Let’s think back to the lab for simple harmonic motion. Consider the setup for t

Question

Let’s think back to the lab for simple harmonic motion. Consider the setup for the simple pendulum. The length of the pendulum is 0.60 m and the bob has inertia 0.50 kg (assume mass of the string is negligible, and small angular displacements). You conduct two experiments (A and B) to investigate the physics of simple harmonic motion. For experiment A, you pull the pendulum 5 , or 36 radians. In experiment B, the angular displacement is 10 , or 18 radians. Define clockwise rotation as positive.

(a) Why do we want to keep the angular displacement “small”?

(b)What is the period of motion for experiment A?

(c) How does the period for experiment A compare to the period for experiment B?

(d)How do the frequencies compare for the two experiments?

(e) What are the maximum kinetic energies and potential energies for both experiments?

(f) In general, where do these occur for the motion of a pendulum?

(g)How do the total energies compare for the two experiments?

(h) For experiment A, write the equation of motion for angular displacement, (t) =??? (use either a sine or cosine and remember that clockwise is positive). Let t = 0 be the instant the bob passes though equilibrium moving in the counter-clockwise direction.

Explanation / Answer

here,
length of string = 0.60 m
inertia of bob,m = 0.50 kg

part A :
motion of simple pendulum is given as :
d^2(Theta)/dt^2 + g*sin(theta) / L = 0 -------------------(a)

If the amplitude of angular displacement is small enough that the small angle approximation (sin(theta) = theta) holds true, then the equation of motion reduces to the equation of simple harmonic motion.

d^2(Theta)/dt^2 + g**theta/L = 0 -----------------------(b)

Simple harmonic solution of above eqn :
theta(t) = (Theta)o * Cos(wt + fi)------------------------(c)

with, w = sqrt(g/L)

Part B:
peroid = 2ip*sqrt(l/g) -------------(1)
Ta = 2pi*sqrt(0.60/9.8)
Ta = 2*3.14* 0.247
Ta = 1.554 s

Part C:
From eqn 1,
The period for a simple pendulum does not depend on the mass or the initial anglular displacement, but depends only on the length L of the string and the value of the gravitational field strength g

So, period of A = period of B

Part D:
Frequency in SHm of pendulum is given as :
w = sqrt(g/l)

For Exp A :
wa = sqrt(9.8/0.60)
wa = 4.041 rad/s

since, length of string doesn't change so,
Wb = wa
Wb = 4.041 rad/s

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