a stream i inch in diam- 
eter with a velocity of 80 feet per second, (a) How 
much work is possessed by the water that flows out 
each minute .? (d) If this energy could all be utilized 
by a water-wheel, what would be its power } 

169. Suppose that the above nozzle drives a water- 
wheel connected with a pump which lifts water 20 
feet. If the efficiency of the whole apparatus is 0.48, 
how much water would be lifted per minute .-* 

170. An impulse water-wheel must provide 3| use- 
ful horse-power; efficiency of wheel is 85% ; water- 
pressure is 60 pounds per square inch ; what size 
nozzle should be used — to the nearest eighth of an 

inch ? 

Work = force X distance 
Force = area X velocity X 62^ 

= area X 8 V60 x 2.304 x 62^ 
Distance = 60 x 2.304 feet. 

171. The five streams shown on the next page are 
being delivered through 100 feet of cotton rubber- 
lined hose- with nozzles i| inches in diameter. The 
full pressure at end of nozzle is 50 pounds per square 
inch. What horse-power is the fire-pump thus 
delivering .? 

172. Through the 100 foot lines of hose there is a 
large loss of pressure. At hydrant the full pressure 
is 75 pounds ; at the nozzle 50 pounds. What horse- 
power is thus lost } 

Force illustrated by two fire streams being delivered by the pump service 
of the large cotton mills of B.B. & R. Knight at Natick, Rhode Island. One 
stream is being held by men in correct position, the other by men who have 
been crowded into an awkward and dangerous position. Pressure shown on 
the gauge at the hydrant was 75 pounds per square inch. 





173. An acrobat weighing 1 50 pounds stands in 
the middle of a tight rope 40 feet long and depresses 
it 5 feet. Find the tension in the rope caused by 
his weight. 


Draw the construction diagram showing the rope, the known 
force of 150 pounds and an arrow to indicate its direction. Then 
draw the stress diagram. Lay off 150 pounds parallel to the known 
force and at a scale of i inch = f^o pounds ; from one end of this 
line ABdrawa full line parallel with one of the forces; from the 
other end a line parallel with the other force. Complete the paral- 
lelogram by drawing free hand the dotted parallel lines AD and BD. 
Put arrows on forces beginning with the known force of 150 pounds 
— this positively acts downward — then the other arrows will follow 
in order around the full-line triangle B to C and C to A. Thus A B 
has become a diagonal of the parallelogram and is the balancing 
force, or what is more properly known as the equilibriant. If this 



force was acting in the opposite direction it would be the resultant 
of the other two forces. 

The full-line triangle ABC constitutes the triangle of forces. It 
is the keynote to the solution of many problems in Forces. The mag- 
nitudes and directions of the forces can be found by scaling from 
this triangle, or by computations involving similar triangles, thus, 
geometry or trigonometry. 

Forces-At-A-Point problems therefore can be solved as follows : 

Draw a construction diagram showing dimensions and loads. 

Draw a stress diagram. 

First, the known force. 

Complete the parallelogram. 

Put arrows on full-line triangle. 

Scale or compute the stresses. 

174. A Speed-buoy is thrown into the water behind 
a ship, and the pull on the buoy by the water is 60 
pounds. The two ropes that connect the buoy with 
the ship make an angle of 15° at their point of attach- 
ment. Find the stresses on the ropes. 

175. Two men pull a body horizontally by means 
of ropes. One exerts a force of 28 pounds directly 
north, the other a force of 42 pounds in direction N. 
42°E. What single force would be equivalent to 
the two } 

176. Three cords are knotted together ; one of 
these is pulled to the north with a force of 6 pounds, 
another to the east with a force of 8 pounds. With 
what force must the third be pulled to keep the whole 
at rest } 

177. Two persons lifting a body exert forces of 44 
pounds and 60 pounds on opposite sides of the ver- 


tical, but each with an indination of 28°. What single 
force would produce the same effect ? 

178. A force of 50 units acts along a line inclined 

at an angle of 30° to the horizon. Find, by construe- c 
tion or otherwise, its horizontal and vertical com- 

179. Explain the boatman's meaning when he says 
that greater force is developed when a mule hauls a 
canal boat with a long rope than with a short one. 
Is the same true of a steam-tug when towing a four- 
master } 

180. Two strings, one of which is horizontal, and 
the other inclined to the vertical at an angle of 30°, 
support a weight of 10 pounds. Find the tension in 
each string. 

181. Two forces of 20 pounds, and one of 2 1 act 

at a point. The angle between the first and second * 
is 120°, and between the second and third, 30°. 
Find the resultant. 

182. Forces of 9 pounds, 12, 13, and 26, act at a 
point so that the angles between the successive 
forces are equal. Find their resultant. 

183. A weightless rod, 3 feet long, is supported 
horizontally, one end being hinged to a vertical wall, 
and the other attached by a string to a point 4 feet ^ 
above the hinge; a weight of 120 pounds is hung 
from the end supported by the string. Calculate the 
tension in the string, and the pressure along the rod. 


184. A weight of loo pounds is fixed to the top of 
a weightless rod or strut 5 feet long whose lower end 
rests in a corner between a floor and a vertical 
wall, while its upper end is attached to the wall by a 
horizontal wire 4 feet long. Calculate the tension in 
the wire, and the thrust in the rod. 

185. A rod AB is hinged at A and supported in 
a horizontal position by a string BC making an 
angle of 45° with the rod ; the rod has a weight of 
10 pounds suspended from B. Find the tension in 
the string and the force at the hinge. (The weight 
of the rod can be neglected.) 

186. A simple triangular truss of 30 feet span and 
5 feet depth supports a load of 4 tons 
at the apex. Find the forces acting on 
rafters and tie rod. 

187. A derrick is set as shown in 
sketch, the load being 8 tons. Find the 
Fig. 12. stress in the boom and the tackle. 

188. A stiff -leg steel derrick, with mast 55 feet 
high, boom 85 feet long, set with tackle 40 feet long, 
as shown in cut, is raising two boilers of 
50 tons weight. Find stresses in boom 
and tackle. (See illustration on page 55.) 

189. Find the stress in tackle and 
compression in boom of towers for six- 
master shown on page 24 when bucket, 
weighing with its load 2 tons, is set in position shown 
by Fig. 13. 



Fig. 14. 

190. A balloon capable of raisin<T^ a weight of 360 
pounds is held to the ground by a rope which makes 
an angle of 60° with the horizon. Determine the 
tension of the rope and the horizontal pressure of the 
wind on the balloon. 

191. A uniform beam 10 feet long, weighing 80 
pounds, is suspended from a horizontal ceiling by two 
strings attached at its ends, and at points 16 feet ^ 
apart in the ceiling. Find the tension in each string. 

192. A boat is towed along a canal 50 feet wide, 
by mules on both banks ; the length of each rope 
from its point of attachment to the bank is 72 feet : 


the boat moves straight down the middle of the 
canal. Find the total effective pull in that direction, 
when the pull on each rope is 800 pounds. 

193. A boat is being towed by a rope making an 
angle of 30° with the boat's length ; the resultant 
pressure of the water and rudder is inclined at 60° 
to the length of the boat, and the tension in the rope 
is equal to the weight of half a ton. Find the re- 
sultant force in the direction of the boat's length. 

194. In a direct-acting steam-engine the piston- 
pressure is 22 500 pounds ; the connecting-rod makes 
a maximum angle of 15° with the line of action of the 
piston. P'ind the pressure on the guides. 

195. A man weighing 160 pounds sits in a loop at 
the end of a rope 10 feet 3 inches long, the other 
end being fastened to a point above. What horizon- 
tal force will pull him 2 feet 3 inches from the verti- 
cal, and what will then be the pull on the rope } 

196. A man weighing 160 pounds sits in a ham- 
mock suspended by ropes which are inclined at 30° 
and 45° to vertical posts. Find the pull in each rope. 

197. Two equal weights, W, are 
attached to the extremities of a 
flexible string which passes over 
w three tacks arranged in the form 
of an isosceles triangle with the 
base horizontal, the vertical angle at the upper tack 
being 1 20°. Find the pressure on each tack. 



198. A rod AB 5 feet long, without weight, is 
hung from a point C by two strings, which are at- 
tached to its ends and to the point ; the string AC is 
3 feet long, and the string BC 2 feet ; a weight of 
2 pounds is hung from A and a weight of 3 pounds 
from B. Find the tension of the strings and the 
condition that these may be in equilibrium. 

199. A weight of 10 pounds is suspended by two 
strings, 7 and 24 inches long, the other ends of 
which are fastened to the extremities of a rod 25 
inches in length. Find the tension of the strings 
when the weight hangs immediately below the middle 
point of the rod. 

200. AB is a wall, and C a fixed point at a given 
perpendicular distance from it ; a uniform rod of 
given length is placed on C with one end against AB. 
If all the surfaces are smooth, find the position in 
which the rod is in equilibrium. 

201. AB is a uniform beam weighing 300 pounds. 
The end A rests against a smooth verti- ^ 
cal wall, the end B is attached to a rope 
CB. Point C is vertically above A, X 
length of beam is 4 feet, rope 7 feet. 
Represent the forces acting, and find the X|^ 
pressure against the wall and the tension ^' 
in the rope. Fig. jg. 

202. A wagon weighing 2 200 pounds rests on a 
slope of inclination 30°. What are the equivalent 
forces parallel and perpendicular to the plane ? 



203. AB is a rod that can turn freely round one 
end A ; the other end B rests against a smooth in- 
clined plane. In what direction does the plane react 
upon the rod } Illustrate your answer by a diagram 
showing the rod, the plane, and the reaction. 

204. A wagon weighing 2 tons is to be drawn up 
a smooth road which rises 4 feet vertically in a dis- 
tance of 32 feet horizontally by a rope parallel to the 
road. What must the pull of the rope exceed in 
order that it may move the wagon t 

205. What weight can be drawn up a smooth plane 
rising i in 5 by a pull of 200 pounds {a) when the 
pull is parallel with the plane t (b) when it is hori- 
zontal .? 

206. A horse is attached to a dump-car by a chain, 
which is inclined at an angle of 45° to the rails; 
the force exerted by the horse is 672 pounds. What 
is the effective force along the rails .? 

~ 207. The angle of inclination of a smooth inclined 
plane is 45° : a force of 3 pounds acts horizontally, 
and a force of 4 pounds acts parallel to the plane. 
Find the weight which they will be just able to 

208. A body rests on a plane of height 3 feet, length 
5 feet. If the body weighs 14 pounds, what force act- 
ting along the plane could support it, and what would 
be the pressure on the plane } 


209. A number of loaded trucks each containing 
one ton, standing on a given part of a smooth tram- 
way, where the inclination is 30°, support an equal 
number of empty trucks on another part, where the 
inclination is 45°. Find the weight of a truck. 

210. Two planks of lengths 7 yards and 6 yards 
rest with one end of each on a horizontal plane, the 
other ends in contact above that plane ; two weights 
are supported one on each plank, and arc connected 
by a string passing over a pulley at the junction of 
the planks ; the weight on the first plank is 2 1 pounds. 
What is the weight on the other, friction not being 
considered 'i 

211. The weight of a wheel with its load is 2 tons, 
diameter of wheel 5 feet. Find the least horizontal 
force necessary to pull it over a stone 4 inches high. 
(When the wheel begins to rise three 
forces are acting : P, W, and R the 
reaction. It is required to find P.) 

212. A rectangular box, contain- 
ing a 200-pound ball, stands on a ^^^' '7- 
horizontal table, and is tilted about one of its lower 
edges through an angle of 30.° Find the pressure be- 
tween the ball and the box. 

213. An iron sphere weighing 50 pounds is resting 
against a smooth vertical wall and a smooth plane 
which is inclined 60° to the horizon. Find the pres- 
sure on the wall and plane. 


214. A beam weighing 400 pounds rests with its 
ends on two inclined planes whose angles of inclina- 
tion to the horizontal are 20° and 30°. Find the 
pressures on the planes. 

215. A thread 14 feet long is fastened to two 
points A and B which are in the same horizontal line 
and 10 feet apart ; a weight of 25 pounds is tied to 
the thread at a point P so chosen that AP is 6 feet — 
therefore BP is 8 feet long. The weight being thus 
suspended, find by means of construction or otherwise, 
what are the tensions of the parts AP and BP of the 

216. AC and BC are two threads 4 feet and 5 feet 
long, respectively, fastened to fixed points A and B, . 
which are in the same horizontal line 6 feet apart ; a 
weight of 50 pounds is fastened to C. Find, by 
means of a line construction drawn to scale, the pull 
it causes at the points A and B. Each of the threads 
AC and BC is, of course, in a state of tension. 
What are the forces producing the tension 1 

217. A boiler weighing 3 000 pounds is supported 

by taqkles from the fore and main yards. If the ^ 
tackles make angles of 25° and 35° respectively with 
the vertical, what is the tension of each .'* 

218. A piece of wire 26 inches long, and strong 
enough to support directly a load of 100 pounds, 
is attached to two points 24 inches apart in the same 
horizontal line. Find the maximum load that can be 


suspended at the middle of the piece of wire without 
breaking it. 

219. A picture of 50 pounds weight hanging ver- 
tically against a smooth wall is supported by a string 
passing over a smooth hook ; the ends of the string 
are fastened to two points in the upper rim of the 
frame, which are equidistant from the center of the 
rim, and the angle at the peg is 60°. Find the tension 
in the string. 

220. A weight W attached by two connecting 
cords of lengths a and b to two fixed points A and B, 
and separated by a horizontal interval r, are in equilib- 
rium under the action of gravity. Required the 
stresses P and O in the cords. 

221. Two equal rods AB and BC are loosely jointed 
together at B. C and A rest on two fixed supports 
in the same horizontal line, and are connected by a 
cord equal in length to AB. If a weight of 12 pounds 
be suspended from B, what is the pressure produced 
along AB and BC, and the tension in the cord } 

222. Two spars are lashed together so as to form 
a pair of shears as shown in sketch. 
They stand with their '* heels " 20 feet 
apart, and would be 40 feet high when 
vertical. What is the tension in the 
guy and thrust in the legs when a load 
of 30 tons is being lifted } 

Suppose that a single leg should replace the Fig. 18. 



two spars. The stress can easily be found in this imaginary leg by 
considering that at A, in this plane, three forces meet, — the imagi- 
nary leg, the back guy, and the vertical load of 30 tons. Then 
consider the three forces at A in the plane of the legs, and thus 
find the stresses in the two equal spars. 

223. When the spars become vertical what stresses 
will exist for the load of 30 tons ? 

Fig. 19. 

224. Figs. 19-20 show a 
pair of shears erected at 
Sparrow's Point, Md., for 
the Maryland Steel Com- 
pany. The two front legs 
are hollow steel tubes 116 
feet long, and inclined 35 
feet out of the vertical. 
The back leg is 126 feet long, and is connected to 
hydraulic machines for operating the shears. How 
much are the forces acting in these legs when a 
Krupp gun weighing 122 tons is being lifted.'* 

225. Each leg of a pair of shears is 50 feet long. 
They are spread 20 feet at the foot. The back stay 
is 75 feet long. Find the forces acting on each 
member when lifting a load of 20 tons at a distance of 
20 feet from the foot of the shear legs, neglecting the 
weight of structure. 

226. Shear legs each 50 feet long, 30 feet apart on 
horizontal ground, meet at point C, which is 45 feet 
vertically above the ground ; stay from C is inclined 



Fig. 20. 

at 40° to the horizon ; a load of 10 tons hangs from 
C. Find the force in each leg and stay. 

227. A vertical crane post is 10 feet high, jib 30 
feet long, stay 24 feet long, meeting at a point C. 
There are two back stays making angles of 45° with 
the horizontal ; they are in planes due north and due 



west from the post. A weight of 5 tons hangs from 
C. Fmd the forces in the jib and stays — ist, when 
C is southeast of the post ; 2d, when C is due east ; 
3d, when C is due south. 

228. The view on opposite page shows one of the 
largest dipper dredges ever built, the " Pan American," 
constructed at Buffalo in 1899 for use on the Great 
Lakes. An A-frame, the legs of which are 57 feet 
long and 40 feet apart at the bottom, is held at the 
apex by four cables which are 100 feet long. The 
boom is 53 feet long and weighs 30 tons. The 
handle, which weighs about 4 tons, is 60 feet long, 
and carries on its end a dipper weighing 16 tons, 
which will dredge up 8 J cubic yards, or about 12 
tons, of material at one load. 

The dipper is operated by a wire rope passing over 
a pulley on the outward end of the boom. In the 
position represented by the outline sketch, the boom 
is inclined to the 

water surface at 
an angle of 30°, 
the dipper is car- 
rying the full 
load, and the han- 
dle is in a hori- 
zontal position 
with its middle point supported at a point on the 
boom 23 feet from the foot of the boom. The apex 
of the A-frame is vertically above the foot of the 
boom. Compute the forces acting in the 100-foot 

Fig. 21. 



back-Stays (considering them to be one rope, in posi- 
tion as per sketch), in the legs of the A-frame,- in the 
boom, and in the wire rope which raises the dipper. 

229- A tripod whose vertex is A, and whose legs 
are AB, AC, AD, of lengths 8 feet, 8.5, and 9 re- 
spectively, sustains a load, of 2 tons. The ends 
B, C, D, form a triangle whose sides are BC 7 feet, 
CD 6 feet, BD 8 feet. Find the stress in each leg. 

Sketch the figure and put on the dimensions. Then draw to scale 
the base BCD, and in this horizontal plane locate the vertices A", 
A-', and h!" of the three faces of the pyramidal-shaped figure that 
is formed by the legs of the tripod. Perpendiculars drawn from A', 
A" and A"' to their respective sides of the triangle BCD will locate 
at their intersection the projection of vertex A. Now pass a vertical 
plane, for example, through AB and the load of 2 tons ; note the in- 
tersection E with line CD. AE can be considered as an imaginary 
leg, and the stress in it can be graphically determined as hereto- 
fore, also the stress in AC and AD. 

30. A tripod with 8-foot legs is to be used for 
lowering a 2-ton water-pipe. How far apart can the 
bottoms of legs be spread, if in an equilateral triangle, 
so that not over i ton stress will come on each leg } 

231. A chandelier of weight 500 pounds is to hang 
under the middle of a triangle 12 feet X 8 X 8. Two 
of the chains are to be 20 feet long. What should 
be the length of the third chain } What stresses 
would exist in chains t 

232. ABCD is a square ; forces of i pound, 6, and 
9 act in directions AB, AC, and AD respectively. 
Find the magnitude of their resultant. 


233. A, B, C, D, are the angular points of a square 
taken in order ; three forces act on a particle at A, 
viz. one of 7 units from A to B, a second of 10 units 
from D to A, and a third of 5 V2 units along the 
diagonal from A to C. Find, by construction or 
otherwise, the resultant of these three forces. 

234. Forces P, 2P, 3P, and 4P act along the sides 
of a square A, B, C, D, taken in order. Find the 
magnitude, direction, and line of action of the result- 

235. A sinker is attached to a fishing-line which is 
then thrown into running water. Show by means of 
a diagram the forces which act on the sinker so as to 
maintain equilibrium. 

236. A uniform rod 6 feet long, weighing 10 pounds, 
is supported by a smooth pin and by a string 6 feet 
long which is attached to the rod i foot from one 
end and to a nail vertically above the pin, 4 feet dis- 
tant. Show by construction the position in which 
the rod will come to rest. 

237. A light rod AB can turn freely round a hinge 
at A ; it rests in an inclined position against a smooth 
peg near the end B ; a weight is hung from the middle 
of the rod. Show in a diagram the forces which 
keep the rod at rest, and name them. 





Fig. 22. 

238. A weight W on a plane 
inclined 30° to the horizontal is 
supported as shown in cut. The 
angles ^ being equal. Find the 
ratio of the power to the weight. 

239. Discuss the action of the 
wind in propelling a sailing-vessel. 

Let AB be the keel, CD the sail. Let the 
force of the wind be represented in magnitude /s,. 
and direction by EF. The component GF 
of EF, perpendicular to the sail, is the effec- 
tive component in propelling the ship ; the 
other component EG, parallel to the sail, is 
useless ; but GF drives the ship forward and 
sidewise. The component GH of GF, perpendicular to AB, pro- 
duces side motion, or leeway ; and the other component HF, along 
the keel, produces forward motion, or headway. 

240. A sailing-boat is being 
driven forward by a force of 300 
pounds as shown in Fig. 24. 
What force is P acting in direction 
of motion of the boat } 

241. Discuss the action of the 
rudder of a vessel in counteracting 

leeway. Show that one effect of the action of the 
rudder is to diminish the vessel's motion. 

Fig. 24. 

242. A thread of length / has its ends fastened to 
two points in a line of length c, and inclined to the 
vertical with angle ; a weight W hangs on the thread ^ 
by means of a smooth hook. Find the position in 


which the weight comes to rest and the tension in the 

243. A smooth ring weighing 40 pounds slides 
along a cord that is attached to two fixed points in a 
horizontal line. The distance between the points 
being one-half length of cord, find position in which 
weight will come to rest and the tension in the string 
near the points of attachment. 

244. A small heavy ring A, which can slide upon 
a smooth vertical hoop, is kept in a given position by 
a string AB, B being the highest point of the hoop. 
Show that the pressure between the ring and the 
hoop is equal to the weight of the ring. 

245. Draw a figure showing the mechanical con- 
ditions of equilibrium when a uniform beam rests with 
one extremity against a smooth vertical wall, and the 
other inside a smooth hemispherical bowl. 

246. A ball 8 inches in diameter, weighing 100 
pounds, rests on a plane inclined 30° to the horizon, 
and is held in equilibrium by a string 4 inches long 
attached to a sphere and to an inclined plane. Rep- 
resent the forces acting, and find their values. 

247. A uniform sphere rests on a smooth inclined 
plane, and is held by a horizontal string. To what 
point on the surface of the sphere must the string be 
attached } Draw a figure showing the forces in action, 



248. A uniform bar of weight 20 pounds, length 12 
feet, rests with one end inside a smooth hemispheri- 
cal bowl, and is supported by the edge of the bowl 
with 2 feet of the bar outside of it. Draw the forces 
producing equilibrium, and find their values. 

The stresses in a roof or bridge truss that carries a uniform load 
are best determined by finding in place of the 
uniform loads equivalent apex loads. And a 
fact that is often obscure to students is, that a 
part of this uniform load is not included in 
our computation of stresses. In the truss of 
Fig. 25 the portions a of uniform load are not 
included in the compressive stresses of A C and 
C B. This fact will be further understood by 
solving the problems that follow. 

Fig. 25. 

249. Two floor beams of 16 feet length meet at 
a post, Fig. 26. The load, 10 feet width of bay for 
each beam, is 150 pounds per square foot. What will 
be the load carried by the post } If it is found that 
the post must be removed so as to give better floor 
space the plan of Fig. 27 could be used. What 
would then be the stress in the short post (3 feet 
long), and in the two rods, and in the floor beams } 

Fig. 26. Fig. 37. 

250. Now if the same conditions exist as in the 
preceding problem, except that the rods, instead of 



being fastened to the ends of the beam are fastened 
to straps on the outside of the wall, what will then 
be the stresses in post, rods, and floor beam ? 

251. The slopes of a simple triangular roof-truss 
are 30^ and 45°, and the span is 50 feet. The trusses 
are set 10 feet apart, and the weight of the roof cov- 
ering and snow is 50 pounds per square foot of roof. 
I-^ind the stresses in tie-rod and rafters. 

10 > < ip;> yv) 





c — < 

Fig. a8. 

Fig. 29. 

The load on any truss would be represented by the shaded area 
in Fig. 28. Find this load and then the apex loads A, C, and B, and 
observe that, according to explanation of preceding problems, the 
loads A and B do not enter into our computations. C alone is re- 
quired. Having found C, the stresses in rafters can be determined. 
Then find the stress that each rafter transmits to the tie-rod. 

252. In a roof of 32 feet span 
and height 12 feet the trusses are 1 b 
10 feet apart, and the members * a ^ f 
EF, GH, come to the middle points 
of the rafters. If the weight of 
the roof-covering and snow is 60 pounds per square 
foot, find the apex loads AO, AB, and BC. 



253. Find the stresses in the 
king-post truss of Fig. 31. Dis- 
tance between trusses is 12 feet. 
^^' ^'* There is a uniform load of 100 

pounds per square foot of roof surface and i 000 
pounds at the foot of the post. 

254. A king-post truss has a span of 18 feet and a 
rise of 9 feet. Compute the stresses due to a load of 
14 000 pounds at the middle. 

255. A floor beam 16 feet long and carrying a uni- 
form load of 200 pounds per linear foot is trussed by 
rods that are \\ feet below middle of beam. Con- 
sider a joint at the middle and find stress in rod. 


The principles of Work can be used to solve nearly 
all problems that belong to the subject of Mechanics, 
but in certain classes of problems shorter methods 
are possible. In the following problems the princi- 
ples of Moments can be used to advantage. 

Definition. — The Moment of a force about a point or axis is 
the product of the force times the perpendicular distance from the 
point to the line of action of the force ; or, briefly; Moment is 
force X perpendicular. 

Clockwise motion will be taken positive ; the opposite direction, 

In beginning the solution of problems always state which point 
or axis the moments are taken about ; thus, " Take moments about 
B," or " Moments about axis B." 


256. A piece of shafting 10 feet long, and weighing 
100 pounds, rests horizontally on ^ '^ ^ ^' 

two horses placed at its ends, ^f £__l_!^_^ 

A pulley weighing 75 pounds is ^' J,^_,,\^ 

keyed 2 J feet from one end. Fi«- 3a. 

How many pounds will a man have to lift at the other 
end to just raise it.-* 

100 pounds, the weight of shaft, acts downward at the middle 
point; 75 pounds, the weight of pulley, acts downward at D, 2 J feet 
from B. Find the required force acting upward. 

Take moments about B, 

+ P X 10 - 100 X 5 - 75 X 2^ = o. 
.*. P = 67.5 pounds. 

257. A uniform lever is 18 inches long, and each 
inch in length weighs i ounce. P'ind the place of the 
fulcrum when a weight of 27 ounces at one end of the 
lever balances a weight of 9 ounces at the other end. 

258. A lever 16 feet long balances about a point 
4 feet from one end; if a weight of 120 pounds be 
attached to the other end, it balances about a point 
6 feet from that end. Find the weight of the lever. 

259. A light rod of length 3 yards has weights of 

15 pounds and 3 pounds suspended at the middle and \/ 
end respectively ; it balances on a fulcrum. Find the 
position of the fulcrum, and the pressure on it. 

260. A stiff pole 12 feet long sticks out horizon- 
tally from a vertical wall. It would break if a weight 
of 28 pounds were hung at the end. How far out 
along the pole may a boy of weight 1 1 2 pounds ven- 
ture with safety } 



261. A man pulls lOO pounds on the end of a 7-foot 
oar that has 2\ feet inside the rowlock. What is the 
pressure on the rowlock, and resultant pressure caus- 
ing the boat to move ? 

262. Find the propelling force on an eight-oared 
shell, if each man pulls his oar with a force of 56 
pounds, and the length of the oar outside the row- 
lock is three times the length inside. 

263. A light bar, 5 feet long, has weights of 9 
pounds and 5 pounds suspended from its ends, and 10 
pounds from its middle point. Where will it balance } 

^264. A weightless lever AB of the first order, 8 
feet long, with its fulcrum 2 feet from B, has a weight 
of 5 pounds hung from A, and one of 17 pounds 
from B. From what point must a weight of 2.5 
pounds be hung to keep the lever horizontal 1 

265. A weight of 100 pounds is supported by a 
rope which passes over a fixed pulley and is attached 
to a 12-foot lever at a point 2 feet from the fulcrum 
which is at the end. What weight must be sus- 
pended at the other end to keep the lever horizontal .'' 

266. Eight sailors raise an anchor, of weight 2 688 
pounds, by pulling on the spokes of a capstain which 
has a radius of 14 inches. If they all pull at equal 
distances from the center and exert a force of 56 
pounds each, what is the distance } 

267. Is there any reason why a man should put 
his shoulder to the spoke of the wheel rather than to 
the body of the wagon in helping it up hill ? 


268. A rod AB, of length 1 5 feet, is supported by 
props at A and B ; a weight of 200 pounds is sus- 
pended from the rod at a point 7 feet from A. Find 
the pressure on the prop at A. 

269. A Hght bar, 9 feet long, to which is attached 
a weight of 150 pounds, at a point 3 feet from 
one end, is borne by two men. Find what portion of 
the weight is borne by each man, when the bar is 

270. A light rod, 16 inches long, rests on two pegs 
9 inches apart, with its center midway between them. 
The greatest weights, which can be suspended sepa- 
rately from the two ends of the rod without disturb- 
ing the equilibrium, are 4 pounds and 5 pounds re- 
spectively. There is another weight fixed to the rod. 
Find that weight and its position. 

271. A light rod AB, 20 inches long, rests upon 
two pegs whose distance apart is equal to half the 
length of the rod. How must it be placed so that the 
pressure on the pegs may be equal when weights 2W, 
3W, are suspended from A, B, respectively ? 

272. The horizontal roadway of a bridge is 30 feet 
long and its weight, 6 tons, may be supposed to act 
at its middle point, and it rests on similar supports 
at its ends. What pressure is borne by each of the 
supports when a carriage weighing 2 tons is one-third 
of the way across the bridge } 



273. "We have a set of hay-scales, and some- 
times we have to weigh wagons that are too long to 
go on them. Can we get the correct weight by 
weighing one end at a time and then adding the two 
weights .?" 

274. A rod, i8 inches long, can turn about one of 
its ends, and a weight of 5 pounds is fixed to a point 
6 inches from the fixed end. Find the force which 
must be applied at the other end to preserve equilib- 

275. A straight uniform lever weighing 10 pounds 
rests on a fulcrum one-third of its length from one 
end ; it is loaded with a weight of 4 pounds at 
that end. Find what vertical force must act at the 
other end to keep the lever at rest. 

276. A weight ot 56 pounds is attached to one end 
of a uniform bar which is ten feet long, and weighs 
20 pounds ; the fulcrum is 2 feet from the end to 
which the weight is attached. What weight must be 
applied at the other end to balance } 

277. AB is a horizontal uniform bar \\ feet long, and 
F a point in it 10 inches from A. Suppose that AB 
is a lever turning on a fulcrum under F, and carrying 
a weight of 40 pounds at B ; weight of lever, 4 pounds. 
If it is kept horizontal by a fixed pin above the rod, 7 
inches from F and 3 inches from A, find the pressure 
on the fulcrum and on the fixed pin. 



278. An ununiform rod, i6 feet long, weighing 4 
pounds, balances about a point 4 feet from one end. 
If, 2 feet from this end, a weight of 10 pounds be 
hung, what weight must there be hung from the other 
end so that the rod may balance about its middle 
point .? 

279. Six men are to carry an iron rail 60 feet long 
and weighing 90 pounds per yard ; each man sustains 
one-sixth of the weight. Two men are to lift from 
one end and the other four by means of a cross-bar. 
Where must the cross-bar be placed ? 

280. A rod 2 feet long, with a weight of 7 pounds 
at its middle point, is placed upon two nails, A and B. 
AB is horizontal and 7 inches long. Find how far 
the ends of the rod must extend beyond the nails, 
if the difference of the pressures on the nails be 
5 pounds. 

281. A davit is supported by a foot- 
step A and a collar B, placed 5 feet 
apart. A boat weighing two tons is 
about to be lowered, and is hanging 
4 feet horizontally from vertical through 
the foot-step and collar. Determine 
the forces which must be acting at A 
and B. 

282. A highway bridge of span 50 feet, breadth 
40 feet, has two queen-post trusses of depth 8 feet ; 
and each truss is divided by two posts into three 
equal parts. The bridge is designed to carry a load 

Fig. 33. 




of lOO pounds per square foot of floor surface. Find 
the stresses developed. 

Find the loads for each truss at the two panel points C and D ; 
then, by the methods of Moments, find the reactions R and R^ ob- 
serving, as explained for problem 251, that at each end half a panel 
of the load goes directly on the abutment and does not affect our 
computation of stresses in the members of the truss. The reactions 
thus known makes it possible to find the two unknown forces 
(stresses in the members) at the abutments. Likewise at foot of 
posts three forces meet in a point. One is known, — the stress in 
post which is equal to load at C or D, — and the other two can be 
found by methods of three forces acting at a point. 

C I 

i 50/«. , 
^7 777-/777 T7:r'. •.:>.'.'.•.' 




— ^ 





— -»1 



VTr.rTZ/. V7. 'TTT/, 777, 7777i 






Pig. 34. Fig- 35- 

283. A king-post truss of 20 feet span, as shown 
in Fig. 35, has a uniform load of 10 X 200 pounds on 
the horizontal member and 10 000 pounds at the foot 
of the post. Determine the reactions and stresses. 

284. A 5 -foot water-pipe is carried across a gully 
by two king-post trusses that are spaced 6 feet apart. 
The pipe when filled with water makes a load of 200 
pounds per square foot. Length of trusses is 40 
feet ; depth, 5 feet. Find the stresses. 

285. A storehouse has queen-post trusses in the 
top story ; 50 feet span, 10 feet depth, lower chord 
divided into 3 equal parts ; trusses 8 feet apart, and 
load 1 50 pounds per square foot. Find the stresses. 


286. A ladder with 21 rungs a foot apart leans 
against a building with inclination of 45°. Find the 
pressure against the building when a man weighing 
1 50 pounds stands on the eleventh rung. 

287. Like parallel forces of 10 and 20 units act 
perpendicularly to AB at A and B ; a force of 15 . 
units acts from A to B. Find the resultant of the 
three forces, and show in a diagram how it acts. 

288. A rod is acted on at one end by a force of 3 
downwards, and at a distance of two feet from this 
end by a force of 5 upwards. Where must a force of 
2 be applied to keep the rod at rest 1 

289. Three parallel forces of i pound each act on 
a horizontal bar. The right hand one acts vertically 
upwards, the two others vertically downwards, at dis- 
tances 2 feet and 3 feet respectively, from the first. 
Draw their resultant, and state exactly its magnitude 
and position. 

290. A rod is suspended horizontally on two points, 
A and B, 1 2 feet apart ; between A and B points 

C and D are taken, such that AC = BD = 3 feet ; a *^ 
weight of 1 20 pounds is hung at C, and a weight of 
240 pounds at D ; the weight of the rod is neglected. 
Take a point O, midway between A and B, and find 
with respect to O the algebraical sum of the moments 
of the forces acting on the rod on one side of O. 

291. A horizontal rod without weight, 6 feet long, 
rests on two supports at its extremities ; a weight of 


672 pounds is suspended from the rod at a distance of 
2\ feet from one end. Find the reaction at each 
point of support. If one support could bear a pres- 
sure of only 1 1 2 pounds, what is the greatest distance 
from the otlier support at which the weight could be 
suspended ? 

292. Three equal parallel forces act at the corners 
of an equilateral triangle. Find the point of applica- 
tion of their resultant.. 

293. Find the center of the three parallel forces 4 
pounds, 6, and 8, which act respectively at the cor- 
ners of an equilateral triangle. 

294. P, Q, R, are parallel forces acting in the same 
direction at the angular points respectively of an 
equilateral triangle ABC. If P = 2O = 3R, find the iy 
position of their center ; also find its position if the 
direction of the force Q is reversed. 

295. Show that if two forces be represented in 
magnitude and direction by two sides of a triangle, 
taken in order, the sum of their moments about every 
point in the base is the same. 

296. Draw a square whose angular points in order 

are A, B, C, D, and suppose equal forces (P) to act v 
from D to A, A to B, and B to C respectively, and a 
fourth force (2P) to act from C to D. Find a point 



such that, if the moments of the forces are taken 
with respect to it, the algebraic sum is zero. 

297. A BCD is a square, the length of each side 

being 4 feet, and four forces act as follows : 2 pounds 

from D to A, 3 pounds from B to A, 

4 pounds from C to B, and 5 pounds 

from D to B. Find the algebraical sum 

of the moments of the forces about C. 


Fig. 36. 

The forces 

act as in the figure. 



perpendicular to DB. 
CM = 


.-. CD2=CM2 + MD2 

= 2CM2. 

.-. CM = 


.-. CM = 

4 = 

2.83 nearly 

.-. Algebraical sum of the moments about C 

= - 2 X DC + 3 X CB + 4 X o — 5 X CM 
= -2 X 4 + 3^X4 +0^5 (2.83) 
= — 8 + 12 "^'14.15 
= — 10.15 units. 

298. ABCD is a square, and AC is a diagonal : 
forces P, Q, R, act along parallel lines at B, C, D, re- 
spectively, Q acts in the direction A to C, P opposite 
direction, and R in opposite direction. Find, and 
show in a diagram, the position of the center when 
Q = 5P and R = 7P. 

299. Draw a rectangle, ABCD, such that the side 
AB is three-fourths of the side BC ; forces of 3, 9, 
and 5 units act from B to A, B to C, and D to A re- 
spectively. Find their resultant by construction or 



otherwise, and show in your diagram exactly how it 

300. Prove that, if parallel forces i, 2, 3, 4, 5, 6, 
are situated at the angles of a regular hexagon, the 
distance of their center from the center of the cir- 
cumscribing circle is two-sevenths of the radius of 
that circle. 

301. Six forces, represented by the sides of a 
regular hexagon taken in order, act along the sides 
to turn the hexagon round an axis perpendicular to its 
plane. Show that the moment of the forces is the 
same through whatever point within the hexagon the 
axis passes. 

302. A triangular table, sides 8 
feet, 9 feet, and 10 feet, is sup- 
ported by legs at each corner, and 
350 pounds is placed on it 3 feet 
from the 8-foot side, 2 feet from 
the 9-foot side, and 2.6 feet 

from the lo-foot side. What are the pressures on 

the legs } 

303. A triangular shaped platform right-angled at 

A, with side AB 10 " =^^ 

feet long, side AC 40 

feet long, is loaded 

with freight at 50 

pounds per square foot ^^s. 38. 

surface. Find the load carried by each of the three 


10 u« 


O, the center of gravity, is at one-third the distance from the mid- 
dle of any base to the opposite vertex. Load equals lo ooo pounds. 

Take moments about axis AB — thus find load carried by C. 
Then take moments about sides AC and BC. 

304. Four vertical forces, 5, 7, 10, and 12 pounds, 
act at the corners of a square of 20-inch sides. Find 
resuhant and its point of application. 

Let A BCD be the square, 

Resultant = 5 + 7+ 10+ 12 
= 34 pounds. 
To find its point of application : 
Resultant of 7 and 10 will be a force 
of 17 pounds acting from point in line 
CB distant /- of 20 inches from B. The 
resultant of 5 and 12 will be 17 pounds 
acting at a point in line AD distant -^^ 
of 20 inches from A. The resultant of 
these two resultants will be a force of 17 
-f 17 pounds, 34 pounds, acting at a point 

half way between them, and at a perpendicular distance from AB of 
\ of [jV X 20 + /y X 20] = 7yV inches. 

305. A floor 20 X 30 feet is supported mainly by- 
four posts, one at each corner. There is a load of 20 
pounds per square foot uniformly distributed, and at 
point O, 5 feet from 30-foot side and 7 feet from 20- 
foot side, there is a metal planer weighing 5 tons. 
Find the load on each post. 

306. Weights 5, 6, 9, and 7 respectively, are hung 
from the corners of a horizontal square, 27 inches in 
a side. Find, by taking moments about two adjacent 
edges of the square, the point where a single force 
must be applied to balance the effect of the forces at 
the corners. 


307. A uniform beam, weighing 400 pounds, is 
suspended by means of two chains fastened one at 
each end of the beam. When the beam is at rest it 
is found that the chains make angles of 100° and 115° 
with the beam. Find the tensions in the chains. 

308. A force of 50 pounds acts eastward and a 
force of 50 pounds acts westward. Will there be 
motion ? 

That depends, as will easily be seen, upon the position of the 
forces. If they act on the two ends of a rope there will be no mo- 
tion. If they act one on the northerly part of a brake wheel and 
one on the southerly part there will be motion, — that of rotation. 

Such forces produce a 

Couple : two equal, opposite, parallel forces not acting in the 
same straight line. 

The tendency to motion by couples is not of translation but of 
rotation. The measure of this tendency is, — 

Moment of a couple equals the product of one of the two forces 
X perpendicular distance between them. 

What is the resultant of a couple of moment 15, 
and a force 3 ? 

309. A brakeman sets up a brake on a freight 
car by pulling 50 pounds with one hand and pushing 
50 pounds with the other ; his forces act tangentially 
to the brake wheel, the diameter of which is \\ feet. 
Another time he produces the same brake resistance 
by using a lever in hand wheel and pulling 25 pounds. 
How far from handwheel must his hands be placed t 

310. When are couples said to be like and when 
unUke 1 When will two unlike couples balance each 


Other? (i) If a system of forces is represented in 
magnitude and position by the sides of a plane poly- 
gon taken in order, show that the system must be 
equivalent to a couple. (2) If the sides of a parallelo- 
gram taken in order represent a system of forces act- 
ing upon a body, express the moment of the couple to 
which the system of forces is equivalent. 

311. Show that a force and a couple in one plane 
may be reduced to a single force. Given in position 
a force of 10 pounds, and a couple consisting of two 
forces of 4 pounds each, at a distance of 2 inches, 
acting with the hands of a clock, draw the equivalent 
single force. 

312. The length of the side of a 
square ABCD is 12 inches. Along 
the sides AB and CD forces of 10 
pounds act, and along AD, CB forces 
of 20 pounds. Find the moment of 
the equivalent couple. 

Moments about D, 

— 12 X 10+12 X2o = moment of equivalent-couple 
12 X 10 = moment of equivalent-couple 

313. Forces P and O act at A, and are completely 
represented by AB and AC, sides of a triangle 
ABC. Find a third force R such that the three 
forces together may be equivalent to a couple whose 
moment is represented by half the area of the triangle. 

314. A tradesman has a balance with arms of un- 
equal length, but tries to be fair by weighing his ma- 


terial first from one scale pan, then from the other. 
Show that he will defraud himself. 

315. A tradesman uses a balance with arms in 
ratio of 5 to 6 ; he weighs out from alternate pans 
what appears to be 30 pounds. How much does he 
gain or lose } 

316. The beam of a balance is 6 feet long, and it 
appears correct when empty ; a certain body placed 
in one scale weighs 1 20 pounds, when placed in the 
other, 121 pounds. Show that the fulcrum must be 
distant about J3 of an inch from the center of the 

317. The weight of a steelyard is 1 2 pounds, its 
movable weight is 3 pounds. Find the distance 
between successive pound graduations, if the length 
of the short arm is 3 inches. 

318. A weight of 247 pounds is attached to one 
end of a horizontal straight lever, which is 22 inches 
long, and may be regarded as having no weight ; 
the force is applied at the other end, and makes an 
angle of 27° with the lever; the fulcrum is 3 inches 
from the weight. F'ind the magnitude of the force 
when it just balances the weight. 

319. A uniform beam rests at a 
given inclination, Q, with one end 
against a smooth vertical wall, and 
the other end on smooth horizontal 
ground : it is held from slipping by 
a string extending horizontally from 


the foot of the beam to the foot of the wall. Find 
the tension in the string and the pressures at the 
ground and wall. 

AB is the beam, AC the wall, BC the string, W the weight of 
the beam acting at its middle point G. 

There are three forces supporting the beam : vertical reaction P, 
horizontal reaction R, and tension in the string F. 

T?.ke moments about B, the point of intersection of two of the 

forces — their lever arms would be zero. 

Rx AC = W X 


Substitute for AC its value BC X tan d, then 


(.)R = 

2 tan 

but R must equal F, both being horizontal resisting forces that main- 
tain equilibrium; likewise P and W must be equal. 

... (2) F = ^^ , and 
^ ' 2 tan e 

(3) P = W 

320. A uniform beam rests with a smooth end 
against the junction of the horizontal ground and a 
vertical wall ; it is supported by a string fastened to 
the other end of the beam and to a staple in the ver- 
tical wall. Find the tension of the string, and show 
that it will be half the weight of the beam if the 
length of the string be equal to the height of the 
staple above the ground. 

321. A uniform rod 8 feet long, weighing i8 
pounds, is fastened at one end to a vertical wall by a 
smooth hinge, and is free to move in a vertical plane 
perpendicular to the wall. It is kept horizontal by a 
string lo feet long, attached to its free end and to a 


point in the wall. Find the tension in the string, and 
the pressure on the hinge. 

322. A uniform beam, 1 2 feet in length, rests with 
one end against the base of a wall which is 20 feet 
high. If the beam be held by a rope 13 feet long, 
attached to the top of the beam and to the summit of 
the wall, find the tension of the rope, neglecting its 
weight, and assuming the weight of the beam to be 
100 pounds. 

323. ABC is a rigid equilateral triangle, weight not 
considered; the vertex B is fastened by a hinge to a 
wall, while the vertex C rests against the wall under 
B. If a given weight is hung from A, find the reac- 
tions at B and C. What are the magnitudes and 
directions of the forces exerted by the weight on the 
wall at B and C 1 

324. A beam AB rests on the smooth ground at 
A and on a smooth inclined plane at B ; a string is 
fastened at B and, passing over a smooth peg at the 
top of the plane, supports a weight P. If W is the 
weight of the beam, and a the inclination of the plane, 
find P and the reactions on the rod. 

Draw the figure. 

The weight W acts at the middle point C, The reaction of the 
ground at A is R, upwards. 

The reaction of the plane at B is Ri, perpendicular to the plane. 

Let the angle BAD = Q. 

The tension of the string at B = tension of the string throughout 

:^ P 

There are four forces acting on the beam, W, R, Ri, P. 
Resolve vertically and horizontally. 


325. A pole 12 feet long, weighing 25 pounds, 
rests with one end against the foot of a wall, and 
from a point 2 feet from the other end a cord runs 
horizontally to a point in the wall 8 feet from the 
ground. Find the tension of the cord and the pres- 
sure of the lower end of the pole. 

326. A light smooth stick 3 feet long is loaded at 
one end with 8 ounces of lead ; the other end rests 
against a smooth vertical wall, and across a nail which 
is I foot from the wall. Find the position of equi- 
librium and the pressure on the nail and on the wall. 

327. A trapezoidal wall has a vertical back and a 
sloping front face ; width of base, i o feet ; width of 
top, 7 feet ; height, 30 feet. What horizontal force 
must be applied at a point 20 feet from the top in 
order to overturn it } Thickness of wall, i foot ; 
weight of masonry in Avail, 1 30 pounds per cubic foot. 

328. Six men using a rope 50 feet long were just 
able to pull over a chimney 7$ feet high. How far 
up from the bottom of the chimney was it advisable 
to attach the rope } 

329. If 1 50 000 pounds is the thrust along the 
connecting rod of the engine, in example %6y 2 J feet 
the crank radius, and the connecting-rod is inclined 
to the crank axis at 150°, show that the moment of 
the thrust about the crank-pin is one-half the greatest 
possible moment. 

330. A trap-door of uniform thickness, 5 feet long 
and 3 feet wide, and weighing 5 hundred weight, is 


held open at angle of 35° with the horizontal by 
means of a chain. One end of chain is hooked at 
middle of top edge of door, and the other is fastened 
at wall 4 feet above hinges. Find the force in the 
chain and the force at each hinge. 

331. The sketch represents a coal wagon weighing 

with its load 4| tons. How 
many pounds applied at 
P by usual methods of hand 
-^B power will just lift the 
wagon when in the posi- 
tion shown in the sketch } 
AE is a rod in tension. CD is a connecting-bar. 
Divide the problem into three parts : 
(a) Draw the forces acting. 

{b ) Find horizontal distance from C to the verti- 
cal through the center of gravity. 

( e) Find force to apply at C parallel to P ; then 
find P. 


332. A rod of uniform section and density, weigh- 
ing 3 pounds, rests on two points, one under each 
end ; a movable weight of 4 pounds is placed on 
the rod. Where must it be placed so that one of the 
points may sustain a pressure of 3 pounds, and the 
other a pressure of 4 pounds ? 


333. Two rods of uniform density 
weighing 2 pounds and 3 pounds re- 
spectively are put together so that the 
3-pound one stands on the middle of 
the other. Find the center of gravity of a- 

the whole. ""=• ''■ 

Take moments about AB, 

+ 3 X J / - 5 X a: = o 

334. A thin plate of metal is in the shape of a 
square and equilateral triangle, having one 
side common ; the side of the square is 1 2 
inches long. Find the center of gravity of 
the plate. 

B Let Gi be the center of gravity of the triangle, Gj 

of the square, G of the whole plate. 
From symmetry EGj GG^O will be a straight line bisecting the 
plate, and 

OG., = 6 inches 
OGi =15.5 inches 
Let 7^ = weight of metal per square inch 

Area of triangle = ^ x 12 x^iz'^ — 6'^ 
= 62.4 square inches 
Weight = 62.4 pounds X w pounds 
Area of square = 144 square inches 
Weight = 144 X w pounds 
Take moments about the axis AB, 
Weight of triangle X OGi + weight of square XOG2— total 
weight X OG = o 
62.4 wx i5.5-|-i44Z£/x6 — (62.4«/-|-i447£/) X OG = o 
.-. OG = 8.86 inches. 



335. A bridge member has two web plates i8 
X I inches, top plate 21 x g, top angles 3x3 and 
I inches thick, bottom angles 4x3 and y^. inches 
thick. Find " eccentricity " — the distance from AB, 
the neutral axis through the center of gravity to C, 
the middle of the section. 




Fig. 45. Fig. 46. 

336. Web plate of Fig. 46 is 10 x \ inches, top 
plate 12 X J, two angles 4 x 3 X |. Find "eccen- 
tricity." (Given in Osborn's Tables (1905) page 24.) 

337. Fig. 47 shows a cross-section of the top chord 
of one of the main trusses in the Portage Canal 
Draw-Bridge at Houghton, Mich. See Engineering 
News of June 15, 1905. In computing the strength 
of this built-up member, it is required to find the 
position of the axis AB that passes through the center 
of gravity of the section. 

I 3ix-3iy| 



s.\ssssss ess 


Fig. 47- 





Fig. 48. 


338. The strength of steel rails is usually com- 
puted by embodying, among other factors, the distance 
from neutral axis, which passes through the center of 
gravity, to the extreme fibres of the section. A 
100-pound rail, of the Lorain Steel Company, has a 
section shown in Fig. 48. Draw the section carefully 
to full scale on bristol board ; then cut it out and 
locate its center of gravity by balancing on a knife 
edge. What is the distance from center of gravity 
to extreme fibres ? 

339. ABC is a triangle with a right angle at A. 
AB = 3 inches ; AC = 4 inches ; weights of 2 
ounces, 3 and 4, are placed at A, B, and C. Find 
the position of their center of gravity. 

340. A uniform triangle ABC of weight W, and 
lying on a horizontal table, is just raised by a vertical 
force applied at A. P^ind the magnitude of this force, 
and that of the resultant pressure between the base 
BC and the table. 

341. A uniform circular disk has a circular hole 
punched out of it, extending from the circumference 
half way to the center. Find the center of gravity 
of the remainder. 

342. A box, including its cover, is made of six 
equal square boards ; where is its center of gravity 
when its lid is turned back through an angle of 180° t i 



343. ABCD is a 

thin rectangular 
plate weighing 50 e; 
pounds, AB is 10 nj 
feet, EC 2 feet ; the ^ i 
plate is suspended 
by the middle point 
of its upper edge 
AB, and then, of ^'^- ^^' 

course, AB is horizontal, but if a weight of 5 pounds 
is placed at A, AB will become inclined to the hori- 
zon. Show how to find the angle of inclination 
either by calculation or by construction. 

344. A circular disk, 8 inches in diameter, has a 
hole 2 inches in diameter punched out of it, the center 
of the hole being 3 inches from the circumference 
of the disk. Find the center of gravity of the remain- 
ing portion. 

345. Find the centers 
of area of the above sec- 
tions of uniform plate 

< 5 in. > 

< 3 > o 

Fig. 50. 

346. Into a hollow cylindrical vessel 1 1 inches 
high and weighing 10 pounds, the center of gravity 
of which is 5 inches from the base, a uniform 
solid cylinder 6 inches long and weighing 20 pounds 
is just fitted. Find the common center of gravity. 








Gi center of gravity of hollow cylinder 
G2 center of gravity of solid cylinder. 
Moments about AB, 

+ 10X5 + 20X3 — 3oXJc=o 

+ 50 + 60 — 30 AT = O 
30^:= I 10 

X = 3§ inches. ^*K- 51. 

347. Give examples of stable and unstable equilib- 
rium. A cone and a hemisphere of the same material 
are cemented together at the common circular base. 
If they are on a horizontal plane, and the hemisphere 
in contact with the plane, find the height of the cone 
in order that the equilibrium may be neutral. (The 
center of gravity of a hemisphere divides a radius in the 
ratio of 3 to 5.) 

348. A thread 9 feet long has its ends fastened to 
the ends of a rod 6 feet long ; the rod is supported 
in such a manner as to be capable of turning freely 
round a point 2 feet from one end ; a v^eight is placed 
on the thread, like a bead on a string. Find the 
position in which the rod will come to rest, it being 
supposed that the rod is without weight, and that there 
is no friction between the weight and the thread. 

349. A circular disk weighs 9 ounces ; a thin 
straight wire as long as the radius of the circle weighs 
7 ounces ; if the wire is placed on the disk so as to be 
a chord of the circle, the center of gravity of the 
whole will be at a distance from the center of the 
circle equal to some fractional part of the radius. 
Find that fraction by construction or calculation. 



350. A cone and a hemisphere are on the same 
base. What height must the cone be in order that the 
center of gravity of the whole solid shall be at the 
center of the common base ? 

r = radius common base. 
h = height of cone. 


The coefficients of friction for various pairs of sub- 
stances have been found experimentally by Morin ; 
these results however can be used only for approxi- 
mate computation ; actual trial should be made for 
specific cases. Average values are: 

Stone on stone 0.40 to 0.65 

Wood on wood 0.25 to 0.40 

Metal on metal, dry . . ... . . 0.15 to 0.30 

well oiled . . . 0.0 1 to 

1. Friction is proportional to normal reaction, R. 

2. Is independent of area of contact. 

3. Is dependent very much on the roughness of surfaces. 

351. Define " coefficient " and *' angle of friction," 
and "resultant reaction." 

R. 352. A weight of 56 pounds is moved 

F^-^-^sib». along a horizontal table by a force of 
I b6iis. I 8 pounds. How much is the coefficient 
Fig. 52. of friction .? 

The pull of 8 pounds is required to overcome friction, and is 
equal to the friction. 

Friction = coefficient x Reaction (perpendicular to plane of table. 


F = /M X R 

= /^i X 56 pounds 
8 = ya X 56 

= f 

353. A 5 X 8-foot vertical gate has a head of water 
against its center equal to 10 feet, or 4 J pounds per 
square inch. The coefficient of friction being 0.40, 
what force is required in raising it to overome the 
friction 1 

354. A horizontal pull of 50 pounds is required to 
slide a trunk along the floor. The coefficient of fric- 
tion is 0.20, and trunk when empty weighs 75 pounds. 
How many pounds of goods does it contain } 

355. A block of stone is dragged along the ground 
by a horse exerting a force of 224 pounds. If /x = 0.6, / 
what is the weight of the block } 

356. A weight of 500 pounds is placed on a table, 
and can hardly be slid by a horizontal pull of 155 
pounds. Find the coefficient of friction, and the 
number of degrees in the angle of friction by measur- 
ing from a drawing made to a scale. 

357. A stone just slides down a hill of inclination 
30°. What is the coefficient of friction t ^ 

358. A block rests on a plane which is tilted till 
the block commences to slide. The inclination is 
found to be 8.4 inches at starting, and afterwards 6.3 
inches on a horizontal length of 2 feet. Find the co- 


efficient of friction when the block starts to slide, and 
after it has started. 

359. A horse draws a load weighing 2 000 pounds 
up a grade of i in 20 ; the resistance on the level is 
100 pounds per ton. Find the pull on the traces 
when they are parallel with the incline. 

360. How much work has a man, weighing 224 
pounds, done in walking twenty miles up a slope of i 
vertical to 40 horizontal 1 What force could drag a 
dead load of the same weight up the same hill {a) if 
friction be negligible, {b) if friction be \ of the 
weight } 

361. Three artillerymen drag a gun weighing 
I 700 pounds up a hill rising 2 
vertically in 17 horizontally. Sup- 
pose the resistance to the wheels 
going up the hill be 16 pounds 

per hundred weight, what pull parallel to the hill must 
each exert to move it } 

When the gun is about to move forward the pull P will be acting 
up the plane, and parallel to it ; the friction F down the plane, hold- 
ing back; the force R perpendicular to inclined plane, partly sup- 
porting the gun, and W the weight of the gun acting vertically down- 
ward. "Weight of gun is given — i 700 pounds. Resolve into com- 
ponents perpendicular and parallel to the plane. The perpendicular 
component will be the supporting force of the plane — its reaction 
R ; the parallel component will be the part of the pull P required by 
weight of the gun. 

362. Find the force which, acting in a given direc- 
tion, will just support a body of given weight on a 


rough inclined plane. The height is to the base of 
the plane as 3 to 4, and it is found that the body is 
just supported on it by a horizontal force equal to half 
the weight of the body. Find the coefficient of fric- 
tion between the body and the plane. 

363. The table of a small planing-machine which 
weighs 1 1 2 pounds makes six single strokes of 4^ 
feet each per minute. The coefficient of friction be- 
tween the sliding surfaces is .07. What is the work 
in foot-pounds per minute performed in moving the 
table ? 

364. A rectangular block ABCD whose height is 
double its base, stands with its base AD on a rough 
floor, coefficient of friction J. If it be pulled by a 
horizontal force at C till motion ensues, determine 
whether it will slip on the floor, or begin to turn over 
round D. 

365. A cubical block rests on a rough plank with 
its edges parallel to the edges of the plank. If, as 
the plank is gradually raised, the block turns over on 
it before slipping, how much at least must be the 
coefficient of friction t 

366. A weight of 5 pounds can just be supported 
on a rough inclined plane by a weight of 2 pounds, or 
can just support a weight of 4 pounds suspended by 
a string passing over a smooth pulley at the vertex. 
Find the coefficient of friction, and the inclination of 
the plane. 


367. Find the least force that will drag a box 
weighing 200 pounds along a concrete floor, the co- 
efficient of friction being 0.50. 

The required force will of course not act horizontally, but instead 
in some direction as P. To find the angle b : 
Resolve vertically 

— P sin (^ — R +, 200 = o 
Resolve horizontally 

+ P cos b — /otR= o 
From these two equations 

p_ 200M _ 

ju. sin <^ + cos b 

When will P be as small as possible ? When m sin b + cos b is 
as large as possible. The student not familiar with the calculus can 
find by trial that the maximum value of denominator, or least value 
of the pull P, will occur when b = tan— ^ /ix, that is, the angle whose 
tangent is m-. 

By the method of calculus, 

M sin b + cos b = x 

Differentiate, noting that m is a constant ; and, to find a critical 
value, which in this case will be a maximum value, place the first dif- 
ferential equal to zero. 

dx 1 • / 

— - = M COS b — s\n b =^ o 


M = tan b 

b = tan ~^ ft. 


p _ 200 X 1 

I X .447 + .894 

= 89 pounds. 

368. By experiment it was found that a box of sand 
weighing 204 pounds required a least pull of iio 
pounds (at angle a) to move it on a concrete floor. 
What was the value of /x .? . 



369. The roughness of a plane of 
inclination 30° is such that a body of 
weight 500 pounds can just rest on 
it. What is the least force required 
to draw the body up the plane ? 

Fig. 55. 
As in problem 367 a will equal the angle of friction, or tan ' /i. 

370. A sled of total weight 3 tons is to be drawn 
up a grade of i vertical to 8 horizontal. The coeffi- 
cient of friction between the sled shoes and the snow 
is o. 10. What angle should the traces make with the 
horizontal } What pull will the horse exert } 

The problems that pertain to the wedge can be solved by the same 
methods that have been used for the inclined plane. The essential 
principles are : Show the conditions by a sketch, indicating carefully 
the position and direction of all forces ; then, (i) resolve parallel to 
plane. (2) resolve perpendicular to plane. Thus for problem 371 

Fig. 56 shows the conditions for one form of wedge. Fig. 57 for 

Observe the directions of R and W. As can be seen in Fig. 56, 
\ P should decrease the value of R, therefore R must act in the 
direction indicated. In previous problems the weight has moved on 
the inclined plane ; here the plane moves. 

Resolve || to top plane (Fig. 56), 

— 0.20 R + 300 cos 3° 35' — W X sin 3° 35' = o. 


Resolve ± to plane, 

+ R-300 sin 3° 35' - W X cos 3° 35' = o. 

Solve these two equations for W. 

To find the pull necessary to withdraw the wedge, sketch another 
figure showing /xR and 5 P in their new positions. Then solve as 
indicated above. 

371. A cotter, or wedge, having a taper* of i in 8, 
is driven into a cottered joint with an estimated pres- 
sure of 600 pounds. Taking the coefficient of friction 
between the two surfaces as 0.2, find the force which 
the wedge exerts at the joint perpendicular to the 
pressure of 600 pounds ; also find the pull necessary 
to withdraw the wedge. 

372. A floor-column with its load of 5 tons is to be 
lifted by two wedges driven towards each other. 
Thickness of each wedge is 2 inches, length, 12 
inches; coefficient of friction, 0.15. Find the force 
that must be equivalent to P in order to drive the 

373. A casting of weight 5 000 pounds is to be 
lifted by an iron wedge that is forced ahead by a 
screw and mechanism that can give an equivalent 
force of 3 000 pounds. If a 12-inch wedge is used, 
what should be its thickness } 

374. A rough wedge has been inserted into a block 
and is only acted on by the reactions. If it is on the 
point of slipping out, and the coefficient of friction 

is — p , what is the angle of the wedge .? 


375. A steel wedge 1 2 inches long, 2 inches thick, 
tapering on both sides to o, is used to wedge up a 
pump plunger weighing 3 000 pounds by means of a 
maul weighing 5 pounds. The coefficient of friction 
is 0.15 and the striking velocity of the maul is 25 feet 
per second. How far will each blow drive the wedge ? 

376. A wheel of weight W rests between two 
planes, each inclined to the vertical at angle a ; the 
plane of the wheel is perpendicular to the line of in- 
tersection of the two planes, which is itself horizontal. 
If ft be the coefficient of friction, find the least couple 
necessary to turn the wheel. 

377. A ladder inclined at an angle of 60° to the 
horizon rests with one end on rough pavement, and 
the other end against a smooth vertical wall ; the 
ladder begins to slide down when a weight is put at 
its middle point. Show that the coefficient of friction 

is — ^ . 

When the ladder begins to slide down, the 
limiting friction would be /xR. 
Resolve vertically, 

W = R 

Resolve horizontally, 

R' =mR 

Take moments about B, and then solve 
for li.. 

If the wall should be rough there would be 
acting at B an upward force of /x'R' that would have to be embodied 
in the above equations. 


378. A uniform ladder weighing loo pounds and 
52 feet long is inclined at an angle of 45° with a 
rough vertical wall and a rough horizontal plane. If 
the coefficient of friction is at each end |, how far up 
the ladder can a man weighing 200 pounds ascend 
before the ladder begins to slip ? 

379. A uniform ladder 30 feet long is equally in- 
clined to a vertical wall and the horizontal ground, 
both rough ; a man with a hod — weight 224 pounds 
— ascends the ladder which weighs 200 pounds. 
How far up the ladder can the man ascend before it 
slips, the tangent of the angle of resistance for the 
wall being \ and for the ground \ ? 

380. A uniform beam rests with one end on a 
rough horizontal plane, and the other against a rough 
vertical wall, and when inclined to the horizon at an 
angle of 30^ is on the point of slipping down ; sup- 
pose the surfaces equally rough, find ya. 

381. A bolt for a cylinder head has 8 threads per 
inch; mean diameter of threads i|- inches, average 
outside diameter of nut 2| inches, inside diameter of 
bearing surface, 1.6 inches. The nut is to be tight- 
ened by a pull on the end of a 3-foot wrench. The 
coefificient of friction for threads and underneath the 
nut being o. 1 5, what pull should be exerted in order 
that the stress in the bolt shall not exceed 50 000 
pounds ? 

Problems pertaining to bolt and nut friction can be solved by 
applying the combined principles of Work and Friction. Thus for 




the above problem suppose that the specified conditions should exist 
for one revolution. This involves no approximation, simply a con- 
venience in numerical figures which otherwise would have to be 
divided by perhaps a hundred or thousand to apply to a fractional 
part of a revolution. Then, 

Work = Work 4- Work + Work 

on wrench on threads under nut of lifting 

The values of work on threads and work under nut can be deter- 
mined near enough for ordinary cases by slight approximations. 
As shown by Fig. 59, \V = .99 R, or usually R may be taken equal 
to W also the length of thread developed (for one revolution) ir x 
\\" \ and in Ffg. 60 the circumference C is one that can be deter- 
mined by the condition that the work done by the friction of all the 
particles outside is the same as that done by all the particles inside. 

Its radius for a section like Fig. 60 is x = -^-^ j- , or, for this 

case, X = I.I I inches, which is approximately the same as would 

TT X 1 J in. 

As the thread advances 
it acts like a wedge. 

Fig. 59. 

Fig. 60. 

result by taking a mean circumference between 1.6 inches diameter 
and 2.75 — or 1 .09 inches. Therefore ordinarily use the mean cir- 
cumference for the position of friction under nut. 

The equation for Work thus becomes : 

For one revolution. 

Wrench Threads Nut 

P(3X i2X2X3l) = 5ooooxo.i5X4.7i + 5ooooxo.i5X 1.09x2x3]^ 


+ 50 000 X 0.125. 

From which equation find P, the pull that should be exerted on 
wrench to produce 50 000 pounds stress in the bolt. 


382. By trial in a 60 000-pound testing machine 
we have obtained with a builder's lifting-jack a stress 
on the machine of 6 000 pounds for a certain pull on 
the end of an 1 8-inch bar. What was that pull ? 
Mean diameter of threads was 1.50 inches, there were 
3 threads to the inch, and diameter of bearing that 
corresponds to the mean circumference of nut de- 
scribed under problem 381 was 1.78 inches. Coeffi- 
cient of friction for threads was 0.15, and for bearing 

383. A locomotive bolt has 10 threads to the inch; 
mean diameter 2 inches, average outside diameter of 
nut 4|- inches, diameter of hole in washer on which 
nut turns 2.2 inches. If length of wrench was 5 
feet, pull 367 pounds, and stress 40 000 pounds, what 
was the value of the coefficient of friction 1 

384. A test of rope friction in our engineering lab- 
oratory at Tufts College has given the following 
result : 

(The weight Tj just moving, and pull T^ resisting any increased 
motion. See Fig. 6i.) 


Weight of 


= 100 






Number of 



Number of 
































Compute the ratios of T, and T^, then plot the results, using a 
scale of I inch = i lap for vertical ordinates, and i inch = ratio of 



10 for horizontal. Sketch the most probable curve for the plotted 
points, observing that it does not necessarily pass through the last 
point, and determine whether or not it should pass through the 

Fig. 61. 

385. Now plot the same results on the specially 
ruled paper of page 108. 

CThis form of ruling was used by the Burr-IIering-Freeman 
Commission on Additional Water Supply of the city of New York 
(1903) for plotting wide ranges of values in a small space [7 million 
to 40 million in a i^-inch space], yet affMiding increased scale for 
the small values. It has not, to my knowledge, been used before 
for mechanics' problems of this sort.) 


Number of Laps 



Determine how this form of ruHng is constructed. 
Plot the points for laps and ratios, and draw the most 
probable straight line through the points as before. 
Should the line pass through the origin .? How do 
the points for i\ laps and 2| compare on this straight 
line with those on the curved line previously plotted } 

386. The lines plotted for the preceding prob- 
lems are sufficient to answer directly many questions 
pertaining to that particular rope and piece of timber. 
For the same conditions, how many laps are needed 
to hold a weight of 300 pounds with a pull of 40 
pounds } 

387. For the same conditions, with 2 laps and a 
pull of 100 pounds, what weight could be lowered 
into the hold of a vessel t 

388. It is evident that the plotted lines of the pre- 
ceding problems would not apply to other cases of 
friction. The value of /a the coefficient of friction 
is contained in the equation of the curves, but can- 
not yet be specified. 

For the purpose of finding the value of the coefficient, as will be 
done later on, it is advisable to determine the slope and position of 
the plotted line ; that is, its equation. Notice that « (the number 


of laps) = c (a constant depending on the slope of the line) x „- 

(the ratio of the two tensions). The value of c (the slope of the 
line) would depend upon the stiffness of rope and roughness of the 
rubbing surfaces. For this particular rope and piece of wood, the 
value of c, according to one plotting that I have, is 2.08. 

What does your plotting indicate for the above ex- 
periment ? 




389. From the above the equation of the line be- 
comes ;/ =z 2.08 los:^. Write your equation and 


transpose so as to write the value of log T,, which I 
find to be log T^ = log T^ + .437 X n. Now this 
equation derived from the laboratory experiment will 
be seen to bear a close relation to the general for- 
mula for rope and belt friction which will now be 

The author's method of analysis is introduced here for the reason 
that he believes it to be more easily understood than the methods 
usually presented in text books. 

At B there would be a tension of Tj, at A, T,„ and at any point, 
C, there would be two tensions, T and T + rt'T. Now if we knew 
the reaction at the point C we could multiply it by /* and obtain the 

friction. To find this reaction R draw the parallelogram of force, 
Fig. 63. Let an arc of the angle dd measured at unit distance from 
the center be arc dQ, then at distance T the arc w^ould be T xarc dQ, 
and for a very small angle — a differential angle — this value of the 
arc would equal R. 

R = T arc dd 
/iR = |uT arc dd. 


Now the friction «R = the difference in tensions t/T. 
dT = AtT arc dB. 

That is, for an infinitesimal arc, the difference in tension = uY x 
the infinitesimal arc. If we take a very large number of small arcs 
we can find the friction at each point, add up and get the sum total 
of friction ; or, summing up by the calculus. 

r ¥-'/'-" 


log,l\ - l0g.T2 =/X^ 

Now 6 = 2 rrn in which n is the number of laps ; and the Naperian 
log can be changed to the common system by multiplying by .4343. 

••• logio T, - log,o T2 = .4343 ^X 2irn 
logTj=logT2+ 2.7288 M^/ 

which is the general formula for rope and belt friction. It contains 
variable quantities : the tension Tj, tension T^, and the number of 
laps //. Any two of these being given the third can be found. 
The formula deduced by experiment in problem 389 was^ 

log T, = log T, 4- .437 «• 
The similarity of this with the general formula is evident. The 
last term must contain the value of fi the coefficient of friction. 
To find this value solve the two equations, and we have 
2.7288 /x« = .437 n 
/x = o. 1 5 

Then if this be taken as the vakie of 11, how many 
laps would be necessary, according to the general 
formula, for 100 pounds to just move 14 pounds.^ 
(Check result with data of problem 384.) 

390. A weight of 100 pounds just m.oves 37 
pounds, both being connected by a plain leather belt 
that encircles one-half of a 14-inch iron pulley that 
does not turn. Plot the point on the paper of page 
108 ; draw the Hne of friction, and write the equation 
of the line. Then compare with the general formula 



and determine the value of the coefficient of friction 
for the plain belt and iron pulley. 

... .iJi 

Fig. 64. 

391. In the same way as by problem 390, after the 
belt had been treated -with " cling-fast " belt dressing 
55 pounds just moved 13. Plot the line and find the 
coefficient of friction. 

Further application of the general belt and rope friction formula 
is seen in the problems that follow. 

392. According to conditions of friction as in prob- 
lem 390, how many turns would have to be taken 
around a capstan in order to lower a barrel of salt, 
150 pounds, into a dory without pulling over 50 
pounds ? 


393. A weight of 5 tons is to be raised from the 
hold of a steamer by means of a rope which takes 
3^^ turns around the drum of a steam-windlass. If 
/A = 0.234, what force must a man exert on the other 
end of the rope t 

394. A man by taking 2\ turns around a post with 
a rope, and holding back with a force of 200 pounds, 
just keeps the rope from surging. Supposing /x = 
0.168, find the tension at the other end of the rope. 

395. A leather belt will stand a pull of 200 pounds. 
It passes around one-half the circumference of a 
pulley that 'is 4 feet in diameter and making 150 
revolutions per minute. What power will it transmit 
if the coefificient of friction between the belt and 
pulley is o. i '^. 

396. A belt laps 150° around a 3-foot pulley, mak- 
ing 130 revolutions per minute; the coefficient of ^ 
friction is 0.35. What is the maximum pull on the 
belt when 20 horse-power is being transmitted and 
the belt is just on the point of slipping.? 

397. A weight of 2 000 pounds is to be lowered 
into the hold of a ship by means of a rope which 
passes over and around a spar lashed across the hatch- 
coamings so as to have an arc of contact of i \ cir- 
cumferences. If /i = 2^, what force must a man 
exert at the end of the rope to control the weight } 

398. A hawser is subjected to a stress of 10 000 
pounds. How many turns must be taken around the // 
bitts, in order that a man who cannot pull more than 


250 pounds may keep it from surging, supposing 
ft = 0.168 ? 

399. A rope drive carrying 20 ropes has a pulley 
16 feet in diameter, and transmits 600 horse-power 
when running at 90 revolutions per minute. Taking 
At = 0.7 and the angle of contact 180°, find the ten- 
sions on the tight and slack sides of the ropes. 

From the data that is given find by the principles of Work the 
force that the belt is transmitting. It is 

Ti-T2= 218.8 pounds 
Substitute this value of Tg in the general formula, also the value 
of \i. and 71 ; then 

log T, == log (Tj- 2 18:8) + .9550 * 

^"^( t, -^218.8 ) = -955 



1\- 218.8 
From which find T^ ; then find T2. 

400. A belt for a dynamo is to encircle half of an 
18-inch pulley. The speed of pulley is to be 1060 
revolutions per minute ; horse-power to be trans- 
mitted, 100 ; coefficient of friction, 0.2 : thickness of 
belt to be || inches, and working strength 300 
pounds per square inch. What should be its width } 

401. A main driving belt is to encircle half of a 
54-inch pulley. The speed of pulley is to be 350 
revolutions per minute ; horse-power transmitted, 520 ; 
coefficient of friction, 0.2. Thickness of belt, accord- 
ing to specifications, is to be || inches, and work- 
ing strength 450 pounds per square inch. What 
should be its width ? 


402. A plain belt without dressing encircling one- 
half of a pulley, when just on the jDoint of slipping 
has a tension of i ooo pounds on the taut side. More 
machinery being put into use, rosin is thrown on the 
belt If the tension on the. slack side remains the 
same as "before, and the belt is just on the point of 
slipping, what horse-power will be transmitted, diame- 
ter of pulley being lO feet, and revolutions per min- 
ute, 140 .^ 

403. A single fixed pulley, 6 inches in diameter, 
turns on an axle 2 inches in diameter ; coefficient of 
friction, 0.2. A weight of 500 pounds 
is lifted by means of this pulley. Find 
the force V that is required. 

Friction causes the axle to creep, as it were, on its 
bearings. S moves a little off center, coming nearer 
to P. When the value of K can be found the fric-. ^i?- 65. 
tion will be determined by multiplying by yu. To find R : 

S = R2 + /i2K2 

as will be evident by plotting a parallelogram of force. 

S = 

P + W 

•. R = 

P+ W 

Vi + /*' 


p-h 500 

/iR = 


0.2 X ^— 


Now to 


Take moments about C, the center 

- P X 3 + 500 X 3 + /xR X I 
P = 570 pounds. 


404. A shaft makes 50 revolutions per minute. 
The load on the bearing is 8 tons, the diameter of the 
bearing is 7 inches, and the average coefficient of 
friction is 0.05. At what rate is heat being gen- 
erated ? 

S = P -f W 
= 8 tons 
P + W 


= 15980 

ftR = 799 pounds 
Work = force X distance 

of friction 

= 799 X (1^0 X 2 X ---^ X 50) 

= 73 500 foot-pounds per minute. 

405. A single fixed pulley, 2 feet in radius, turns 
on an axle i inch in radius ; the weight of the pulley 
is 80 pounds. A weight of 500 pounds is lifted by 
means of this pulley. What force P is required } 
The coefficient of friction between axle and bearing 
is o. I ; the rope is flexible, and without weight, and 
P acts vertically. 

406. Find the horse-power necessary to turn a shaft 
9 inches in diameter making 75 revolutions per min- 
ute, if the total load on it is 12 tons and 11 = .015. 

407. Let P and W be inclined to each other at an 
angle of 90° ; radius of pulley is 6 inches ; radius of 
axle I inch ; coefficient of friction, 0.2. Determine 
the relation of P and W in case of incipient motion. 



408. A horizontal axle 10 inches in diameter has a 
vertical load upon it of 20 tons, and a horizontal pull 
of 4 tons. The coefficient of friction is 0.02. Find 
the heat generated per minute, and the horse-power ^ 
wasted in friction, when making 50 revolutions per 

409. The shaft of a i 000-kilowatt dynamo is 25 
inches in diameter, makes ibo revolutions per min- 
ute, and carries a total load of 45 000 pounds. The 
coefficient of friction being 0.05, find the horse-power 
lost in heat that is generated by friction. 

410. Find the horse-power absorbed in overcoming 
the friction of a foot-step bearing with flat end 4 inches / 
in diameter, the total load being i^ tons, the number 
of revolutions 100 per minute, and the average coeffi- 
cient of friction 0.07. 

force X distance 

Work = \\> X (fj X {-.^) X 2 TT X 100. 

of friction 

The distance being obtained by consideinng a circumference as in 
problem 381, outside of which the work is the same as that inside. 
For a bearing with a flat end that circumference has a radius of two- 
thirds of R. 

411. Calculate the horse-power absorbed by a foot- 
step bearing with flat end 8 inches in diameter when 
supporting a load of 4 000 pounds, and making 100 
revolutions per minute, coefficient of friction 0.03. 

412. A 1 50-horse-power turbine has an oak step 

6 inches in diameter and with conical end tapering ^ 
45°. If the load on the step be 2 tons, and the 


coefficient of friction between the wood and its metal 
seat be 0.3, find the horse-power thus absorbed at 
65 revolutions per minute. 

To resist the load of 2 ^tons would require a pressure of 2.83 tons 
by the 45° slope of the foot-step. The mean circumference would 
be as in preceding problems, distant two-thirds Rfrom center. 

413. The shaft of a vertical steam turbine has a 
conical foot-step bearing 3.5 inches in diameter, and 
length 3 inches. Total load on shaft, i 500 pounds ; 
speed 2 500 revolutions per minute ; coefficient of 
friction, 0.07. Find the horse-power that tends to 
" burn out " the foot-step. 




414. A body moving with a velocity of 5 feet per 
second is acted on by a force which produces a con- 
stant acceleration of 3 feet per second. What is the 
velocity at the end of 20 seconds } 

Velocity gained = acceleration per second x number of 

V — a y. t 
= 3 X 20 

= 60 feet per second. 
Final velocity = 60-1-5 

= 65 feet per second. 

415. The initial velocity of a stone is 12 feet per 
second ; this velocity decreases uniformly at the rate 
of 2 feet per second. How far will the stone have 
traveled in 5 seconds } 

416. Two trains A and B moving towards each 
other on parallel rails at the rate of 30 miles and 
45 miles an hour, are 5 miles apart at a given instant. 
How far apart will they be at the end of 6 minutes 
from that instant, and at what distances are they from 
the first position of A t 

417. Two trains, 130 and no feet long, pass each 
other in 4 seconds when going in opposite directions. 
The velocity of the longest train being double that of 1/ 
the other, find at what speed per hour each is going. 



418. Two trains going in opposite directions pass 
each other in 3 seconds. One train is 142 feet long 
and the other 88 feet long. When going in the same 
direction one passes the other in 1 5 seconds. How 
fast is each train going ? 

419. The velocity of a train is known to have been 
increasing uniformly; at one o'clock it was 12 miles 
per hour ; at 10 minutes past one it was 36 miles per 
hour. What was it at yl minutes past one.** 

420. A train moving at the rate of 30 miles an 
hour is brought to rest in 2 minutes. The retarda- 
tion is uniform. How far did it travel } 

A railroad train is moving at 30 miles an hour. In each second, 
then, it moves 44 feet. Its velocity for each second during the time 
/ may be represented as in Fig. 66 by lines of equal length, and the 
area of the rectangle, or vt, represents the distance passed over. 
This is an illustration of uniform motion. 

When a railroad train starts from a station, and by uniform gain 
in speed attains a velocity of 30 miles an hour, the distance passed 



Fig. 66. Fig. 67. Fig. 68. 

over may be graphically represented as in Fig. 67. The area would 
represent said distance. 

421. Similarly, what condition of speed of the rail- 
road train would Fig. 6'^ represent } 

Note that the area of Fig. 68 is v^f -^ \ {v — v^ t, or vj + \ afi. 

422. A stone skimming on ice passes a certain 
point with a velocity of 20 feet per second, then suf- 


fers a retardation of one unit. Find the space passed 
over in the next lo seconds, and the whole space 
traversed when the stone had come to rest. 

423. On the New York Central and Hudson River 
Railroad test tracks near Schenectady, an electric loco- 
motive hauled 9 Pullman cars at a running speed of 
60 miles per hour. The average acceleration from 
start to full speed was 0.5 miles per hour per second. 
The retardation on applying air brakes was 0.88 feet 
per second per second. These results were obtained 
by carefully timing the train at measured stations. 
Total distance was 4 miles. What was the total time .'* 

" ^ 424. A train is running at the rate of 60 miles an 
hour when the steam is turned off ; it then runs on a 
level track for 3 J miles before stopping. If friction 
be the constant retarding force, find its value in 
pounds per ton. Also how far does the train run in 
3 minutes from the instant steam is turned off } 

425. A body acted on by a constant force begins 
to move from a state of rest. It is observed to move 
through 55 feet in a certain 2 seconds, and through 
yy feet in the next 2 seconds. What distance did it 
describe in the first 6 seconds of its motion 1 

426. A steamer approaching a wharf with engines 
reversed so as to produce a uniform retardation is 
observed to make 500 feet during the first 30 seconds 
of the retarded motion and 200 feet during the next 
30 seconds. In how many more seconds will the 
headway be completely stopped ? 

MOTION. 123 

427. Two bodies are let fall from the same point 
at an interval of 2 seconds. Find the distance be- 
tween them after the first has fallen for 6 seconds. 

For I St body, s =\gf' 

= ^ X 32 X 6- 

= 576 feet 
For 2d body, s = \gf^ 

= Kx 32 X 4' 
= 256 feet 
.'. distance apart = 576-256 
= 320 feet. 

428. A stone is projected vertically upwards with 

a velocity of 80 feet per second from the summit of A- 
a tower 96 feet high. In what time will it reach the 
ground, and with what velocity ? 

429. A hammer of 10 tons weight falling from a 
height of 4 feet drives a wooden pile and comes to 
rest in ^J^- second. How far does it drive the pile ? 
And assuming the force is uniform find it. 

430. A stone is dropped into a well, and the sound 
of its striking is heard 2y'2 seconds after it is dropped ; 
the velocity of sound in air is i 200 feet per second. 
What is the depth of the well .-^ 

Let s = depth of well. 

.'. time for sound to come up = seconds. 

I 200 

Tinie for stone to fall is found from formula 
s^ }ygr- 

^ 2S s J , ^ 

.*. ^ = — = — r J and / = — . 
^ i^ 4 


Time for stone to fall + time for sound to come up = 2y' 

I 200 4 12 
s + 300 ys = ^ 100 
s ± 150- 4- 300 Vi- = 3 100 ± 150^ 
Vi- = — 310 an inadmissible value, 
or \/s = + 10 

.*. s = 100 feet, depth of well. 

431. A stone is dropped from a tower of height a 
feet ; another is projected upwards vertically from the 
foot of the tower ; the two start at the same moment. 
What is the initial velocity of the second if they meet 
halfway up the tower .? 

432. A stone is dropped into a well, and the sound 
of the splash is heard y.y seconds afterwards. Find 
the distance to surface of the water, supposing the 
velocity of sound to be i 120 feet per second. 

433. A bucket is dropped into a well and in 4 sec- 
onds the sound of its striking the water is heard. 
How far did the bucket drop ? 

434. A balloon has been ascending vertically at a 
uniform rate of 4^ seconds, and a test ball dropped 
from it reaches the ground in 7 seconds. Find the 
velocity of the balloon and the height from which 
the ball was dropped. 

435. From a balloon that is ascending with velocity 
of 32 feet per second, a ball drops and reaches the 
ground in 17 seconds. How far up is the balloon ? 




436. A ball is let fall to the ground from a certain 
height, and at the same time another ball is thrown 
upwards with just sufficient velocity to carry it to the 1/ 
point from which the first one fell. When and where 
will they meet? 

437. A cake of ice slides down a smooth chute 
that is set at an angle of 30° to the horizon. Through 
how many feet vertically will the cake of ice fall in 
the fourth second of its motion? 

The acceleration for a body falling vertically is^, 32 feet per sec- 
ond per second. The acceleration component measured along a 
30°-plane is 32 x sin 30°, or 16 feet per second per second. 

= 72 feet, for 3 seconds 
= 128 feet for 4 seconds 
Therefore space along plane in the 4th second 
= 56 feet 

438. A cable car "• runs wild " down a smooth track 

of inclination 20 to the horizontal. How far does it ^' 
go during the first 8 seconds after starting from rest ? 

439. A body is projected up a plan of 30° incli- 
nation with a velocity of 80 feet per second. How 
long before it will come to rest ? How far will it 
go up the plane. 

440. A body is sliding with velocity u down an in- 
clined plane whose inclination to the horizon is 30°. - 
Find the horizontal and vertical components of this 

441. A stone was thrown with a velocity of 33 feet 
per second at right angles to a train that was going 


30 miles an hour. It hit a passenger who was sitting 
on the opposite side of the car that was 9 feet wide. 
How far in front of him should be the hole in the 
window .-* 

442. A deer running at the rate of 20 miles an 
hour keeps 200 yards distant from a sportsman. How 
many feet in front of the deer should aim be taken if 
the velocity of the bullet be i 000 feet per second t 

443. A boat is rowed at the rate 
of 5 miles an hour on a river that 
runs 4 miles an hour. In what di- 
V rection must the boat be pointed 
to cross the river perpendicularly .-* 
With what velocity does it move t 

Let OX be 4 units in length to represent 
the velocity of the stream. 
Draw OM perpendicular to OX. The resultant velocity is to be 
in the direction OM. 

With center X and radius of 5 units describe an arc cutting OM 
in P. 

Join XP, and complete the parallelogram of velocities OXPQ. 

OQ is the required direction. 
The angle QOP = sin-' i. 
Therefore the boat must not be rowed straight across, but up 
stream at an angle of 53° 10'. 
To find the resultant velocity : 

OP2 = OQ' - QP2 

= 5^-4^ 
= 25-16 

= 9 

•■•. ^^ = ^ 
.*. the boat crosses the river at the rate of 3 miles an hour. 

MOTION, 127 

444. A river flows at the rate of 2 miles per hour. 
A boat is rowed in such a way that in still water its 
velocity would be 5 feet per second in a straight line. 
The river is 3 000 feet wide ; the boat starting from 
one shore, is headed 60° up-stream. Where will it 
strike the opposite shore } 

445. A bullet moving upwards with a velocity of 
I 000 feet per second, hits a balloon rising with 
velocity 100 feet per second. Find the relative 

446. A train at 45 miles an hour, passes a carriage 
moving 10 yards a second in the same direction along 
a parallel road. Find the relative velocity. 

447. To a passenger in a train, raindrops seem to 
be falling at an angle of 30° to the vertical ; they are 
really falling vertically, with velocity 80 feet per 
second. What is the speed of the train t 

448. Two roads cross at right angles ; along one 
a man walks northward at 4 miles per hour, along the 
other a carriage goes at 8 miles per hour. What is 
the velocity of the man relative to the carriage } 

449. A steamer is going east with a velocity of 
6 miles per hour ; the wind appears to blow from the 
north ; the steamer increases its velocity to 1 2 miles 
per hour, and the wind now appears to blow from the 
north-east. What is the true direction of the wind 
and its velocity } 


450. A ship is sailing north-east with a velocity of 
10 miles per hour, and to a passenger on board the 
wind appears to blow from the north with a velocity 
of 10 V2 miles per hour. Find the true velocity of 
the wind. 

451. A fly-wheel revolves 1 2 times a second. What 
is the angular velocity of a point on the rim taken 
about the center } 

452. A broken casting flies along a concrete floor 
with initial velocity of 50 feet per second. The 
coefficient of friction being \ what will be its velocity 
after 3 seconds } 

One of the axioms for problems in Motion is, that 
P the force : W the weight =^ a : g. 
The force producmg motion : the total weight moved = the accel- 
eration produced by the force : the acceleration that gravity would 

For the above example the force producing motion (or in this 
case retardation) is W x § and 

W X J ; W = « : 32 

a = 16 feet per second per second 
After 3 seconds the velocity would be 

z' = 50 — 16 X 3 
= 2 feet per second. 

453. A locomotive that weighs 100 tons is increas- 
ing its speed at the rate of 100 feet a minute. What 
is the effective force acting on it .? 

454. An ice boat that weighs i 000 pounds is 
driven for 30 seconds from rest by a wind force of 
100 pounds. Find the velocity acquired and the 
distance passed over. 


MO TJON. I 29 

455. A 5 -pound curling iron is thrown along rough 
ice against a friction of one-fifth of its weight ; it 
comes to rest after going a distance of 40 feet. What 
must have been its velocity at the beginning ? 

456. The table of a box-machine weighs 50 pounds 
and is pulled back to its starting position, a distance of 
6 feet, by a falling weight of 20 pounds. What time, 
neglecting friction, will thus be used in return 
motion } 

457. A body whose mass is 108 pounds is placed 
on a smooth horizontal plane, and under the action 

of a certain force describes from rest a distance of \/ 
1 1^ feet in 5 seconds. What is the force acting.? 

458. Two bodies A and B, that weigh 50 pounds 
and 10 pounds, are connected by a string ; B is placed 
on a smooth table, and A hangs over the edge. 
When A has fallen 10 feet, what is the accumulated 
work of the bodies jointly, and what of them severally } 

459. A 500-volt electric motor imparts velocity to 
an 8-ton car so that at the end of 20 seconds it is 
moving on a level track at the rate of 10 miles an ^ 
hour ; the total efficiency of the motor and car is 60 
per cent. What amperes are necessary ? 

460. Show that to give a velocity of 20 miles an 
hour to a train requires the same energy as to lift it 
vertically through a height of 13.4 feet. 

461. What force must be exerted by an engine to 
move a train of weight 100 tons with 10 units of accel- 
eration, if frictional resistances are 5 pounds per ton "i 


'^ 462. A train that weighs 60 tons has a velocity of 
40 miles an hour at the time its power is shut off. 
If the resistance to motion is 10 pounds per ton, and 
no brakes are applied, how far will it have traveled 
when the velocity has reduced to 10 miles per hour? 

The retardation a will be found to be o,i6 feet per second per 
second ; the total loss in velocity is 44 feet per second. Then find 
the time, and lastly the space by observing that space = average 
velocity X time. 

463. A locomotive running on a level track brings 
a train of weight 1 20 tons to a speed of 30 miles an 
hour in 2 minutes. The resistance to motion of the 
train being uniform and equal to 8 pounds per ton, 
what will be the required horse-power at the draw-bar 
and what the distance from the starting point when 
the speed of 30 miles an hour is attained .? 

464. A freight train of 100 tons weight is going 
at the rate of 30 miles an hour when the steam is 
shut off and the brakes applied to the locomotive. 
Supposing the only friction is that at the locomotive, 
the weight of which is 20 tons, what is the coefficient 
of friction if the train stops after going 2 miles .? 

20 X /^ : 100 = ^ (which can be found from the data 
given in the problem) -.32. 

^ 465. A train of 100 tons, excluding the engine, 
runs up a i % grade with an acceleration of i foot per 
second. If the friction is 10 pounds per ton, find 
the pull on the drawbar between engine and train. 

Total force = force for acceleration -{- force for lifting 
+ force for friction. 

MOTION. 131 

466. A body is projected with 
a velocity of 20 feet per second 
down a plane whose inclination is 
25° ; the coefficient of friction be- Fig. 70. 

ing 0.4. Determine the space traversed in 2 seconds. 
P : W = ^ : ^ 
(.423 - -3625) X W : W = ^ : ^. 
The space traversed, 

^ = V/ -f- \ af. 

467. A body slides down a rough inclined plane 100 
feet long, the sine of whose angle of inclination is 0.6 ; 
the coefficient of friction is \. Find the velocity at 
the bottom. If projected up the plane with a velocity 
that just carries it to the top, find that velocity. / 
The forces acting down the plane 

= \V X sin a — W X cos a X ^. 

468. An electric car at the top of a hill becomes 
uncontrollable and '' runs wild " down a grade of i 
vertical to 20 horizontal a distance of \ mile. The 
resistance to friction being 20 pounds per ton and the 
total weight of car and passengers 50 tons, how fast 
will the car be going when it reaches the foot of the 
hill ? 

469. Two weights of 120 and 100 pounds are sus- 
pended by a fine thread passing over a fixed pulley 
without friction. What space will either of them pass 
over in the third second of their motion from rest } 

Observe that the force producing motion is in this case 20 pounds, 
and the total weight moved is 220 pounds. Then a =2.92 feet per 
second per second. 


470. A man who is just strong enough to Hft 150 
pounds can hft a barrel of flour of 200 pounds weight y 
when going down on an elevator. How fast is the 
velocity of elevator increasing per second ? 

471. A cord passing over a smooth pulley carries 
10 pounds at one end and 54 pounds at the other. 
What will be the velocity of the weight 5 seconds 
from rest, and what will be the tension in the cord } 

After computing the acceleration that the two weights would 
have, find the equivalent force, or tension, that would be required to 
cause said acceleration on the lo-pound weight, which is the one 
that is being moved. We have a == 27, and 
P : 10 = 27 : 32 
P, the tension = 8.4 pounds. 

472. Two strings pass over a smooth pulley ; on 
one side both strings are attached to a weight of 5 
pounds, on the other side one string is attached to a 
weight of 3 pounds, the other to one of 4 pounds. 
Find the tensions during motion. 

473. Weights of 5 pounds and 1 1 are connected 
by a thread ; the 1 1 -pound weight is placed on a smooth 
horizontal table, while the other hangs over the edge. 
If both are then allowed to mpve under the action of 
gravity, what is the tension of the thread } 

474. A lo-pound weight hangs over the edge of a 
table and pulls a 45-pound box along ; the coefficient 
of friction between the table and the box is 0.5. 
Find the acceleration and the tension in the string. 

\) 475. An engine draws a three-ton cage up a coal- 
pit shaft at a speed uniformly increasing at the rate 

MOTION, 133 

of 5 feet per second in each second. What is the 
tension in the rope ? 

476. A balloon is moving upward with a speed 
which is increasing at the rate of 4 feet per second per 
second. Find how much the weight of a body of 10 
pounds as tested by a spring balance on it, would 
differ from its weight under ordinary circumstances. 

477. An elevator of 300 pounds weight is being 
lowered down a coal shaft with a downward accelera- 
tion of 5 feet per second per second. Find the ten- 
sion in the rope. 

478. An elevator, starting from rest, has a down- 
ward acceleration of ^ ^ for i second, then moves 
uniformly for 2 seconds, then has an upward acceler- 
ation of ^ ^ until it comes to rest, {a) How far does 
it descend } {b) A person whose weight is 140 pounds 
experiences what pressure from the elevator during 
each of the three periods of its motion .? 

479. A weight of 10 pounds rests 6 feet from the 
edge of a smooth horizontal table that is 3 feet high. 
A string 7 feet long passes over a smooth pulley at 
the edge of the table and connects with a lo-pound 
weight. If this second weight is allowed to fall in 
what time will it cause the first weight to reach the 
edge of the table 1 

480. A body is projected with a velocity of 50 feet 
per second in a direction inclined 40° upward from 
the horizontal. Determine the magnitude and direc- 


tion of the velocity at the end of 2 seconds {g being 
taken equal to 32.15). 

Let ACE be the path of the projectile. The vertical velocity 
which the body possessed when it started from A carried it to the 
summit C of the trajectory, where it had zero vertical velocity, and 
when it reached E it would possess its initial velocity, which would 
be « X sin a. (In Prob. 480 a is 40°.) The constant horizontal 
velocity would be u x cos a. 

The vertical velocity acquired in falling from the highest point to 
the horizontal AE would be g x /, 

.*. ^ X / = ?/ X sin « 
and the time from A to highest point 

_ ?/ X sin a 

g . 
and the total time of flight 

2 ?/ X sin a 

The range AE 

= the horizontal component of veloc- 
ity X the time of flight 
2 u sin a 

ti X cos a X 
?/2 sin 2 a 


A g 

The above explanation and formulas will be of material assist- 
ance in solving the problems that follow. In all of these problems 
the resistances of the atmosphere are neglected. 

481. A bullet is fired with a velocity of i 000 feet 
per second. What must be the angle of inclination, 
in order that it may strike a point in the same horizon- 
tal plane, at a distance of 15 625 feet } 

482. From the top of a tower a stone is thrown up 
at an angle of 30°, and with a velocity of 288 feet per 
second ; the height of the tower is 160 feet. Find 
the time required for the stone to reach the ground, 
and the distance it will be from the tower. 



483. From a train moving at 60 miles per hour a 
stone is dropped ; the stone starts at a height of 8 
feet above the ground. Through what horizontal 
distance will the stone go while falling t 

484. A stone from a quarry blast has a velocity of 
200 feet per second, in a direction inclined at an 
angle of 60° to the horizontal plane. To what height 
will it rise, and how far away will it strike the ground .'* 

485. A bullet is fired with a velocity of which the 
horizontal and vertical components are 80 and 120 
feet per second respectively. Find the range and 
greatest height. 

486. The top of a fortification wall is 50 feet above 
the level of a city. From a man-of-war in the bay 
300 feet below the top of the wall and distant hori- 
zontally 3 000 feet, a projectile is fired with velocity 
of I 000 feet per second. The projectile just clears 
the wall. Where will it land inside the city } 



000 X / X sin rz — ^ gi- = 300 
I 000 X / X cos a = 3000 

Eliminate / and solve for a (a = S° 28'). 


Find the greatest height h, then d being known will enable one 
to find the time for the projectile to fall that height or to pass hori- 
zontally over the distance /. To / add half the range and thus find 
the distance from man-of-war to where the projectile will land inside 
the city. 

487. A ball is discharged with an initial velocity 
of I 100 feet per second. How many miles is the 
greatest possible range 1 

488. A cannon ball is fired directly from a hill 
that is on the coast and 900 feet high : find the time 
which elapses before it strikes the sea. 

489. A projectile is fired horizontally from the top 
of a hill 300 feet high to a ship at sea. Its initial 
velocity is 2 000 feet per second and its weight 500 
pounds. What will be its range, and what will be the 
energy of the blow that it strikes } 

490. What velocity must be given to a golf ball to 
enable it just to clear the top of a fence at 12 feet 
higher elevation and 100 yards distant, if the ball is 
struck upv/ards at an angle of 45° .? 

491. The explosive force of a shell is to be regulated 
by proper charging so that a required velocity can be 
attained. Find what velocity will be required for 
it just to clear a fortification wall the top of which 
is distant horizontally i mile and at elevation 300 feet 
above the gun. Angle of projection is 45°. 

492. A rifle projects its shot horizontally with a 
velocity of i 000 feet per second ; the shot strikes the 



OF THE *^ 



ground at a distance of i 000 yards. What is the 
height of the rifle above the ground ? 

493. What is the pres- 
sure exerted horizontally 
on the rails of an engine 
of 20 tons weight going 
round a level curve of 600 
yards radius at 30 miles 
an hour ? 

Centrifugal^ W 7^^ 
force g r 

Fig. 72. 
To derive the above formula : 

Let A and B be two positions of the engine. At A the velocity, 

which is 30 miles an hour, would be in the direction of tangent v, 

and at B the same velocity would be in direction of tangent v. In 

going from A to B the direction of the velocity has changed, and the • 

measure of this change is the centrifugal force. 

Its value depends upon the rate of change of motion. To find 

the rate, from A draw AD to represent the velocity of position B. 

Then CD will represent the velocity of B relative to A, and its 

value will be AB x -, as found from similar triangles ACD and 

ABO ; and AB = velocity (on the curve) X time, so that CD, the 


velocity of B relative to A = 7'/ x -, and the rate of change a = 

vt Y. - -=r t = -. Thus knowing the rate of change or acceleration 
r r 

a the centrifugal force c can be found. 

tr : W = a \ g 

~ g r 

494. A train of 60 tons weight is rounding a curve 
of radius one mile, with a velocity of 20 miles an 
hour. What is the horizontal pressure on the rails ? V 


495. A 24-ton engine is rounding a curve of 400 
yards radius ; the horizontal pressure on the rails is 
4.84 tons. What is the velocity of the engine ? 

496. The rim of a pulley has a mean radius of 20 
inches ; its section is 6 inches broad and \ inch thick. 
It revolves at 200 revolutions per minute. What is 
the centrifugal force per inch length of rim 1 

497. The mass of the bob of a conical pendulum is 
2 pounds, the length of the string is 3 feet, the angle 
of inclination to vertical is 45°. What is the tension } 

The three forces acting on the bob are : its weight downward, the 
tension in the string, and the centrifugal force outward. 

498. The mass of the bob is 20 pounds, the length 
of the string is 2 feet, the tension of the string is 
5007r^ pounds weight. How many revolutions per 
second is the pendulum making t 

499. If a conical pendulum be 10 feet long, the 
half angle of the cone 30°, and the mass of the bob 
12 pounds, find the tension of the thread and the 
time of one revolution. 

500. A ball is hung by a string in a passenger car 
which is rounding a curve of i 000 feet radius, with 
a velocity of 40 miles an hour. Find the inclination 
of the string to the vertical. 

501. A ball is hanging from the roof of a railroad 
car. How much will it be deflected from the vertical 
when the train is rounding a curve of 300 yards 
radius at speed of 45 miles an hour } 



502. Find the speed at which a simple Watt Gov- 
ernor runs when the arm makes an angle of 30° with 
the vertical. Length of arm from center of pin to 
center of ball, 18 inches. (Fig. y^.) 

Fig. 73. 

503. Find the speed of a cross-arm governor when 
the arms make an angle of 30° with the vertical. The 
length of the arms from center of pin to center of 
ball is 29 inches ; the points of suspension are 7 
inches apart. (Fig. 74.) 

504. The rotating balls on a centrifugal governor 
make 160 revolutions per minute ; the distance from 
the center of each ball to the center of the shaft is 
4.5 inches. The balls are of cast iron and 2\ inches 
in diameter. Find the centrifugal force of the gov- 

505. Find the speed in revolutions per minute of 
a cross-arm governor when the arms make an angle 
of 30° with the vertical, the length of the arms from 
center of pin to center of ball being 24 inches, and 
the points of suspension being 6 inches apart. 


506. Find the tension in each spoke of a six-spoked 
flywheel, 8 feet in diameter and weighing 1344 pounds 
when making 200 revolutions per minute assuming all 
its mass collected at its rim, and that by reason of 
cracks in the rim, the spokes have to bear the whole 
of the strain. 

507. The flywheel, which burst in the Cambria Steel 
Company's mill Jan. 21, 1904 killing three men and 
seriously injuring nine more, is reported to have 
weighed 50 tons, and to have been about 20 feet 
in diameter. If rim weighed 35 tons and its weight 
was acting at a mean diameter of 19 feet what centri- 
fugal force did each of the 8 spokes and its portion of 
the rim have to withstand when the wheel was 
"racing" at 150 revolutions per minute.-* 

508. A man standing on a street corner was injured 
by a horse-shoe that was thrown from the front wheel 
of a passing automobile. The rubber tire caught up 
the horse-shoe by a protruding nail and carried it 
around to the top of the wheel when it was thrown off 
with full force. Diameter of wheel was 30 inches, and 
speed of automobile was 20 miles an hour. What 
velocity would the rim of the wheel thus give to the 
horse-shoe t 

509. The automobile of the above problem was turn- 
ing the corner in a curve of 100 feet radius, the road 
being level. Therefore what two centrifugal forces 
were acting at the instant the i -pound horse-shoe was 
at the top of the wheel t What were their values t 


MOTION. 141 

510. If the horse-shoe was thrown with the full 
velocity of the rim and horizontally from the top of it 
how far away would it land on level ground ? 

511. As the man was standing 6 feet from the track 
of the automobile how far from him must have been 
the wheel when the horse-shoe was thrown off, pro- 
vided it was thrown tangentially to track ? 

512. A locomotive that weighs 35 tons runs at 40 
miles an hour on a level grade round a curve of 3 300 
feet radius (about 1° 44'). What centrifugal force is 
produced ? What should be the elevation of the outer 
rail for a standard gage track of 4 feet lof inches ? 

513. In the case of problem 512 the railroad con- 
template putting on a 60-ton locomotive and run- 
ning at maximum speed of 60 miles an hour. What 
lateral pressure will the spikes of the rails, if not 
changed, then have to withstand ? 

514. A stone weighing four ounces is whirled 
around the head 90 times a minute. If the sling is 
3 feet 6 inches long what will be the pull in it .? 

515. Given / the length of a simple pendulum, 

'^X - the time of an oscillation : show how to find 

^ g 
approximately the height of a mountain when a 

seconds pendulum, by being taken from sea level to its 

summit, loses n beats in 24 hours. If n = 15, what 

is the height of the mountain, the radius of the earth 

being 4 000 miles ? 



516. At sea-level a pendulum beats seconds. At 
the top of a mountain it beats %6 360 times in 24 
hours. What is the height of the mountain t 

517. A pendulum of length 156.556 inches oscil- 
lates in two seconds at London. What is the value 
of p;} 

518. An 800-pound shot is fired from an 81 -ton 
gun, with a muzzle velocity of i 400 per second : a 
steady resistance of 9 tons begins to act immediately 
after the explosion. How far will the gun move 1 

An impulsive force is a very large force that acts on a body for so 
short an interval of time that the body has practically no motion, but 
receives a change of momentum ; and this change of momentum 
measures the Impulse or effect produced by the Impulsive Force. 

In the above problem the impulsive force, or action on the shot 
to drive it forward, is equal to the reaction on the gun to drive it 

Action = reaction 
Momentum before = rhomentum after 
Momentum of gun backward = momentum of shot forw'ard 
W W 

g s 

and this simple formula, wath a knowledge of the piinciples of work, 
will solve many problems that involve questions of momentum of 
two or more bodies. 

For the above problem : "^ 

2W0 X I 400 = 81 X ?? 

V = Yt ^^^^ P^^" second, velocity of gun 

at beginning of its motion. 
s = average velocity X time. 

To find the time : Motion has been retarded by a force of 9 tons 
and the weight thus retarded is 8r tons. Find a the rate of retarda- 
tion ; then since the velocity of Yr equals a y^t, t can be found and 
lastly the space. 

MOTION. 143 

519. A 56-pound ball is projected with a velocity 

of I 000 feet per second from an 8-ton gun. What ^ 
is the maximum velocity of recoil of the gun ? 

520. A one-ounce bullet fired out of a 20-pound 
rifle pressed against a mass of 180 pounds, kicks the 
latter back with an initial velocity of 6 inches per 
second. Find the initial velocity of the bullet. 

521. A shell bursts into two pieces that weigh 12 
pounds and 20. The former continues on with a 
velocity of 700 feet per second, and the latter with (^ 
a velocity of 380 feet per second. What was the 
velocity of the shell when the explosion occurred ? 

522. A man weighing 160 pounds jumps with a 
velocity of \6\ feet per second into a boat weighing 
100 pounds. With what velocity will boat move away ? 

523. A freight train weighing 200 tons, and travel- 
ing 20 miles per hour, runs into a passenger-train of 
50 tons standing on the same track. Find the ve- 
locity at which the broken cars of the passenger train 
will be forced along the track, supposing e — \. 

Momentum before = momentum after. 
Now with this formula combine a second law, namely: 
The differences in velocities before x some constant = the differ- 
ence in velocities after. 

w w , w w ^ 

1. — u ■\ u — — V -\ V 

g g g g 

2. (7/ — u)e = V — t'' 

(u and ;/' are velocities before impact ; v and ?/' after impact.) 

Solving these equations, and 

_ Ww + W'// - ^W (u - 7/') 
^' ~ W -f w 

_ -Wu + Wu' + W<f (u - 1/) 

w + w 


524. A freight car of 20 tons weight is switched 
on to a siding with velocity of i 5 miles an hour, and 
collides with another car of 10 tons weight that is 
moving in the same direction at 5 miles an hour. If 
the coefficient of impact is J, find the velocities of 
the cars after they collide. 

525. A ball of mass 4 pounds and velocity 4 feet 
per second meets directly a ball of mass 5 pounds 
with opposite velocity of 2 feet per second ; e = \. 
Find the velocities after impact. 

526. An 8 -pound bowling ball going 12 feet a 
second overtakes and strikes directly a lo-pound ball 
going 6 feet a second. Find their velocities after 
striking when the coefficient of impact e is |. 

527. A body weighing 10 pounds, and moving 
at the rate of 1 5 feet a second, strikes another body 
B weighing 20 pounds, and moving at the rate of 10 
feet a second, in the direction at right angles to that 
of A's motion. The bodies are to be treated as points, 
and the impact is supposed to take place in the direc- 
tion of A's motion. Find the velocities and directions 
of the motions of the bodies after impact, the restitu- 
tion being perfect (coefficient of elasticity = i). 

Plot a figure to show the conditions. The- velocity of each in a 
direction perpendicular to a line through their centers is unchanged 
by the impact. 

Note that u of formula for problem 523 becomes tt X cos 45° ; 
u' becomes vf X cos 90°; v, v cos 180°; and v\ 7/ cos 45°. 

REVIEW. 145 


Many of the review problems that follow have 
been prepared from actual engineering conditions. 
They are classified somewhat according to their sub- 
ject matter, but are not given in graded order, nor 
with solutions, as they are intended especially for 
students who have already pursued a course in 
Mechanics, or for engineers in practice who may 
wish to "brush up a bit." 

528. One of the largest chimneys in America is that 
of the Clark Thread Co. at Newark, N.J. Its height 
is 335 feet, interior diameter 1 1 feet, outside diameter 
at base 28^ feet, at top 14 feet. Find the work done 
in raising the material from the ground to its place in 
the chimney. 

The volume may be determined by considering the whole cone 
and subtracting the top and the core. The average height to raise 
the material is found to be 1 19.07 feet ; the average weight of mate- 
rial is 130 pounds per cubic foot. 

529. By tests at the U.S. Naval Academy a concrete pile of 
conical shape 19 feet long and tapering from 6 inches at the' point to 
20 inches at the head, was driven \ inches by 2 blows from a 2 100- 
pound hammer falling 20 feet. The same hammer and fall by 
two blows drove a wooden pile also 19 feet long, but 9^ inches at the 
point and 11 inches at the head, a distance of 5^5^ inches. The re- 
port says : " This shows the comparative bearing power." 

What were the resistances {a) of the concrete pile 
{b) of the wooden } 



Fig. 75. Half-way Home. 

530. A 60-inch McCormick water turbine at the 
Talbot Mills in North Billerica, Mass. was tested by 
the author and engineering students in December, 

The quantity of water entering the turbine was, 
by measurements with a Price current meter, found to 
be 7 946 cubic feet per minute, with speed gate open 
40 per cent ; net fall given by difference in reading 
of gauges in foreway and raceway, 10.7 feet. How 
many horse-power was the water delivering to the 
turbine, that is, what was the " input .-^ " 

531. The power available for manufacturing pur- 
poses was measured by a friction brake. (See Fig. 
'j6}j Length of arm was 12.00 feet, revolutions of 
pulley 100 per minute, net reading on platform 
scales for the above test was 400 pounds. What 



horse-power was developed, that is, what was ''the 
output?" "The in-put" by problem 530 being 
161 horse-power, what was at that time the efficiency 
of the turbine with its set of bevel gears and 50 feet 
of horizontal shafting ? 

Fig. 76. 

532. The strap underneath the pulley in Fig. 76 is tightened 
by the nuts A and B on top of the lever. When the brake is in use 
one nut can be tightened by a pull of 50 pounds on a 3-foot wrench, 
the other by 25 pounds. 

Which nut should tighten harder and what will 
be the approximate tension in that end of the strap, 
the threads being 6 to the inch with mean diameter 
of i^ inches, and the hexagonal nut having a mean 
outside diameter of 2 J inches? Use 0.25 as the co- 
efficient of friction between the nut and its washer, 
and 0.15 for the threads. 

533. For a bridge like Fig. 77 with lower chord a 
para])()la 200 feet span and 30 feet rise what would be 
the stresses at the abutments and middle wlicn the 
vertical reaction at each end is 600 000 pounds ? 



Fig. 77. The Driving Park Bridge at Rochester, N. Y. 

534. The platform of a suspension foot-bridge 100 
feet span, 10 feet width, supports a load, including 
its own weight, of 150 pounds per square foot. The 
two suspension cables have a dip of 20 feet. Find 
the force acting on each cable close to the tower, and 
in the middle, assuming the cable to hang in a para- 
bolic curve. 

535. Find analytically the stress in the cable at 
a horizontal distance of 30 feet from the center. 

536. The weight of a fly-wheel is 8 000 pounds and 
the diameter, 20 feet ; diameter of axle, 14 inches ; 
coefficient of friction, 0.2. If the wheel is discon- 
nected from the engine when making 27 revolutions 
per minute, find how many revolutions it will make 
before it stops. 


REVIEW. 149 

537. Experiment shows that a weight can lift only 
three-quarters of its own weight by means of a rope 
over a single pulley, this being on account of the 
stiffness of the rope and the friction of the axis. 
Hence show that the mechanical advantage of four 
such pulleys arranged in two blocks is about 2.05. 

Fig. 78. Concrete dam falling into place, Niagara, N. Y. 

Height of trestle is 20 feet, of dam 50 feet, cross section of dam 
7 feet, 4 inches square, and base of trestle i foot wider on the water 
side. Tower was tipped by three heavy jacks until it fell, as shown 
in cut. (See Eng. Record of Nov. 18, 1905.) 

538. How many feet out of plumb would the top 
of tower move before starting to fall > 



539. What foot-pounds of energy did the falling 
tower possess at the instant it passed the level of the 
base ? 

540. A bolt I \ inches in diameter, 9 threads to an 
inch, head 2 inches outside diameter, is tightened up 
by a wrench 12 inches long, and pull on end of 50 
pounds. /A for threads is 0.2, for nut 0.3. Find the 
stress in the bolt. 

541. In a certain piece of street railway construction I obsei"ved 
that the bolts in the fish plates were being tightened unusually hard, 
and the workmen told me that such bolts often broke. By test we 
found that a pull of 170 pounds was being used at a distance of 3.75 w 
feet from center of nut. There were 9 threads to the inch, mean 
diameter 0.86 inches, average outside diameter of nut 1.4 inches, 
diameter of inside bearing 0.89 inches. The coefficient of friction 

for the threads was about 0.2 and for the nut 0.3. 

Find stress in bolt, and then the stress per 
square inch at root of thread which was 0.60 inches 
in diameter. 

542. The mean diameter of the threads of a J 
inch bolt is 0.45 inches, the slope of the thread .07, 
the mean circumference of nut 0.38 inches and the 
coefficient of friction 0.16. Find the tension in the 
bolt when tightened up by a force of 20 pounds on 
the end of a wrench 1 2 inches long. 

543. The floating cantilever crane shown in the 
illustration is hoisting a 90-ton turret off the U. S. 
Monitor "Florida." Distance from middle of crane 
to the legs is 45 feet; from legs to turret 15 feet. 
Vertical height of legs 60 feet, distance aj^art at top 



10 feet, at bottom 50 feet. Find the stresses for this 
case in the legs, and in the inclined members at 
center considering a joint at that point. 

544. The load of 90 tons is supported by two large 
steel blocks. Each block has four 4-foot steel sheaves 
thus using 8 strands of \\ inch cast-steel rope. The 

Fig- 79. 

speed of vertical travel is 50 feet per minute, {a) What 
will be the velocity of travel of the hoisting strand ? 
{b) What will be the stress per square inch in each 
rope of each block } 

545, Launching data for nine of our recent warships was given 
in papers read before the Society of Naval Architects and Marine \, 


Engineers at New York, November, 1904, and published in abstract 
by Engineering News, Dec, 22, 1904. 

For the " California," built by the Union Iron Works of San 
Francisco, and launched April 28, 1904 : 

Total moving weight, ship, cradle, etc. ... 6 062 tons 

Mean slope of upper end of ways i in 27.6 

Ship started '' very slowly." It moved the first foot in 1 1 seconds. 

From the above the initial coefficient of friction has 
been computed as .036. Check this result. 

546. Total travel of the ship, in preceding prob- 
lem, was 'j'^6 feet; total time, i minute 16 seconds ; 
Travel to point of maximum velocity, 320 feet ; time 
to maximum velocity, 41.5 seconds; amount of maxi- 
mum velocity, 22.1 feet per second. The weight of 
ship exclusive of cradle was 5 980 tons. Means for 
checking were rope stops (72 were broken of total 
strength 30 tons), anchors, and mud banks. Find the 
average resistance that the means of checking must 
have afforded. 

547. Consult the reference of problem 545 (Eng. 
News, Dec. 22, 1904), and copy Fig. 6 in note-book. 
Explain why the velocity and distance curves start 
horizontally from the origirf. In the acceleration 
curve what was the cause of the rise between 15 
and 30 seconds of elapsed time .? 

548. From curve of problem 547 what was the 
velocity at time of 41.5 seconds } The acceleration } 
What was the acceleration at 5 5 seconds } From that 
data compute the velocity for 5 5 seconds. 



549. Coal from a barge (Figs. 80 to 82) is hoisted 
to a steeple tower where it is run into a car and by the 
action of gravity alone the car goes down, a grade 
294 feet long in 24 seconds. It strikes a cross-bar, 
or " stopper " which is pushed back a distance of 30 
feet while the car empties and for an instant comes to 
rest. The weight of the car is 2 000 pounds and of 
the coal 4 000. If the car empties uniformly during 
the 30 feet, what is the average force of resistance 
that the cross-bar exerts ? 

Fig .82. 

550. The method of stopping the car of problem 549 may be 
understood by referring to Figs. 80 to 82. The car, going down 
the grade, picks up the cross-bar C, which is clamped to the wire 
cable AB. As the cross-bar is pushed along the cable moves, and the 


pulley E, around which the cable passes, as shown in the enlarged 
sketch E", goes from E, its initial position, to E', its final position. 
The travel of this pulley raises a triangular frame, that is partly filled 
with broken stone, from the position shown dotted to that shown by 
full lines in Eig. 82. The mass of stone is 5.5 feet X 4.8, as shown, 
and I ^ feet thick. The space is one-third voids ; weight of stone, 
150 pounds per cubic foot ; weight of wooden frame, i 000 pounds. 

Show that the force exerted parallel to the cable 
is, for the initial position with pulley at E, about 
220 pounds; for the final position E' about 1925 

551. When the cross-bar and wire cable of the pre- 
ceding problems move through a distance of 30 feet 
the travelling pulley E goes from E to E' as described. 
The diameter of pulley being i^ feet what will be the 
distance E E' t What would be the distance if pulley 
were 2 feet in diameter t 

552. One-fourth of the energy possessed by the 
mass of stone when in the final position E', Fig. 82, is 
lost in friction, and three-fourths of it is utilized to 
"kick" automatically the empty car up the incline 
from C back to its starting point A. If this energy is 
expended on the car in a return distance of 30 feet 
what will be the maximum velocity of the car as it 
starts back 1 

553. When the bridge of Fig. 83 carries a crowd 
of people making a load of 150 pounds per square foot 
what will be the reactions for the middle truss and 
the stresses in the inclined and horizontal members 
at the abutments ? 



A modern highway bridge over the main tracks of the Reading 
raihoad at Seventeenth and Indiana Streets, Philadelphia. Three 
Pratt trusses, with 24-feet clear roadways and two lo-foot sidewalks 
outside. Concrete abutments. The middle is 135 feet of inches 
from center to center of end pins, and 22 feet from ienter to center 
of chords. Equal panels. 


554. A bridge of the type shown in Fig. 83 is sued 
for a double-track raih'oad. Length of bridge is 150 
feet, there are 6 equal panels, and height of trusses 
is 25 feet. With loading of Fig. 85 and the second 
driver of forward locomotive placed at the first panel 
point from abutment, what will be the stresses in the 
inclined member and the first panel of lower chord of 
the truss which is carrying two-thirds of the loadings .? 

555. Span 1 1 of the Benwood Bridge on the Balti- 
more and Ohio Railroad is 347 feet long. It was 
reconstructed in 1904 and designed to carry the 
heavy loading shown in Fig. 85. What would be the 
reactions for a train half-way across the bridge 1 

556. With the forward truck just going off the 
bridge, what will be the reactions } 

hfitl. The locomotive of the Empire State Express 
has four drivers and a total weight of 124 000 
pounds ; the weight on the drivers is 84 000 pounds ; 
the coefficient of friction between wheels and rails is 
0.18. Find the total weight of itself and train which 
it can draw up a grade of i in 100, if the resistance 
to motion is 1 2 pounds per ton. 

558. From the following data given by a General 
Superintendent, determine what revolutions per minute 
the locomotive drivers were making : 

" The driver wheels are 42 inches in diameter, each pair being 
geared to a 200 HP, 625 volt-motor, with ratio of gearing 8i to 19, 
providing for a total tractive effort at full working load on 8 motors 
of 70 000 lbs. and at starting of 80 000 lbs., assuming 25% tractive 
coefficient, giving a nominal rating of i 600 HP. The free running 
speed of these locomotives is about 20 miles per hour." 







i c 










-Tc- 25000 


-*- 60 000 
-■^- 50000 
-y- 50000 

--51- 50000 


— X- 32 500 


-^- 32500 


-^- 32500 
-)c- 32 000 

-^- 26 000 


-X- 60 000 

-4- 50000 

-^- 50000 

-*- 50000 

-^- 32 500 
-"k- 32500 

-^^- 32 500 

-r- -S- 

Dw«. AxU 

in. ft. Load 

in lb: 

C-r" ^*- 32000 

Q 2 

-!" 00 O 

o a {H 

K 2 W 

P o ^ 

h « 

C » oi 
o o rt 

Fig. 84 shows a consolidated locomotive typical of modern de- 
sign and development ; Fig. 85, two such locomotives and their 
train load as used in computations of modern railroad bridges. 


559. Also from the foregoing data determine what 
revolutions the motors were making. What amperes 
were supplied to the 8 motors ? 

560. An enormous freight locomotive — a Mallet 
duplex compound — designed as a " mountain helper " 
was put in service on the Baltimore and Ohio Railroad 
in January, 1905. This locomotive has drawn 36 steel 
cars weighing 702 tons, and i 66^ tons of lading, up 
a 1% grade, and with an average speed of 10^ miles 
an hour ; weight of locomotive with tender and an 
average amount of coal and water is 225 tons. What 
horse-power without friction was developed for haul- 
ing the above total load t 

561. What per cent of the work done in the preced- 
ing problem would be paying work t 

562. The draw-bar pull of this Mallet Compound 
has been found to be 74 000 pounds. When running 
with conditions according to problem 560 what fric- 
tional resistances would exist .^ 

563. At the testing-plant of the Pennsylvania Railroad at the 
St. Louis Exhibition in 1904 a freight locomotive of type two-cylinder 
cross-compound consolidation (2-8-0), size 23 &35 X 32, made by 
the American Locomotive Company for the Michigan Central Rail- 
road, gave the following data : Driving wheels 5 feet 3 inches in 
diameter, total weight 189000 pounds, on drivers 164500; maxi- 
mum tractive effort, sand being used and locomotive acting as a 
compound, was 31 838 pounds. 

According to the above data what would be the 
coefficient of friction between drivers and rails .•* 



564. With speed of 15.01 miles per hour, 80.18 
revolutions per minute, piston speed of 428 feet per 
minute, the indicated horse-power was 734.9 ; dyna- 
mometer horse-power 675.7. Find the draw-bar pull 
and the per cent of indicated horse-power that was 
lost in friction. 

565. A car is supported on four 36-inch wheels 
with 4-inch axles and coefficient of friction 0.05. 
What traction will be required to move the car on 
a level track with a total weight of 20 ton on the 
axles .-* What energy will be lost in friction per 
minute with the car moving 30 miles an hour } 


S S ^ 


Fig. 86. 

A block of maple wood 8 inches long and 2x3 inches in cross- 
section is being tested in a 60 000-pound Olsen testing machine. 


The load on the test piece at the time of failure is 45 40c pounds. 
The plane of fracture is shown by the white line on the test-peice, 
Fig. 86. This plane makes an angle of 23° with the horizontal. 

566. What would be the pressure in the direction 
of the plane at 'the time of breakage ? Find the 
number of square inches thus resisting this pressure 
and then the stress per square inch — which is known 
in applied mechanics as the Shear. 

567. A horse is pulling a 300-pound cake of ice up 
a plank run which makes an angle of 40° with the 
horizontal. There are two single pulleys which have 
efficiencies of 80% each. Coefficient of friction on 
plank run 0.05. What pull must horse exert } 

568. A ring of weight W is free to slide on a 
^ smooth circular wire that stands in a 
\ \ vertical plane. A string attached to 
^->^lc the ring passes over a smooth pin at 

A the highest point of the circle and 
w sustains a weight P. Determine the 
Fig. 87. position of equilibrium. 

569. CD is a vertical wall. A is a point of sup- 
port 12 feet from the wall. ED is a uniform bar 32 
feet long resting on A and against the wall CD. All 
the surfaces are smooth. Find the position of equi- 
librium of the bar. 

570. Fig. 88 shows a Carson-Lidgerwood cable way in use at 
Hartford, Conn., lowering a 12-foot length of 36-inch cast-iron pipe. 
The pipe is part of an intercepting sewer that passes under a 
river by means of an inverted siphon. Weight of pipe is 3 tons, 
span of cableway 300 feet, and the design of cableway is such that, 




Fig. 88. 

as customary, the sag by a full load in the middle is allowed to be 
^Q of the span. 

The practical method used by manufacturers of trench machinery 
for computing the total stress in cableways of this sort is to con- 
sider the condition of maximum loading, namely with the load in 
the middle of the span, and for that condition to divide half the 

span by double the sag [ ^^ — ) • This gives a factor which mul- 





tiplied by the total load (in this case 3 tons + weight of cable) gives 
the required stress in the main cable- 

How does the result obtained by the above prac- 
tical method check with result obtained by the 
method of parallelogram of forces, considering the 
total load of 3 tons -f- weight of cable as acting at 
middle of span ? 

571. Each tower for the above cableway is 30 feet high, and con- 
sists of a vertical timber frame, or bent, that is formed by two 8 X 
10 inch legs spread 9 feet apart at the bottom and 2 feet at the top. 
The main cable passes over an iron saddle at the top of the tower 
and is fastened to a " dead-man " anchorage that is buried 60 feet 
from the foot of the tower. The timber frame is kept vertical by 
steel guys made fast at top of tower and to the same " dead man." 

What horizontal . stress do these guys have to 
provide for when the main cable is free to slip on the 
saddle ? What vertical load do they add to the tower } 

572. The pull on the hoisting ropes causes an 
additional stress that would be, for this case, equiva- 
lent to a vertical load of about 3 tons on one tower. 
Add the three vertical stresses due to main cable, 
steel guys, and hoisting ropes and then find the stress 
in each leg of the tower. 

573. A 12-inch Pelton water-motor of 3 horse- 
power is tested by a friction brake that encircles 
three fourths of a 4-inch pulley on the motor and has 
a lever arm that extends 22 inches from center of 
pulley to scales. The scales read 5 pounds when 
motor is making i 1 50 revolutions per minute. What 
horse-power is being developed t 



574. A 6-inch water-pipe that is 600 feet long is 
deHvering 750 gallons of water per minute ; the water 
is shut ojf by uniformly closing a 6-inch vaK'e in 3 
seconds of time. How much will the static pressure 
near the valve be increased ? 

|< 69 «. H ' 

Fig. 89. 

575. A water-works tank is on a trestle which 
stands on uneven ground as shown in diagram. 
The tank weighs 30 000 pounds. A strong wind 
gives a pressure of 40 pounds per square foot. Find 
the stress in the plane of the legs DB (a) when the 
tank is empty ; {b) when the tank is full, {c) Find 
how much water will prevent the tank from over- 


The wind acting on the curved surface of the tank causes a 
pressure that may be taken as 0.6 of that on a vertical section 
through the middle of the tank. 



576. Name two advantages of the hemispherical 
(or similar) bottom over a flat bottom as represented 
in Fig. 89. A tank 20 feet high on the sides and 20 
feet in diameter with hemispherical bottom will hold 

how many gallons of 
wat er ? Wh at wi 1 1 b e 
the wind pressure if 
taken as in preceding 
problem ? 

577. A clause in pro- 
posed specifications for 
water valves requires that 
'' a valve shall stand with- 
out injury a pull of 175 
pounds on a wrench that 
is in lenglh \\ times the 
radius of the wheel." In 
considering these specifica- 
tions an engineer and in- 
spector asks if this test 
unduly strains a valve ? The 
following analysis can be 
made relative lo an S-inch 
outside screw-and-yoke 
valve : Diameter of hand 
wheel is 14 inches, diam- 
eter of spindle, if inches, 4 
threads to the inch (mean 
bearing diameter i^ inches). 
Valve seats taper from 4 inches to 2 inches in diameter of valve 
— 8 inches. Bearing of hand wheel has a mean diameter of 2 inches. 
The coefficient of friction for the bearing of hand wheel, the threads, 
and the face of valve against its seat, may be taken as 0.15. 

Find the stress in the spindle caused by the pull 
of 175 pounds as indicated above. 

Fig. 90. A modern form of water-tank. 
(Erected at St. Elmo, 111., by the Chicago 
Bridge and Iron Works.) 



To find this stress it is only necessary to consider 
conditions that affect the friction of the hand wheel. 
The resistances at the valve have no affect on the 
stress in the spindle which in any case is subjected to 
that part of the stress that is transmitted by the hand 
wheel. To compute this stress consider one revolu- 
tion of the hand wheels. 

Work = Work +Work + Work 

of pull of lifting on threads on bearing 

Substitute and find the unknown term W, the stress. 

Fig. 9'. 

Fig. 9a. 



578. Find the pressure against the seat of the valve 

(no water pressure being considered) when a pull of \/ 
i^j^ pounds is applied as in problem 577. 

579. When a water pressure of 100 pounds per 
square inch acts on one side of the 8 inch valve in 
problem 578 and the test of 175 pounds pull is 1/ 
applied what normal pressure exists on each side of 
the valve ? 


A length of water pipe 

Spinot end 

2.25 in. -*J 

8 Half Sectiqn of an 18-incli water pipe 

Fig. 93. 

Dimensions and weights for cast-iron water pipe are given in the 
*' Standard Specifications for Cast-iron Pipe and Special Castings," 
issued Sept. 10, 1902, by the New England Water Works Asso- 

REVIEW. 167 

580. The dimensions for an 18-inch pipe of class D 
designed for a hydrostatic pressure of 300 pounds per / 
square inch are represented in Fig. 93. Find the 
weight of portion E D and then the total weight of 
the whole length of pipe. 

581. A roof has triangular trusses 12 feet apart. 
Weight of roof covering and snow equals 30 pounds 
per square foot, and the floor gives a load equivalent 
to 20 000 pounds concentrated at the foot of a vertical 
rod at the center of the truss ; length of truss is 
40 feet, height 10 feet. Find the stresses in rafters 
and tie-rod. 

582. A triangular jib-crane ABC carries at A 
60 000 pounds, the line of action being parallel to BC 
which is vertical. AB = 10 feet, BC 8 feet, AC 
1 1 feet. Find the amount and kind of stresses acting 
in AB and AC. 

583. A derrick with mast 40 feet long and boom 
55 feet long, set at 60° from the horizontal, is lower- 
ing into water a wrought-iron pipe 12 feet long, 60 
inches internal diameter, 66 inches external diameter. v 
Density of wrought-iron is J.Z. Find the stresses 

in boom and tackle when the pipe is in air, and also 
when it is in water. 

584. Is the retaining wall shown in Fig. 94 safe 
against overturning by the earth pressure acting as 
represented.? Does the resultant pressure between 
the weight of the masonry, taken at 170 pounds per 



cubic foot, and the earth pressure cut the base *' within 
the middle third ?" 

I i. 

Fi2. 94- 

585. Fig. 95 shows a retaining wall of masonry as 
built at Northfield, Vt. As in the preceding prob- 
lem, is the wall safe against overturning ? Where 
does the resultant cut the base ? 

586. The waste gate of the canal for the Nashua 
Manufacturing Company at Nashua, N.H., is about 
7 J feet high and 4^ feet wide. When this gate is 
closed there is usually a head of 10 feet of water on 
its center. The coefficient of friction of this wooden 

gate against an ordinary metal 
seat is taken as 0.40 and the 
weight of the gate is i 000 
pounds. What force in tons 
\W11 be required to lift it } 

587. How wide on top should 
be the dam shown in Fig. 96 to 
withstand the reservoir pres- 




/ Concrete Dam 


liO lbs. per cu. ft. 


Fig. 96. 

sure with factor of safety of 8 ? 



588. Two Indians wanted to divide a birch log, 
that was 30 feet long and tapering from 8 inches in 
diameter to 1 2, so that each would have one-half. A 
school teacher told them to balance it and saw it open 
at that point. At what point should it be cut } 

589. A 20 pound shot is fired from a 2 000-pound 
gun of length 10 feet ; the muzzle velocity of the shot 
being i 200 feet per second how far will the gun \J 
recoil up an incline rising i vertically to 15 on the 
slope? How long will it take the shot to travel 
through the gun } 

Fig. 97. 

590. The engine and geared drum shown in the illustration are 
used for hoisting ore to the top of a blast furnace. The engine cyl- 



inder is 12 inches in diameter, makes a 15-inch stroke, and 300 revo- 
lutions per minute. The mean effective pressure of the steam being 
100 pounds per square inch, what horse-power is developed ? The 
ratio of gears is 5.6, to i ; the diameter of drum, \\ feet. The effi- 
ciency of engine, geared drum and rest of mechanism is about 85 
per cent. 

Therefore, under the above conditions, what force 
will the engine give to the cable for drawing the 
loaded " skip-car " up the incline to the top of the 
blast-furnace ? 

591. The drum of a hoisting engine is 4 feet in 
diameter. The angle between the engine crank and 
connecting rod is 60°. Length of crank i foot, con- 
necting rod 5 feet. Steam pressure on the piston 
100 000 pounds which just balances a load W that is 
being hoisted. Determine the load W, the compres- 
sion in the connecting rod and the side pressure 
against the cross-head guide. 

592. Sixteen horse-power is to be transmitted by a 
belt which embraces | of the circumference of a 
20-inch pulley that makes 1 20 revolutions per minute ; / 
coefficient of friction is 0.35. Find {a) the tension in 

the two sides of the belt when slipping is just pre- 
vented and {b) the width of belt required, thickness 
being f inches, and working stress 300 pounds per 
square inch of section. 

593. Find the width of a belt necessary to transmit 
10 horse-power to a pulley 12 inches in diameter, so 
that the greatest tension may not exceed 40 pounds 

REVIEW. 171 

per inch of width when the pulley makes i 500 revo- 
lutions per minute, and the coefficient of friction is 

594. A test was made, Aug. 15, 1905, of the new 
steam plant of the Wolff Milling Company at New 
Haven, Mo. The engine had a high-pressure cylin- ^ 
der 1 2 inches in diameter ; low pressure, 24 inches ; 
length of stroke, 36 inches. The revolutions were 

77.1 per minute; the indicated horse-power of high- 
pressure cylinder, 85.74, low pressure, 66.19. What 
were the mean effective pressures in the two cyl- 
inders } 

595. The engine and boiler test of problem 594 
was continued 10 hours. During that time 177.72 
barrels of flour (each weighing 196 pounds) had been 
made, and 3 685 pounds of coal had been burned 
in the boilers ; cost of coal per ton of 2 000 pounds 
was $2.90. Find the cost of coal for each barrel of 
flour made, and the pounds of coal burned per hour 
per indicated horse-power. 

596. A Columbus Gas Engine tested as shown by 
Fig. 98, Oct. 21, 1905, gave the following data : Revo- 
lutions during 15 minutes 3 688, explosions i 516, net 1/ 
load on brake arm 30 pounds, length of arm 5 feet 
3.024 inches. Four indicator cards taken during the 
test gave average areas 0.69 square inches, length 
3.00 inches. Stiffness of spring that was used 300. 



The diameter of engine cylinder is 'j.'j^ inches and 
the length of stroke 1 1 inches. Find the efficiency. 

In computing the horse -power of a gas engine by indicator cards 
the number of explosions corresponds to the number of revolutions 
for an ordinary engine. 

Fig, 98. 

597. What would be the indicated horse-power of 
the gas engine, shown on page 17, and which has a 
piston 12 inches in diameter and a crank 8 inches 
long ? The engine works at 1 50 revolutions a minute, 
there is an explosion every 2 revolutions, and the 
mean effective pressure in the cylinder is 62 pounds 
per square inch. 

598. The speed of the governor shaft AB is 500 
revolutions per minute. The lo-pound ball is to 
be replaced by two 20-pound balls that revolve in 





\ \ 


\ \ 

\ \ 



Fig. 99. 


the lo-pound 

planes distant i foot and 4 feet 
from the plane of the lo-pound 
ball. Take the distance AC as 
7 inches and find R and Rj, the 
distances at which the 20-pound 
balls will revolve from the gov- 
ernor shaft when their centrifugal 
forces have the same moment 
about the speed controller at A 
one alone. 

599. In the preceding problem could the distance 
AC be changed and still have the moments of the two 
20-pound balls balance the 10.^ If so what is one 
such distance "^ 

600. A rope manufacturer's catalogue states : 

" The breaking strength of rope may be taken as 7 000 x diam- 
eter squared. For a constant transmission the best results are 
obtained when the tension on the driving side of the rope is not 
more than ^^^ of the breaking strength ; and the tension on the driv- 
ing side is usually twice the tension on the slack side." 

Find the horse-power of a 2-inch diameter rope, of 

weight per foot 0.34 X diameter squared, that runs 

at 3 000 feet per minute when centrifugal force is 


Observe that the tension on slack side = centrifugal force of 
belt -f ^ of difference in tension. 






Senior Mechanical and Mining Engineers. 

March, 1905. 

1. {a) Define five different units of force, {b) A 
balloon is ascending with a speed which is increasing 
at the rate of 4 feet per second in each second. Find 
the apparent weight of 10 pounds weighed by a spring 
balance in the balloon. 

2. A weight of 20 pounds rests 7 feet from the 
edge of a smooth horizontal table 4 feet high. A string 
8 feet long passes over a smooth pulley at edge of the 
table and connects with a lo-pound weight. If this 
second weight is allowed to fall, in what time will the 
first weight reach the edge of the table. 

3. {a) A cord passing over a smooth pulley carries 
10 pounds at one end and 54 at the other ; what will 
be the tension in the cord > {b) A shopkeeper uses 
a balance with arms in ratio of 5 to 6. He weighs 
out from alternate pans what appears to be 60 pounds. 
How much does he gain or lose } 

4. {a) Define a force couple. Show that a force 
couple cannot be replaced by a single force, {b) Show 

* Preparatory Studies : About 20 weeks of Mechanics in a three hour a week 
course, and the present course of 10 weeks with three hours a week preparation. 


how to find the resultant of any number of non-con- 
current forces acting on a rigid body. 

5. {a) Find the force of attraction of a homoge- 
neous sphere on a particle within the sphere, {b) The 
mass of the sun is 300 000 times the mass of the 
earth, and its radius is 100 times the radius of the 
earth. How far will a stone fall from rest in one 
second at surface of sun t 

6. {a) A uniform rod 8 feet long, weighing 18 
pounds, is fastened at one end to a vertical wall by a 
smooth hinge. It is kept horizontal by a string 10 
feet long, attached to its free end and to a point in 
the wall. Find the tension in the string and the 
pressure on the hinge, {b) A uniform rod AB, 
20 inches long weighing 20 pounds, rests horizontally 
upon two pegs whose distance apart is 8 inches. 
How must the rod be placed so that the pressure on 
the pegs may be equal when weights of 40 and 60 
pounds are suspended from A and B, respectively } 

7. Find by the principle of virtual work the con- 
dition of equilibrium for a differential screw consider- 
ing friction. 

8. A uniform ladder 70 feet long is equally inclined 
to a vertical wall and the horizontal ground. A man 
weighing 224 pounds ascends the ladder, which weighs 
448 pounds. How far up the ladder can the man 
ascend before it slips if the coefficient of friction for 
the wall is \ and for the ground \ ? 


9. Find the work lost by a shaft with a truncated 
pivot, bearing an end thrust. 

10. A belt passing around a drum has an angle of 
contact a and a coefficient of friction /i. Find the 
horse-power which can be transmitted. 

11. Two rough inclined planes are placed end to 
end. A body of lOO pounds rests on one of the 
planes, which has an inclination of 6o°. A string 
attached to this body passes over a smooth pulley at 
the apex of the planes and holds another body on the 
second plane of inclination, 30°. If coefficient of 
friction for each plane is ^, find the weight of second 
body to just hold the first from sliding down the plane. 

sine 30° = .5 cosine 30° = .86 



Examination at Mid-Year, Feb. 6, 1905.* 

Answer any eight questions. 

1. In tests of cast-iron fly wheels {Eng. Nezvs, Dec. 
15, 1904) record is given of one as follows : Diameter 
of wheel 4 feet, stress in each arm due to the centri- 
fugal force of its portion of the rim 1680 pounds, 
weight of same portion of rim yh pounds. Find 
bursting speed in miles per hour. 

2. A leather belt treated with dressing has coeffi- 
cient of friction on an iron pulley of 0.3. The belt 

* Preparatory studies : Physics lectures one year, laboratory one-half year, 
mechanism one-half year, and present course of half year with three class hours per 
week and two hours of preparation for each. 


encircles 200° of a pulley 10 feet in diameter. When 
running at 140 revolutions per minute the belt must 
transmit 300 horse-power. How wide should belt 
be if it is designed to stand 100 pounds per inch of" 
width t 

3. A wall derrick has a vertical post 9 feet high, 
at top a horizontal member 15 feet long, and 3 feet 
back from the load of 10 tons at outward end is a 
brace 13 feet long connecting with the vertical post 
at a point 4 feet up from ground. Find stresses. 

4. A highway bridge 80 feet long has supports 
2 feet from one end and 10 feet from the other. 
Uniform load on bridge is 300 pounds per linear 
foot. A road roller of 10 tons weight is half-way 
across ; what load is then on each abutment } 

5. A large type of locomotive recently put in ser- 
vice on the N. Y. C. & H. R. R. has developed ap- 
proximately 2 000 horse-power. How heavy a train 
could this locomotive draw, at speed of 40 miles an 
hour, up a 2 per cent grade — {a) without wind or 
frictional resistances, {b) with resistances of 20 pounds 
per ton acting t 

6. A train of 400 tons starts from a station and 'on 
a level track attains a speed of 40 miles an hour in 
one minute. Neglecting resistances, what would be 
the draw-bar pull } 

7. A stiff-leg steel derrick with vertical mast 
55 feet high, boom 85 feet long, set with tackle 


40 feet long is raising two boilers of 50 tons total 
weight. Find stresses in boom and tackle and in 
back stay which makes an angle of 30° with vertical. 
'If mast be made of two members joined at top and 20 
feet apart at bottom what stresses must they sustain ? 

8. What would be the total horse-power of pumps 
working 12 hours per day to supply the City of Med- 
ford, 21 600 population, with 100 gallons of water 
per day (for each person) and forced against 60 pounds 
pressure (equals a height of 138 feet) t The efficiency 
of engines and pumps is to be 80 per cent. 

9. A shell can be fired with velocity of 2 000 feet 
per second ; neglecting resistances, how near to shore 
can a man-of-war be in order to have its shells just 
clear a fortification wall 500 feet above sea level, angle 
of projection being 30° t 

10. Derive the formula for centrifugal force. The 
20th Century express attains a speed of 60 miles per 
hour. When rounding a curve of 4 000 feet radius 
how much should the outer rail be elevated to avoid 
lateral pressure } (Center to center of rails is 4 feet 
I of inches.) 

11. Define acceleration, work, moment of a force, 
coefficient of friction. Find the least force necessary 
to pull a packing case of 300 pounds weight along a 
horizontal floor. Coefficient of friction 0.58. 

Total number of problems taken during the half-year has been 
about 195. 



Examination at Mid-Year, Feb. i, 1906* 
Division a answer any 8 questions. Division ^ answer No. 11 and 
7 others. 

1. In a direct-acting steam engine the piston pres- 
sure is 22 500 pounds ; the connecting-rod makes a 
maximum angle of i 5° with the hne of action of the 
piston. Find the pressure on the guides. 

2. An iron wedge having faces of equal taper that 
make an angle of 10° is being forced under an iron 
column which is supporting a load of 5 tons. The 
coefficient of friction for the iron surfaces is 0.18. 
What force is needed to push the wedge forward } 

3. An electric car that is filled with passengers 
and weighs 25 tons goes up a grade of i in 100 at 
the speed of 10 miles an hour. The total resistances 
to traction are 30 pounds per ton. What horse-power 
must be supplied when the efficiency of the mechan- 
ism is 60 per cent } For an electro motive force of 
500 volts what amperes would be necessary.^ 

4. A shaper head that weighs 500 pounds makes 
its forward stroke of 12 inches in 6 seconds. The 
resistances of cutting and of machinery are equivalent 
to a coefficient of friction of 0.5. At what rate is 
work being done .^ 

* Preparatory studies same as for examination of 1905 and given at bottom of 
page i?6. 


5. Coal is hoisted from a barge to a tower where it 
is run into a car that goes down a grade 294 feet long 
in 24 seconds. It strikes a cross-bar or '' stopper " 
which is pushed back a distance of 30 feet while the 
car empties and for an instant comes to rest. The 
weight of the car is 2 000 pounds and of the coal 
4 000. If the car empties uniformly during the 30 
feet what is the average force of resistance that the 
cross-bar exerts } 

6. A highway bridge of span 48 feet, width 40 feet, 
has two queen-post trusses of depth 9.2 feet ; and 
each truss is divided by two posts into three equal 
parts. The bridge is crowded with people making a 
load of 1 50 pounds per square foot, and also an elec- 
tric car one-third the way across the bridge causes an 
additional load equivalent to a concentrated load of 
20 tons. Find the stresses in chords and posts. 

7. The head plate of a Buckeye 
engine is to be hoisted by a con- 
tinuous rope that passes through eye 
bolts that are 5 feet apart, and 
through a chain-hoist hook that is 
3 feet above the plane of the eye 
bolts. The rope is free to slip, and 
the plate weighs 500 pounds. Find 
the total pull that tends to break the eye bolts. 

8. The center of a steel crank-pin that weighs 
16 pounds is 12 inches from the center of the engine 


shaft. The shaft makes 190 revolutions per minute. 
Find the centrifugal force caused by the pin. 

9. The San Mateo Dam in California was designed 
for a height of 170 feet, width at top 25 feet, at base 
176 feet, with a uniform batter on the water side 4 to 
I, and on the back side near the top 2 J to i, then a 
curve of radius 258 feet to near the bottom where the 
batter is i to i. The material throughout is concrete 
of weight 1 50 pounds per cubic foot. Compute 
approximately the factor of safety of such a section 
against overturning. 

10. Define moment of a force and illustrate by an 
example. Also define and illustrate "resolve parallel 
and perpendicular to plane," a couple and three other 
important terms or equations of Mechanics. Show 
how to find the least force necessary to pull a box 
along a horizontal floor. 

11. ~U -\ « = — V -\ V 

g g g S 

Z' — 7<' = e {1/ —u) 

Tell what the above formulas mean. 

2 u sin a " 
" Horizontal range = u cos a X * 

How obtained ? 

A bullet is fired with a velocity of i 000 feet per 
second. What must be the angle of inclination in 
order that it may strike a point in the same horizontal 
plane at a distance of 15 625 feet .-* 

Total number of problems taken during the half-year, about 180. 



First Course in Mechanics 

1. Find the components of a force of 500 pounds along 
lines inclined to it by (^) 0° ; {b) 24°; {c) 30°. Algebrai- 
cally only. 

2. Find the moment of (300 pounds 48° (- 4, 6)) 
about {a) (4, 6) ; {b) (o, o) ;(.;(- 4, 6) ; {d) (3, - 7). 
Algebraically only. 

3. A uniform body in the shape of an isosceles triangle 
with base of 60 feet and altitude of 20 feet weighs 200 
pounds. It is supported at points in its base 20 feet 
and 60 feet respectively from the left end. Forces of 20 
pounds and 40 pounds act vertically upward and down- 
ward respectively from points bisecting the left and right 
sloping sides respectively. 

Determine the pressures upon the supports. 

4. A rectangle, 10 inches by 8 inches, has one corner 
at the origin, two sides coincident with (9Jf and (9F, and 
a corner at (10, 8). Two forces, of 20 pounds each, act 
one along the upper edge of it toward the right, the other 
along the lower edge toward the left. Two more forces, 
of 40 pounds each, act respectively upward along left 
edge and downward along the right edge. 

{a) Is the body subject either to translation or rotation ? 
(J?) If any further forces be needed to cause equilibrium 
state the value of the simplest system that will do it. 


I S3 

5. Find the center of gravity of a plane figure of five 
sides with corners at (o, o), (5, o), (4, 5), (4, 3), (14, 3), 
(8, o). 

Solve both algebraically, and graphically, using in the 
latter case the general string polygon method. 

6. A sphere weighing 1000 pounds rests between two 
smooth planes which are inclined to each other by 30°, the 
less steep of which is inclined 10° to the horizontal. 

Determine the pressure on each plane algebraically. 

7. A plane rectangular frame 60 feet high and 10 feet 
wide stands on two supports, one at each of the lower 
corners. A horizontal wind force of 4000 pounds is 
applied at 30 feet from the ground and a load of 6000 
pounds rests at the middle of the top. 

If the thrust of the wind be assumed to be resisted 
equally by the supports, determine the remaining forces at 
the supports. 

8. Determine graphically stresses in all of the bars of 
the given truss. Show numerical results upon large free- 
hand sketch of truss. 

9. Determine algebraically stresses in Q, 7?, and S of 
truss of last question without finding other stresses. 

10. Determine the reactions {H^, If., V^, V.) at the 
supports of the given three-hinged arch. 


sin a 

cos a 

tan a 


Sin a 

cos a 



1. 00 













1. 16 
























1. 00 


I. CO 





In preparing this new edition two opposing sugges- 
tions have been offered to me : one that I should give 
all the answers to the problems, the other that I 
should give none. I have taken the middle ground, 
and am giving about half of the answers, believing 
that this method will serve both for engineers in prac- 
tice and others who wish to know that their results 
are correct, and for college classes where it is often 
preferred that some of the answers be omitted lest 
the student place too much dependence on them. It 
is generally agreed, I think, that with students the 
advice frequently given by Professor Merriman in his 
excellent text books should be emphasized, namely : 
that the answers are not the main part of a problem. 
In fact, the student is urged not to consult the answer 
at the beginning of a problem, and then aim merely 
to get that numerical result. First an understanding 
of the problem should be obtained, then a diagram 
representing the data should be drawn, and an esti- 
mate of the answer based on experience should be 

Furthermore, in my own classes I require that the 
solutions shall be carefully made in special note books, 


and that the student's method of analysis shall be 
plain, concise, and easily understood ; for the ability 
to reason soundly and to demonstrate clearly should 
be leading aims in the study of Mechanics. 

Work. (3) 3 000 foot-pounds. (6) 320 men. (10) 
169 000 foot-pounds. (ij.) 19 800 000 foot-pounds, 
(15) 352 000 foot-pounds. (17) 104.8 foot-pounds. (20) 
104 167 foot-pounds. (22) 20 foot-pounds. (25) 125 
pounds. (28) 120 000 pounds. (32) 1.5 1 inches. (34) 
0.54 pounds. (36) 28.5 pounds. (38) 112 pounds. 
(40) 522.5 pounds. (42) 6000 foot-pounds; ratio 3 : 2. 
(44) 12.9 man-power. (46) 9^\- horse-power. (48) \\ 
horse-power. (50) 36 j\ horse-power. (53) 25 horse- 
power. (55) 435 horse-power. (58) 139 kilowatts. 
(60) 67.8 amperes. (63) About 80 pounds per inch of 
width. (66) 4 inches. (68) 4 horse-power. (70) 12 
miles an hour. (72) 15 pounds per ton. (74) i 000 
horse-power. (77) %% against friction; 23 520 000 foot- 
pounds wasted. (79) 10.5 horse-power theoretically. 
(81) 1 40 horse-power. (82) 107 horse-power. {^^ 97.5 
horse-power. (86) i 665 horse-power. (87) 13.2 horse- 
power. (89) 2 566 looms. (91) 0.061. (92) 12.6 
horse-power. (95) 450 horse-power. (97) 62 pounds 
per square inch. (100) 5.6 horse-power. (loi) i 594 
horse-power. (104) 9448100ms. (106) 5 million horse- 
power. (108) 0.39. (no) 157 horse-power. (113) 
132 no 000 foot-pounds, or 132 million Duty. (116) 
10 500 cubic feet. (118) 3.44 machines. (121) 21 
hours 18 minutes. (124) 39 600 pounds. (125) 0.14 
horse-power; 97 cubic inches. (127) 184' pounds. (129) 
88 a R/P strokes per minute. (131) 156 tons. (133) 265 
pounds. (137) 1 1-2 pounds per ton. (140) i 783 


amperes. (142) 393 pounds; 19.6 per cent. (145) 4.5 
feet. (147) I 856 horse-power. (149) 12 758 foot- 
pounds; I 450 pounds. (152) 0.17 horse-power. (156) 
549 pounds. (158) I 250 tons. (159) 15 625 feet 
height. (162) 750 pounds. (164) 44.3 turns. (166) 
Ratio I to 1.94. (167) Average of 28.75 foot-pounds. 
(169) 62.5 cubic feet. (171) 15.2 horse-power. Force. 
(173) 300 pounds. (176) 10 pounds. (178) 43 units ; 
25. (181) 29 pounds. (183) 150 pounds; 90. (186) 
Rafters 6.32 tons; tie rod 6 tons. (188) '"Boom 77.3 
tons; tackle 36.4. (191) 50 pounds. (193) 580 pounds. 
(194) 6 030 pounds. (196) 117. 1 pounds; 82.8. (199) 
2.8 pounds ; 9.6. (200) cos^^ = bla, b being distance 
from C to AB and 2 a the length of the rod. (203) 
Perpendicular to plane. (205) 1020 pounds; i 000. 
(209) 2.45 tons. (211) 1. 15 tons. (213) 86.6 pounds; 
100. (217) I 460 pounds; i 990. (218) 77 pounds. 
(220) D being area of triangle, Y = ^ b {c^ -\- c^ — b^^ -r- 
4^D; Q=W«(ft2^^2_^2y4^2?. (222) In guy 10.3 
tons; in legs 18.6 tons. (224) Back stay, 90 tons; legs, 
100. (226) 7.8 tons; 6.5. (228) Back stays, 81 tons; 
A-frame 36; wire rope 29; upper boom 46; lower 51; 
guy or tackle 49.5. (229) 1.16 tons ; 0.55 ; 0.53. (232) 
14.2 pounds. (233) 13 units at tan~^ j^ with AB. (234) 
2 V2^ parallel to CA at distance from AC of 3 V2/2 X 
AB. (238) Ratio i to V3. (242) Tension = W / -^ 
2 \Ip — c^ smO. (246) 115. 5 pounds; 57.7. (252) 6250 
pounds; 5000, (257) 6 inches from end. (259) 9 
inches from middle; 18 pounds. (263) 5 inches from 
the middle. (266) 7 feet. (268) io6| pounds. (270) 
3.5 pounds; ^ inch from middle. (272) 4^ tons; 3§. 
(276) 6.5 pounds. (278) 9.5 pounds. (280) 11 inches 


and 6. (282) Posts 40 tons; lower chord, 37,5 and 32.5 ; 
upper chord 32.5. (284) Post 6 tons; tie 12.8; chord 
12.5. (286) 75 pounds. (287) 33.6 pounds. (290) 
540. (292) On line bisecting vertical angle § from ver- 
tex. (293) 2 V3rt'/9, '^a/T,, 4 V3 ^ from the sides, if 
each side = 2 a. (294) 6 V3^/ii, 3 ^f^a/ii, 2 V3 <?/ii 
from sides; outside the triangle at distance (i^2>^/Sy 
— 3^3 ^/5» 2 V3 (i/s. (296) Any point of line parallel to 
CD passing through X which is in BC produced so that 
CX = 2 BC. (299) 5 units acting parallel to BD, cutting 
BC produced at X, so that 4 CX = BC. (302) 124 
pounds, 92, 134. (306) At point 15 and 16 inches from 
adjacent sides. (309) 2] feet from rim. (313) If D be 
the middle point of BC, R is represented in magnitude 
by 2 AD, and acts through X parallel to DA, X being in 
DC orDB, so that DX =BC/8. (315) He loses i pound. 
(317) I inch. (318) 85.9 pounds. (321) 15 pounds 
each. (323) At C force is horizontal, and = W V3/2 ; 
at B tan"^ V3/2 to vertical and = W V7/2. (326) Length 
of stick from nail to wall v 3 : pressure := S^ ;^ ounces 
and 8 V -y^ _ I. (327) 18900 pounds. (328) 35.3 
feet. (331) P= 15 000 pounds. (332) | of length from 
end where pressure is 4 pounds. (335) 1.33 inches. 
(339) I inch from AC, i^ from AB. (342) It divides 
the face to which the cover is hinged in ratio of i to 
2. (345) From left-hand edge 2.84 inches; 5.36 inches. 
(348) 2 cos^ = 3 cos (tt — ^)/3, being angle with hori- 
zontal. (350) /i = r^^. (352) f (355) 373 pounds. 
(357) 1/^/3- (359) 200 pounds. (362) ^\. (366) I; 
incUnation, tan~^ |. (369) 433 pounds. (371) 1140 
pounds; 314. (374) 60°. (376) /a VVr/(i 4- -y) sin a, 
r being radius, and W the weight of wheel. (378) 47 


feet. (380) 1/V3. (382) 104 pounds. (388) About 
2.08. (393) 65 pounds. (394) 2 800 pounds. (396) 
898 pounds. (398) 2>\' (400) 13 inches. (401) 37.6 
inches. (405) 504 pounds. (406) 1.92 horse-power. 
(407) W/P = 0.95 or 1.05. (408) 137 thermal units. 
(410) 0.44 horse-power. (412) 3.5 horse-power. Motion. 
(415) 35 feet. (417) 13/^ miles per hour ; 27^- (419) 
30 miles an hour. (422) 150 feet; 200. (424) 13 
pounds per ton. (425) 99 feet. (426) 5 seconds. (428) 
6 seconds; 112 feet per second. (431) V<^^ feet 
per second . (433) 231 feet. (435) 4080 feet. (436) 
In '^hl2 g seconds, and 3/^/4 feet from ground. (438) 
350 feet. (440) 7W3/2; u/2. (442) 17.6 feet. (444) 
About J mile up-stream. (446) 245 miles an hour. (449) 
Northwest 6 V2 miles an hour. '(45^) 24 tt. (454) 65.5 
miles an hour; 0.27 miles. (457) 300 pounds. (459) 24.3 
amperes for maximum velocity. (462) 1.9 miles. (464) 
0.014. (467) 16 V5 feet per second. (469) 7.25 feet. 
(470) 8 feet per second. (473) 3tV pounds. (475) 
6938 pounds. (478) {a) ^^- g feet; {b) 70 pounds, 140 
and i86f. (479) i second. (481) 15° or 75°. (483) 
44 V2 feet. (486) 3903 feet inside of city. (487) 7.16 
miles. (489) 8 600 feet ; 31 250 000 foot-pounds. (494) 
0.3 tons. (497} 2.83 pounds. (500) Tan-^ ^VVo- (503) 
43 revolutions per minute. (506) 5^ tons. (507) 321 
tons. (510) 23.ifeet. (514) 3.4Pounds. (515) 3666! 
feet. (519) 3.1 feet per second. (521) 496.8 feet per 
second. (525) — i feet per second; +2. (527) A 
returns 5 feet per second ; B moves at 45° with its course 
and velocity of 10 V2. Review. (529) (a) 58 tons- 
(531) 56.7 per cent. (534) 60 000 pounds close to tower ; 
47000 in middle. (535) 52222 pounds. (536) 17 


revolutions. (538) 7.25 feet. (541) In bolt 27500 
pounds. (543) In leg 63.24 tons ; in inclined members, 
18.75 tons. (545) 0.036. (549) I 250 pounds. (552) 
27.8 feet per second. (557) 472 tons. (558) 160. 
(559) 682 revolutions per minute; 1910 amperes for the 
8 motors. (560) i 453 horse-power. (563) 0.19. (564) 
16 879 pounds; 8.1 per cent. (566) 2 720 pounds per 
square inch. (568) cos d = P 2 W. (571) Horizontal 
stress 1.8 tons; vertical additional 0.9 tons. (573) 2.0 
horse-power. (574) 35 pounds. (575) {a) 41 800 pounds ; 
{h) 458000 pounds; {c) i 195 gallons. (577) 7419 
pounds. (578) 30 862 pounds. (579) 25 800 pounds, 
35 900. (580) I 780 pounds. (583) In air, boom 16.57 
tons; in water, boom 14.45 tons. (5S6) 4.V tons. (587) 
15.2 feet. (589) 33.75 feet; ^V second. (590) 9 518 
pounds. (592) {b) 19.2 inches. (594) 54.2 pounds; 
10.4. (596) 80,6 per cent, (600) 42 horse-power. 



Falling Bodies : Velocity Acquired by a Body Falling a 
Olveu Helgbt. 
























































p. see. 

p. sec. 

p. sec. 

p. sec. 

p sec. 

p. sec. 








































































































20 6 






























































































































































































































































































12. C 


























3 .5 


































































4 25 








62 7 



























































































































Reprinted from Kent's Mechanical Engineers' Pocket-Book. 

Functions of Angles 







cos 1 1 












I 000 1 
















I. 0014 
















1 .0038 


1 1 .430 





O.I 05 1 

















1 .0098 








1. 0125 

































4 4454 







1 .0306 
























































1. 07 I 2 
















1 .0864 








1 .0946 








I. 1034 
















1. 1223 








1. 1326 










1 .8040 














1. 1666 

1. 94 16 







1. 1792 

1. 887 1 







1. 1924 

I. 8361 
































1. 6616 

























1. 1918 




















































30 miles 

an hour 

=: 44 feet per second 

I ton 

•= 2 000 pounds 

I fathom 

= 6 feet 

I knot 

= 6 080 feet 

I cubic foot of water 

= 62^ pounds 

= 7^ gallons 

I gallon 

of water 

= 8^ pounds 

I pound 

of watei 

: pressure 

= 2.304 feet head 

I British thermal 

1 unit 

= 778 foot-pounds of energy 

g, acceleration ol 


= 32 feet per second per 


unless otherwise specified 

e the base of Napierian system of 


= 2.7 1828 1828 

I horse-] 


= 746 watts 

I kilowatt 

= 1.34 horse-power 

Watts = volts X amperes 







60° 90 






















2 ° 










y/^ Infinite 





- 1.732 

Sin = 


Cos = 

Base ^^^ Perp 
Hypot Base 

Tan = 


Cot = 

Sec- ,.' 
Tan Cos 

Cosec = 


Vers = 

I — cos a\b => Sin A 



• B)^^inACosB-\-Cc 

Sin {A 4 

)sASinB c^Va^ + l>^-2abCosC 





Acceleration, 5, 119 
Angular velocity, 128 
Answers to problems, 
Automobile, 22, 140 
Axle friction, 115 


Belt friction, 108, 1 70 
Bicycles, 140 
Bolt Friction, 104, 150 
Bridges, 75, 77. 78, 147, 154 

Cast-iron pipe, 166 
Center of gravity, 4, 90 
Centrifugal force, 5, 137 
Centroid, 4 
Chimney, 145 
Coal, unloading, 23, 153 

wagon, 90 
Coefficient of friction, 96 
Components, 3, 4, 98, 100 
Concurrent forces, 3 
Cooper's loading, 157 
Couples, 4, 84 

Dam falling, 149 
Davit, 77 
Definitions, 2 
Derricks, 54, 167 
Dipper dredge, 64, 65 
Drum for hoisting, 169 

Electric car, 131 

current, 19 
motors, 41, 129 

Energy, 1,44 

Equilibriant, 3 
Examination papers, 174 

Flailing bodies, 123 
Fire engines, 37 
streams, 49 
Floor-posts, 83, 102 
Floating cantilever, 151 
Fly wheels, 48, 140, 148 
Foot-pounds, 7 
Foot-step bearings, 117 
Force problems, 51 
Forces at a point, 5 1 
Fortification wall, 135 
Friction coefficients, 96 

problems, 96 
Friction of angles, 191 

Gas engines, 16, 171 
Governors, 139, 172 
Gravity, acceleration of, 44, 123, 

Horse-power, 2, 16 

Impulse, 5, 142 
Indicator cards, 26 

Ladders, 79, 103, 104 

Launching data, 45, 151 

Least pull, 100 

Levers, 73 

Locomotives, 22,41,122, 128, 130, 

141, 156 
Logarithmic-decimal paper, 108 



Moments, 3, 72 
Momentum, 142 
Motion problems, 119 

Parallel forces, 72, 80 
Parallelogram of forces, 3 
Pendulum, 141 
Pile driver, 12 

resistance, 145 
Plates, structural, 91 
Projectiles, 46, 134 
Pulleys, 13,21 
Pumps, 35, 38 

Rail sections, 92 
Relative velocity, 126 
Resolution of forces, 100 
Restitution, coefficient, 5 
Resultant, 3 
Retaining walls, 89, 167 
Review problem, 145 
Roof trusses, 70 
Rope friction, 106, 114, 173 

Sailing vessel, 68 

Shears, 61, 62, 63 

Ship resistance, 40, 151 

Sound velocity, 123 

Steam engines, 25, 56, 89, 171 

turbine shaft, 1 18 
Steel rails, 93 
Structural plates, 92 

Trench machine, 160 

Tripod, 66 

Trusses, 70, 71, 72, 77, 78 

Unit values, 192 

Velocities, 119 

of falling bodies, 190 
Water gates, 97, 164, 168 

motor, 20, 162 

turbine, ^iZi 14^ 

power, i,T,, 38 
Water-works tanks, 163 
Wedges, loi 
Wire rope, 173 
Work problems, 7 




This book is due on the last date stamped below, or 

on the date to which renewed. 

Renewed books are subject to immediate recall. 


jAN21'6[) - I>P Wt 

0E cn<*'^ 


^i^h '67-^HNl 




Univeistty of California 

General Library