The amplitude of oscillation of a simple pendulum is increased from 1$\degree$to 4$\degree$.Its maximum acceleration changes by a factor of:
A)1/4
B)1/2
C)2
D)4
E)16

Question

Quizplus · Accepted Answer

The maximum acceleration of a simple pendulum is given by the formula a_max = g\sin	heta, where g is the acceleration due to gravity and 	heta is the maximum angle of oscillation. Since g is constant, we only need to consider the change in 	heta.

The amplitude of oscillation is directly proportional to 	heta, so if the amplitude is increased by a factor of 4, then 	heta is also increased by a factor of 4.

Using the formula for maximum acceleration, we have:

a_max1 = g\sin	heta1
a_max2 = g\sin	heta2 = g\sin(4	heta1)

Dividing a_max2 by a_max1, we get:

a_max2/a_max1 = (g\sin(4	heta1))/(g\sin	heta1) = 4\cos	heta1

Since 	heta1 = 1 degree, \cos	heta1 = \sqrt{1 - \sin^2	heta1} \approx 0.9998 (using a calculator).

Therefore, a_max2/a_max1 \approx 4*0.9998 = 3.9992, which is approximately equal to 4.

Hence, the factor by which the maximum acceleration changes is 4, and the answer is D).

The Amplitude of Oscillation of a Simple Pendulum Is Increased °\degree°

The Amplitude of Oscillation of a Simple Pendulum Is Increased $\degree$