Question

Two identical balls A and B, each of mass 0.1 kg 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 centers of the balls can move in a circle of radius 0.06 m. Each spring has a natural length 0.06 π m and spring constant 0.1 N/m. Initially both the balls are displaced by angle π /6 radian with respect to the diameter PQ of the circle and released from rest.

(a) Calculate the frequency of oscillation of ball B. 

(b) Find the speed of the ball A when A and B are at the two ends of diameter PQ 

(c) What is the total energy of the system

Answer 1

Here the two balls connected by the springs are free to oscillate along the length of the springs, so the time period will depend on the reduced mass of the two-ball system.

(a) Restoring force on A or B

Where ∆x is compression in the spring at one end? 

Effective force constant = 2k

Where µ is reduced mass of system.

(b) P and Q are equilibrium position. Balls A and B at P and Q have only kinetic energy and it is equal the potential energy at extreme positions.

Potential energy at extreme position

When the balls A and B are at points P and Q respectively

(c) Total potential and kinetic energy of the system is equal to total potential energy at the extreme position=3.94 x 10^-4J.