A block of mass M 1 is constrained to move along with a moveable pulley of mass M 2 which is connected to a spring of force constant k , as shown in the figure. If the mass of the fixed pulley is negligible and friction is absent everywhere, then the period of small oscillations of the system is

Step-by-step solution

If the mass M 1 is displaced by x and has speed v , spring will extend by x / 2 and M 2 will have speed v / 2 . Energy of this system can be written as - M 1 gx + 1 2 M 1 v 2 + M 2 g x 2 + 1 2 M 2 v ⁡ 2 2 + 1 2 k x 2 2 = const. - M 1 gx + 1 2 M 1 v 2 + M 2 2 gx + 1 8 M 2 v 2 + 1 8 k x 2 = const. Differentiate wrt time - M 1 gv + M 1 v dv dt + M 2 g 2 v + 1 4 M 2 v dv dt + 1 8 k x.v = 0 ⇒ - M 1 g + M 1 dv dt + M 2 g 2 + 1 4 M 2 dv dt + 1 8 k x = 0 M 1 + M 2 4 dv dt = - k 4 x + M 1 g - M 2 g 2 a = dv dt = - k 4 M 1 +