A tank contains two immiscible liquids of densities 6 and 2 . The higher density liquid is filled up to a height L/2 from the bottom. A thin rod of density and length L is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is 2 n L g , where g is the acceleration due to gravity. The value of n is:
A. A
Let A be the cross-sectional area of the rod. Mass of the rod, M = A L. Weight of the rod, W = A L g, acting at a distance L/2 from the hinge. When the rod is displaced by a small angle , the restoring torque is provided by the buoyant forces from the two liquids. Buoyant force due to the lower liquid, F_ B1 = (6 ) A (L/2) g = 3 A L g. This force acts at the center of the immersed part in the lower liquid, which is at a distance L/4 from the hinge. Torque due to F_ B1 is _ B1 = F_ B1 (L/4) = (3 A L g) (L/4) = 3 4 A L^2 g . Buoyant force due to the upper liquid, F_ B2 = (2 ) A (L/2) g = A L g. This force acts at the center of the immersed part in the upper liquid, which is at a distance L/2 + L/4 = 3L/4 from the hinge. Torque due to F_ B2 is _ B2 = F_ B2 (3L/4) = ( A L g) (3L/4) = 3 4 A L^2 g . Destabilizing torque due to gravity is _g = W (L/2) = ( A L g) (L/2) = 1 2 A L^2 g . Net restoring torque, _ net = _ B1 + _ B2 - _g = ( 3 4 + 3 4 - 1 2 ) A L^2 g = A L^2 g . Moment of inertia of the rod about the hinge, I = M L^2 3 = A L^3 3 . The equation of motion is I = - _ net A L^3 3 = - A L^2 g = - 3g L This represents simple harmonic motion with ^2 = 3g L . Time period, T = 2 = 2 3 L g . Answer: 3 = 1.73
Related: Physics — Oscillations · All PYQ Banks