Calculate the angular frequency of the system shown in figure. Friction is absent everywhere and the threads, spring and pulleys are massless. Given that m A = m B = m.

Step-by-step solution

Let x 0 be the extension in the spring in equilibrium. Then equilibrium of A and B give, T = kx 0 + mg sin θ ...(i) and 2T = mg ...(ii) Here, T is the tension in the string. Now, suppose A is further displaced by a distance x from its mean position and v be its speed at this moment. Then B lowers by x 2 and speed of B at this instant will be v 2 Total energy of the system in this position will be, E = 1 2 k x + x 0 2 + 1 2 m A v 2 + 1 2 m B v 2 2 + m A gh A - m B gh B or E = 1 2 k x + x 0 2 + 1 2 m v 2 + 1 8