Two uniform ropes having linear mass densities m and 4m, length l. Each are joined to form a closed loop. The loop is hanging over a fixed frictionless small pulley with the lighter rope above as shown in the figure (in the figure equilibrium position is shown). Now if the point A joint, is slightly displaced in downward direction and released. It is found that, the loop perform angular SHM with the period of the oscillation equal to 2N. The value of N (take $l = \frac{150}{π^{2}}$ metre) is _________.

Difficult

Solution

Let point A is displace in the downward direction by x.
Net imbalance forces = $m_{t o t a l} a$
$\begin{array}{l} [4 m g (l + x) + m g (l - x) - {4 m (l - x) g + m (l + x) g}] \\ = (4 m \times 2 l + m \times 2 l) a \\ a = - (\frac{3 g}{5 l}) x; ω = \sqrt{\frac{3 g}{5 l}} \Rightarrow T = 2 π \sqrt{\frac{5 l}{3 g}} = 10 s \end{array}$

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