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 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 0.1N/m. Initially both the balls are displaced by an angle $θ = π / 6$ radian with respect to the diameter $PQ$ of the circle and released from rest. The frequency of oscillation of the ball B is

Moderate

Solution

As here 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.
$T = 2 π \sqrt{\frac{m_{r}}{K_{eff .}}}$
Here $m_{r} = \frac{m_{1} m_{2}}{m_{1} + m_{2}} = \frac{m}{2} \Rightarrow K_{eff .} = K_{1} + K_{2} = 2 K$
$∴ n = \frac{1}{2 π} \sqrt{\frac{K_{eff .}}{m_{r}}} = \frac{1}{2 π} \sqrt{\frac{2 K}{m} \times 2} = \frac{1}{π} \sqrt{\frac{K}{m}} = \frac{1}{π} \sqrt{\frac{0 .1}{0 .1}} = \frac{1}{π} Hz$

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