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 0.06m 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, of a spring of effective spring constant.
$\begin{array}{l} T = 2 π \sqrt{\frac{m_{r}}{K_{e f f}}} \\ Here m_{r} = \frac{m_{1} m_{2}}{m_{1} + m_{2}} = \frac{m}{2} \\ \Rightarrow k_{e f f .} = k_{1} + k_{2} = 2 k \\ ∴ n = \frac{1}{2 π} \sqrt{\frac{k_{e f f}}{m_{r}}} = \frac{1}{2 π} \sqrt{\frac{2 k}{m} \times 2} \\ = \frac{1}{π} \sqrt{\frac{k}{m}} = \frac{1}{π} \sqrt{\frac{0.1}{0.1}} = \frac{1}{π} Hz \end{array}$

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