Simple Harmonic Motion 4 Question 15
15. 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 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 figure) and released from rest.
(a) Calculate the frequency of oscillation of ball
(b) Find the speed of ball
(c) What is the total energy of the system?
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Answer:
Correct Answer: 15. (a)
(b)
(c)
Solution:
- Given, mass of each block
and
Radius of circle,
Natural length of spring
In the stretched position elongation in each spring
Let us draw FBD of
Spring in lower side is stretched by
Restoring torque of this force about origin
Since,
Eq. (i) can be rewritten as
(a) Frequency of oscillation,
Substituting the values, we have
(b) In stretched position, potential energy of the system is
and in mean position, both the blocks have kinetic energy only. Hence,
From energy conservation
Substituting the values
(c) Total energy of the system,
or
or