Q. Two identical balls and , each of mass , are attached to two identical massless springs. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in the figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius . Each spring has a natural length of and spring constant , Initially, both the balls are displaced by an angle with respect to the diameter of the circle (as shown in the figure) and released from rest.

The frequency of oscillation of balls is

Question

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Solution:

Given, Mass of each block A and B, m = 0.1 kg
Radius of circle,
Solution
The natural length of spring (Half circle) and spring constant,

In the stretched position elongation in each spring

Let us draw FBD of A
Solution
Spring in lower side is stretched by 2x and on upper side compressed by 2x. Therefore, a force 2kx on each block would be exerted by each spring.
Hence, a restoring force, F = 4kx will act on A in the direction shown in figure.
Restoring torque of this force about origin

or ..... (i)
Since, , each ball executes angular SHM about origin O.
Equation. (i) can be written as

or
or
Frequency of oscillation,

Substituting the values, we have