A cylindrical cork of uniform density floats in a liquid of density _1. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density _2, then the similar oscillation has time period 2T. The value of _2/ _1 is :
A. 1/4
Let m be the mass of the cork and A be its cross-sectional area. When the cork is depressed by a small distance x in a liquid of density _L, the extra buoyant force provides the restoring force: F = -A _L g x The acceleration of the cork is: a = - A _L g m x Comparing this with the standard equation of SHM a = - ^2 x, we get: = A _L g m The time period of oscillation is: T = 2 = 2 m A _L g This shows that T 1 _L . For the first liquid of density _1: T 1 _1 For the second liquid of density _2: 2T 1 _2 Dividing the two equations: 2T T = _1 _2 2 = _1 _2 Squaring both sides: 4 = _1 _2 _2 _1 = 1 4 Answer: 1/4
Related: Physics — Oscillations · All PYQ Banks