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Full text of "Equation of Motion for a Pendulum in a Resistive Medium and a Determination of the Resistive Constant k"

Equation of Motion for a Pendulum in a Resistive 
Medium and a Determination of the Resistive Constant k 

By Patrick Bruskiewich 



Abstract 

The period of a pendulum in a resistive medium is derived. The value for the resistive 
constant k is determined using the fractional variation in the period of the pendulum. This 
paper was written in 1981. The manuscript remained lost in the author's papers for years, 
until recently rediscovered with other manuscripts. 

1.0 Introduction 

Consider a pendulum consisting of a particle of mass m, supported by a string of length l 
(of negligible mass) in a gravitational field with strength g, which swings in a resistive 
medium with a resistive force that is proportional to the velocity. [1] 

The equation of motion is 



ml 2 — — = -mgl sin 6 - kl 2 — 
dt 2 dt 



where k is a coefficient of resistance. Using the small angle approximation sin# « we 
find that 

dt m dt I 

which is a second order ordinary differential equation with constant coefficients. 



When k = we recover the familiar expression for the period of the pendulum t q 



dt 1 I ° \g 



1.3 



A solution of the full equation of motion is the familiar 



9{t) = 



kt 

e 2m 



cos 




f_k_\ 

\lm j 



\ 



+ a 



J 



1.4 



where Q is an initial angle and a is an initial angular phase. 



The pendulum in a resistive medium has a period r given by 

2n 
t - 



g_ 



< k ^ 



2m J 



1.5 



By inspection, when k = we recover the familiar expression for r 



2.0 Determination of the Resistive Constant k 



The traditional approach to determine the resistive constant k is in terms of the decay of 
the angle with time. Such a measurement is both inaccurate and imprecise and requires 
specialized instruments, as anyone who tried this approach can attest. 



Since chronometers are very accurate and precise a better approach is to determine k in 
terms of the fractional variation in the period of the pendulum. 



Case 1: In the small k realm 



( k ^ 



K 2mj 
medium can be approximated by 



« — the period t of the pendulum in the resistive 



2n 



r - 



g 

I 

f 
2n 



' k ^ 



\2m j 



g 
vv/y 

f 



1- 



( k \ 



g 



2m 



\ Am j 



o 



1 + - 



^ k ^ 



V 



2g 



\2m j 



2\ 



J 



2.1 



By inspection we see that the expression for the period of the pendulum in a resistive 
medium is longer than that for a period not in a resistive medium. 

The fractional variation in the period of the pendulum is defined as 



Sr= T -^ 







In terms of the fractional variation we find for the small k realm 



2.2 



St = ^~ l 



T o 2 g 



\2m j 



2.3 



Solving for the resistive constant k in terms of the fractional variation in the period of the 
pendulum, 



k = 2m. 



m 



St 



= 4x 



(,A 428i 



\ T o J 



2.4 



Case 2: the period r of the pendulum in the resistive medium is given by 



T - 







1- 



' ik v 
\4nm j 



2.5 



In terms of the fractional variation in the period of the pendulum St 



(St + 1) 



f \ 

T 



V r oy 



1- 



< T»k * 



\47imj 



2.6 



Solving for k we find 



k = 47T 



f \ 
m 



V r o J 



f(^ + l) 2 -l 
(Sr + lf 



2.7 



3.0 Example Determination of the Resistive Constant k 



A precise series of measurements show that for a pendulum in a resistive medium 



<&■ = (1.006 ±0.001) 

m = (0.0241±0.0005)A:g 
r = (2.020 ± 0.0007) s 



2.8 



This yields a value for the resistive constant k of 



k = An 



0.0241 



V 2.02 j 
= 0.130^ _1 



|(2.006) -1 
(2.006) 2 



2.9 



References: 



,rd 



[1] H.B. Phillips, Differential Equations, 3 ru ed. John Wiley and Sons, 1951, p. 131-132 



1981 Patrick Bruskiewich