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I The Project Physics Course Programmed Instruction Vectors 1 The Concept of Vectors Vectors 2 Adding Vectors Vectors 3 Components of Vectors INTRODUCTION You are about to use a programmed text. You should try to use this booklet where there ore no distractions — a quiet classroom or a study area at home, for instance. Do not hesitate to seek help if you do not understand some problem. Programmed texts require your active porticipa- tion and ore designed to challenge you to some degree. Their sole purpose is to teach, not to quiz you. This book is designed so that you can work through one program at a time. The first program. Vectors 1, runs page by page across the top of each page. Vectors 2 parallels it, running through the mid- dle part of each page, and Vectors 3 similarly across the bottom. This publication is one of the many instructional moterials developed for tho Project Physics Course. These ma- terials include Texts, Handbooks, Teacher Resource Books, Readers, Programmed Instruction Booklets, Film Loops, Transparencies, 16mm films and laboratory equipment. Development of the course has profited from the help of many colleagues listed in the text units. Directors of Project Physics Gerald Holton, Department of Physics, Harvard University F. James Rutherford, Chairman, Department of Science Education, New York University Fletcher G. Watson, Harvard Graduate School of Education Copyright (?) 1974, Project Physic* Copyright (^C^ 1971. Project Physics All Rights Reserved ISBN 0-03-089642-8 012 OOK 987 Project Phyiics it a registered trademark A Component of the Protect Physics Course Distributed by Molt, Rinehart and Winston New York — Toronto Cover Art by Andrew Ahlgren Vectors 1 The Concept of Vectors You are familiar with signs such a s \^ONEJ^{^!!j [SUBWAyJ that indicate a direction. You have also seen signs which give a magnitude such as OR MAXIMUM 35 TONS CAPACITY This program is about quantities that hove both a direction and a numerical value. These are called vectors and they are very important in physics. You are already familiar with some ex- amples of vectors. This port of the program will start with these examples. Vectors 2 Adding Vectors Adding vectors is on important technique for you to understand and be able to use. After going through this set of programmed materials you will be able to add two or more vectors together and obtain the resultant vec- tor. The next three sample questions represent the kinds of questions you should be able to answer after you have finished Vectors 2. if you can already answer these frames, you need not take Vectors 2. 't that case you can go on to Vecto--- 3. Vectors 3 Components of Vectors When we use a vector to represent a physical situation, we may wish to find the component of that vector in a given direction. This is Part III of the series of programmed instruction booklets on vectors. In this part, you will learn how to separate vectors into components and how to obtain a vector from its components. The two sample questions that follow illustrate the objectives of this part of the program. Vectors 3. If you find that you can answer these two questions correctly, you need not work through the program. INSTRUCTIONS 1. Frames: Each frame contains a question. Answer the question by writing in the blank space next to the frame. Frames ore numbered 1, 2, 3, . . . 2. Answer Blocks: To find an answer to a frame, turn the page. Answer blocks ore numbered Al, A2, A3, . . . This booklet is designed so that you can compare your answer witfi the given answer by folding back the page, like this: 1 - 1 U- "75^, 2 ; 1 3. Always write your answer before you look at the given answer. 4. If you get the right answers to the sample questions, you do not hove to complete the program. INSTRUCTIONS: Same as for Program 1, above. INSTRUCTIONS: Same as for Program 1, o' we. Sample Question A Answer Space Complete this sentence if you can: A scalar quantity can be expressed by (i) quantity must be expressed by both (ii) , but a vector Sample Question A Given are two vectors, X and Y, represented by the arrows drawn here. (i) Draw an arrow to represent the vector sum (resultant). (ii) Give its magnitude / Answer Space 2 units Sample Question A An arrow is shown that represents a force vector F. (i) Draw Fy, the component of F in the y-direction. (ii) Draw Fx, the component of F in the x-direction. Answer to A (i) a number (with or without units) (ii) number (with or without units) and a direction. Answer to A (i) X * Y (ii) 3.7 units Answer A i 7 F y /! / L F ( Answer Space Sample Question B It is important to be able to distinguish between vector and scalar quantities in equations. (i) List all of the vector quantities in the equation T= mT+ 6P! (ii) List all of the scalar quantities in the same equation. Sample Question B Answer Space Three forces acting on an object, 0, can be repre- sented by arrows as drawn below. What is the resultant force on the object, that is, what is the vector sum of the three forces? Sample Question B Given Vx and Vy: (i) Construct and draw v. (ii) Give the direction and magnitude of v. scale: 50 Answer to B (i) T, 7. and P^ (ii) m and 6 Answer to B Resultant Resultant Force, F„ shown. An; (i) (ii) 45 "^ below Korizontol, 50 m sec. If your answers to the sample questions were correct, the remainder of the program is optionol. Sample Question C Answer Space Suppose the wind is blowing from the northeast at 12 m/sec. Draw an arrow that represents this wind velocity to the scale given. 10 15 20 -I I Sample Question C Answer Space Forces F,, F2 and F3 (from the last frame) are shown acting on a car. You found the resultant force by adding these vectors together tip-to-tail as shown at the left. What should the magnitude of F, have been if you wanted the resultant force to be zero? Draw the vector B that must be added to A to give C. Answer to C Your answer is correct only if the arrow you draw points in the same direction as this one and is the same length. If you answered all 3 sample questions correctly, you are ready for the Vectors 2 program. If not, begin with question 1 on the next poge. Answer to C /^ New F, F, , F; ♦• F3 = if the mognitude of F, is 3.5 units. A1 Now turn the page to begin Vectors 1. Remember to proceed through the book from left to right, confining your attention to the top frame on each page. Now turn the page to begin Vectors 2. Remember, left to right, middle frames only. Draw two perpendicular vectors that add to give F. -U JUDDCII WASHINGTON, D. C :^g^jyj^vj| ooMmvnoN lau. 5:^ sg D^^' Wi n®^!K □ M] C^SPm^^^ LimJ ^&( t-«MlJ ;r!-f Scale (meters) 500 ■ I . 1000 V»cton Part I ^ Mop of CantTol Section o* j — 1 ( — I I — ^ WoAifiBtoo, O.C.,U.S.A. ^gS§^ snr The Parallelogram Law A vector is an entity having both magnitude and direction; vectors also have the property of addition by the parallelogram law as shown here, where A and B represent two vector quantities. It can be drawn either The vector sum of A t B is C and can be drown in two ways. Both ways of drawing the parollelo- grom low shown above are equivalent, but the "tip-to-tail" method on the right will be shown to- the more powerful since it can be extended easily to more than two vectors. There are many physical quantities which hove both direction and magnitude and odd to- gether according to the parallelogram low. In Part I of the vectors program the displacement vector was introduced, and Port II will begin with the addition of displacements. A2 possible solutions: NOTE; There are on infLnit* number of solutions. Questions 1 through 16 require the map of Washington, D.C., shown to the left. Find the location of the Lincoln Memorial and the Jefferson Memo- rial on the map of Washington, D.C. A straight line is shown be- tween the memorials. According to the scale of the map, the dis- tance between the Lincoln and Jefferson Memorials is meters. (Hint: One way to use the scale on the map is to copy it off the edge of a piece of paper which can be placed along any line you wish to measure.) Read the panel on the opposite page. You learned in Part I of the program that a vector quantity has both magnitude and direction. What other property will a vector quantity have? Martha walked from the post office to the bus stop. Her displacement is represented by the arrow marked D on the map. (i) How many blocks / Po^t y east did she walk? \omceJ (ii)How many blocks south did she walk? / M W Oak St, Elm St, Park St.! A1 about 1700 meters, measuring center to center A1 Vector quantities add according to the paral- lelogram low. A3 (!) 6 blocks east (ii) 2 blocks south From the compass directions on the mop we can see that the Jefferson Memorial is located 1700 meters of the Lincoln Memorial. Let us use vectors to represent a trip around the city block. The first leg of the trip starts at intersection A, and is represented by dAo, the displacement vector drawn from A to B. (i) What is the magnitude of the vector ■^AB? (ii) What is its direction? Scale: 1 cm = 100 m W-^ -^ E The diagram below shows that A + B = C. Two vectors which add to give a third vector are called components of that vector. In this example, (i) are components of (iii). and (i A2 southeast A2 (i) 250 meters (approx.) (ii) north A4 (i) A (or B) (ii) B" (or A) (iii)C Locate the White House, and find the distance and direction of the White House from the Jefferson Memorial. On the panel draw the second leg of the trip around the block, namely from B to C. (i) Give the direction and magni tude of the displacement vector dof-- (ii) Give the total distance traveled on the first two legs of this trip. The two paths marked / post ^ 1 and 2 yield the same \officeJ displacement vector D. Also, the easterly and southerly components must add to give D independently of the ^ path. ~ c5 ^ ^ -^ What is the magnitude of the southerly component of D? A3 approximately 2100 meters to the north A3 (i) a few degrees North of East 170 meters (ii) 420 meters (A to B = 250 m, B to C = 170 m) A5 2 blocks One of the important concepts of physics is that of displacement: it is the straight line distance and direction between the initial and final locations of an object. Use the map of Washington, D.C., to answer the following questions: (i) What building will you reach if you start at the Washington Monu- ment and travel 2600 meters due east? (ii) What was your displacement? Draw the vector dip between points A and C. (This goes diago- nally across the block.) (i) Give the magnitude and direc- tion of d A p . (ii) What is the difference (in meters) between the distance traveled from points A to B to C, and the rrxignitude of the vector d^C ? The dashed line represents the actual path Martha took from the post office to the bus stop. Her displacement D does not de- pend on her path and the components of D likewise do not depend on her path. /^ — ^ I / post Y What is the mag- '^ office^ nitude of the compo- nent of D in the easterly direction? A4 (i) the U.S. capifol (ii) 2600 m east from the Washington Monument A4 (i) 330 m a few degrees North of NortKeost (ii) 90 m difference A6 6 blocks (i) What would be your displacement if you traveled from the Capitol to the White House? (ii) What IS the dispiacement if something is moved from the White House to the Washington Monument? The displacement vector from A to C, dxr / is the resultant of adding d^g and dg^. The displacement vector d^^p is the resultant of adding ^aq and (ii) What is the resultant of 'BC jnd 6qq? (iii) Draw the resultant of dgp and df-Q on the diagram at the right. The vector F represents the force exerted by the rope on the wagon. We can separate the force into vertical and horizontal components. (i) Draw the component of F in the vertical direction. Label it F^. This component tends to lift the wagon. (ii) Draw the component of F in the horizontal direction. Label it • ^. This component of the force is responsible for the motion of the wagon along the ground A5 (i) 2900 m, approximately northwest (octually 290' from north) (ii) 1100 m south (octually slightly east of south) A5 CO dcD (ii) dgD A7 Tra^^te?^ A displacement can be represented ^Krj^^ by an arrow in a mop. The length of ^<i the arrow represents a scale drawing » of the actual displacement. ra^H^FI IIBl What displacement is shown? ^0 AVI ^5[ The final leg of the trip around the block, from intersection D to A, is given by the displacement vector ^DA- Draw the vector sum of d/-Q and ^DA- The arrow labeled Fgrav represents the force of gravity on this railroad hopper car. The component of Fgrav P®''" pendicular to the track is balanced by the opposite force of the track on the wheels. (i) Draw the component of Fgrav that is perpendicular to the track. Label it Fj^. (ii) Draw the component of Fgrav that is parallel to the track. Label it F„. A6 White House to Washington Monument (1100 m south) A6 A8 grov (i) Draw an arrow on the map to represent the displacement of a person who has walked from the Washington Monument to the Jefferson Memorial. (Hint: If you are not sure how to do this, recall the definition of displacement in Frame 4.) (ii) Draw a broken line on the map to show the shortest path for walking on dry ground from the Washington Monument to the Jefferson Memorial. (iii) Is the path length the same as the displacement? (iv) Does the choice of path change the displacement? The four legs of the trip around the block can be represented by the four separate vectors shown here. 'AB ^CD What is the sum of these four vectors? Here is an expanded diagram from Frame 8 The magnitude of Fg^^^ is 120,000 N. (i) Find the magnitude of Fj_. (ii) Find the magnitude of Fn . scale: '■■■■' 50,000N grav (iii) no (it changes the path length, but not the displacement, which is defined as the straight- I ine distance.) A7 i S^^JIS A7 A9 (i) 120,000 N (ii) 30,000 N On the map of Washington, D.C., there is on arrow which indicated that the displacement of New York City from Washington is • distance? direction? 8 If the vector C is the sum of vectors A and B, we can write: (i) Given A and B as shown, draw the vector sum C. (ii) Find the direction and magnitude of C by measuring the scale drawing. 10 . L J t In general, components of a vector are constructed as the sides ot a parallelogram which has the vector as the diagonal. The angle betwee the sides of the parallelogram may be any value; however, the physical analysis is often easiest if this is chosen to be 90°. The preced- ing examples of the wagon and the hopper car illustrate the use- fulness of components that are at right angles. As an example of non-per- pendicular components, take the vector Fgrav from before and re- solve it into components in the q and r directions. Label the components F and F^,. Be sure to draw these components as vectors. 2 units Scole: I 1 • A8 320 Icm northeast A8 (i^ (ii) direction: 43° from A. magnitude: 5.? units. A10 9 1 Note that the distance scale at the bottom of the map is for | measurements inside Washington, and the displacement to more re- | mote places such as New York City is represented with another | scale. It is not essential that the arrow representing a displace- | ment vector be drawn to the same scale as the map. | Pittsburgh, Pennsylvania, is approximately 320 kilometers to the | northwest of Washington. Draw an arrow by which you can repre- | sent this displacement. 1 (Use the same scale as the arrow showing the displacement of 1 New York City.) 1 Two arrows representing the vectors S and T are drawn i separately. S and T cannot be added without shifting them so | that they touch. The most useful way to make this shift is so I that the pointed "tip" of one touches the blunt "tail" of the 1 other. 1 (i) Redraw S and T with • the tip of S touching j ■ the tail of T. . / t 1 / ! (ii) Drew the vector sum / i of S and T on the tip- 1 to-tail drawing. . 11 1 The previous frames have shown that a vector may be resolved into components along any chosen axis. - Now, given the components, it can be seen thot a ■ vector is the (vector) sum of its components. ■ ■* 1 b 1 y Given B and B , find B. | X A9 A9 10 / (ii) All 10 Quantities that have both magnitude and direction ore called vectors. Quantities that have a magnitude but no direction are called scalars. Is the displacement shown below a scalar or a vector? 10 (!) Shift the arrow representing the vec- tor Z so that its tail is touching the tip of Q. (ii) If R = Q + Z, draw an arrow repre- senting R. 12 The ground exerts a pe-pendicular force Fj. on the skier and the cable pulling the skier exerts a force Fn . The friction between the skis and snow is negligible. (i) Construct and draw the arrow repre- senting the net force (Fnet) o^ ^^^ of the cable and the ground on the (ii) What is the direction and magnitude of the net force? A10 vector A10 (i) *a r^ 00 'a / r^* (ii) vertical (upward) 22 units of fore* 11 Quantities that have only a magnitude are called scalars. Those quantities that have both magnitude and direction are called vectors. 100 150 Is the position of the 50 meter mark on the scale a vector or a scalar? 11 iTr F+ G^, Find h" by adding F and G with the tip-to-tail method in both of these ways: (i) shifting F to the tip of G. (ii) shifting G to the tip of F. (iii) Do both procedures give the same result? 13 The diagram shows a particle striking a barrier and rebounding elastically. (i) Resolve each of the velocity vectors into components which are perpendicular to the wall and parallel to the wall. (ii) Which component of velocity did not change during the interaction? All scalar All (i) (ii) (iii) Yes A13 The component of velocity parallel to the wall does not change during the interaction. (ii) V 12 A scalar quantity can be expressed by a single number (with or without units), but a vector must have both 12 The clear advantage of using the tip- to-tail method of graphically adding vectors can be seen when three or more vectors are to be added We have already seen this in the example of the city block. The ad- dition is performed by making a "chain" of vectors. Then the sum (or resultant) is found by drawing the arrow from the tail of the first to the head of the lost arrow in the chain. Draw the resultant for U + P + S 14 Here is the same event again. Describe the change of the component of velocity peqtendicular to the wall. A12 magnitude and direction A12 ^*P.c^ A14 The component of velocity perpendicular to the wall reverses direction but does not change in magnitude. 13 Are the following pictures representations of vectors, of scalers, or of neither? (i) (ii) 450 Miles Son Froncisco to San Diego To Chicago 13 (1) Redraw U, E and Y tip-to-tail. (ii) Draw the vector sum of U + E + Y. 15 A ball has components of velocity Vx and Vy as shown in the diagram. (i) Construct and draw v. (ii) Give the direction and magnitude of v. X "^ N ) V \ X J i Y scale: 50 m/sec A13 (i) vector (a displacement) (ii) neither (only direction) A13 NOTE: As the reduced sketches below indi< any sequence of V, E. ond Y will give the s resultant. / / / / / A15 (i) (ii) 45^" below horizontal. 50 m sec 14 1 On the map of Washington, D.C., there is an orrow representing | the wind velocity. The arrow indicates that the wind is blowing from | th«» (i) nt n <:p<>ed of (ii) | 14 j (i) Redraw M, N and tip-to-tail. • (ii) Draw the vector sum W, ^ ■ where M + N + = W. ~~^y^ ' You hove now completed all three programs in this book. Understanding and being able to use vectors should be helpful to you in many ways. If ever you wish to refresh your memory on Vectors, you can cover up the answer space with a sheet of blank paper and quickly run through the frames again. A14 (i) southeast (ii) 9 m/sec (about 20 miles/hr) A14 NOTE: Any sequence of M, N, and will give the some W. 15 1 The speed and direction of the wind is a vector quantity, and . therefore it can be represented by an arrow drawn to scale. Suppose ■ the wind changed and is now coming from the west at 18 m/sec. ■ Draw the new wind direction, and indicate the new wind speed by i making the arrow of the proper length (using the other wind arrow i as a guide). 1 15 j / 1 (i) Redraw the vectors J, K and L v j tip-to-tail. \ j (ii) T+ 1< + r = m! Draw the V^^^ ^ 1 arrow representing M. | | J j (iii) Does the order in which you ' redraw the vectors affect M ? T . A15 wind speed = 18 rrv/$ec m (This is twice as long as the length shown for a wind speed of 9 rv sec.) A15 (i) (ii) (iii) 1 16 1 To the same scale what is the length of the arrow needed to repre- ' sent a wind speed of 27 meters/sec? 16 [ Given M, , Mj, M3 as shown, and ^ M, ^ M2 + M3 = M4 ^ '"^Z 1 FindM^^. ^ ^^3 / 1 \ 1 A16 three times the length for 9m sec A16 M4 is zero 17 Whenever we encounter a physical quantity — such as speed force, energy, or whatever — it is useful for us to know whether or not it involves direction. Those quantities that involve direction as well OS magnitude ore called (i) (ii) Does the pull each team exerts on the rope in the tug-of-war involve a direction? 17 If a", ^ A^ ♦ A3 = "5 and A, and A2 are as shown, con- struct the vector A3 that satisfies this equation. A17 (i) vectors (ii)yes A17 18 When we encounter a physical quantity that is a scalar we mean it has no (i) (ii) Is the diameter of the water wheel shown here a vector or a scalar? 18 Force is a vector quantity. Each of the cars shown here is exerting a force on the large wooden box. Below each car draw an arrow to indicate the direction of the force each car exerts on the object to which it is hitched. A18 (i) direction (ii) scolar A18 1 19 1 Four boys ore shown pushing a | car. The force each boy exerts on the | car is a | (i) qiinntity^ nnd the number of boys pushing the car is a . (ii) qiinntity. 19 Suppose the small cor (1' force the other car (2) exerts. 1 2 pulls with half the i m-H II v I Draw arrows represe nting the fo ce each car exerts. 1 A19 (i) vector (ii) scalar A19 (I) (2) NOTE: These arrows con be of ony length except thot (1) must be |ust one-half the length of (2). 20 j When writing one usually draws a small arrow over the symbol i used for vector quantities. For example, in the equation i F = m a, 1 F represents a vector quantity, the force, and a represents an ac- | celeration in the same direction as F. The letter m represents a a scalar, mass. 1 (i) List all vector quantities in the equation • T = ma + 6N ■ (ii) List all of the scalar quantities in the same equotion. • 20 j (i) What is the sum of the two pulls of the cars, namely the j resultant force exerted on the box by both cars pulling together? Assume the pulling forces; F, = 5 units (to the • left) F2 = 10 units (to the 1 left) 1 (ii) Draw the resultant force ( Fp ) ? I A20 (i)T,^N (Did yoo put the orrows over the symbols?) (ii) m, 6 A20 (i) 15 units of force to the left (ii) (1) (2) Resolton* % 21 The negative of a vector quantity is represented by an arrow in the reverse direction. For example if A is represented by X ., -r. . ,, -X ifTis >/ ^. then —A is represented by draw — B. 21 Two cars are shown pulling on a wooden box. The pulling force of each car is represented by the vectors F, and F2 (note the units). (i) Construct the vector sum Fp of these forces using the tip-to-tail method. (If you are not sure how to do this, refer to Frame II.) (ii) What is the direction and magni- tude of the sum Fp ? (iii) Write an equation to represent the relation between F) , Fj and Fp . A21 y Did you draw - B to the proper length? It is a vector in the direction opposite to B but having the same magnitude. A21 (i) (ii) to the left and a few degrees below horizontal, magnitude about 15 units (iii)F, F, = F. 22 1 If -C is \^ give a full label to: >^ ' 22 [ Suppose that two cars were pulling • an object, and that each is exerting a . force represented by the arrowns shown ^ ■ '"'■ - - ''%^. 1 (i) Find the vector sum Fi + Fz- ^*"^v,.^ | (ii) Draw an arrow representing a force \. ^/^ 1 vector F3 such that F, + F2 + F3 = 0. ^^^ (iii) If F3 is the force exerted on the ^9^ object by a third car, what is the ^W^ ! resultant force on the object? ' A22 (iii) 23 j This ends Vectors 1. | You have learned to distinguish between vectors and scalars. You | have drawn vector quantities to scale, and you hove learned that a | negative vector is in the opposite direction from the corresponding 1 positive vector. 1 You are now ready to learn to add vector quantities. See the pro- ' gram Vectors 2. It begins at the front of this book and occupies ' the middle of each page. • 23 1 Three forces acting on an object can be represented | by arrows as drawn below. 1 Draw an arrow to represent the resultant force Fp | on the object. 1 ^ 'i A . ! (Hint: If you are not sure how to do this, refer to Frame 15.) ■ A23 24 1 Forces F,, Fj and F, (from the last frame) are sfiown acting on object C. You found the resultant force Fp by adding these three vectors together ""tip-to-tail" in Frame 23. What mcanitude ihouic F. hove tn order to make the resuitcnt force zero? i • 1 , ^' A . Force scale ^^^^ 2 units A24 3 units 25 Given are two vectors, X and Y, represented by the arrows drawn here. (i) Draw on arrow to represent the vector sum. 1 ^ (ii) Give its magnitude. ^ y 1 / \ 2 units 1 t A25 (i) $om X + Y (ii) 3.7 units I This ends Vectors 2. I You have learned how to add two or more vectors together and to I draw the resultant vector. Also, given two vectors, you have • practiced finding a third vector that would just balance the first I two vectors so that the sum of the three was zero. I If you would now like to learn about components of vectors, see ■ the program Vectors 3. It begins on the bottom part of the first , page of this book. , 0-03-089642-8