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 of bent in the form of a

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 of 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 0 . 06 m . Each spring has a natural length of 0 .06 π m and force constant 0 .1 N m - 1 . Initially, both the balls are displaced by an angle θ = π 6 rad , with respect to the diameter PQ of the circle, and released from rest. The frequency of oscillation of the ball B is

TestHub · Accepted Answer

As the two masses are connected by two springs, this problem is equivalent to the oscillation of a reduced mass $m_{r}$ , of a spring of effective spring constant $K_{e f f}$ .

$T = 2 π \sqrt{\frac{m_{r}}{K_{e f f}}}$

Hence, $m_{r} = \frac{m_{1} m_{2}}{m_{1} + m_{2}} = \frac{m}{2}$ , $K_{e f f} = K_{1} + K_{2} = 2 K$

$∴$ $n = \frac{1}{2 π} \sqrt{\frac{K_{e f f}}{m_{r}}} = \frac{1}{2 π} \sqrt{\frac{2 K}{m} \times 2} = \frac{1}{π} \sqrt{\frac{K}{m}} = \frac{1}{π} \sqrt{\frac{0.1}{0.1}} = \frac{1}{π} Hz$

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