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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$\pi$ m and force constant 0.1N/m. Initially both the balls are displaced by an angle $\theta = \pi /6$ radian with respect to the diameter $PQ$ of the circle and released from rest. The frequency of oscillation of the ball B is
$\pi \,Hz$
$\frac{1}{\pi }Hz$
$2\pi \,Hz$
$\frac{1}{{2\pi }}Hz$
Solution
(b)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\pi \sqrt {\frac{{{m_r}}}{{{K_{eff.}}}}} $
Here ${m_r} = \frac{{{m_1}{m_2}}}{{{m_1} + {m_2}}} = \frac{m}{2}$ ==> ${K_{eff.}} = {K_1} + {K_2} = 2K$
$n = \frac{1}{{2\pi }}\sqrt {\frac{{{K_{eff.}}}}{{{m_r}}}} = \frac{1}{{2\pi }}\sqrt {\frac{{2K}}{m} \times 2} $$ = \frac{1}{\pi }\sqrt {\frac{K}{m}} = \frac{1}{\pi }\sqrt {\frac{{0.1}}{{0.1}}} = \frac{1}{\pi }Hz$
