Oscillations

JEE Main level, single correct, medium

9 of 194

Question 9 of 194

A cylindrical block of mass MM and area of cross section AA is floating in a liquid of density ρ\rho and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ____\_\_\_\_.

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Answer: (D) 2πMρAg2 \pi \sqrt{\frac{M}{\rho A g}}

When the block is depressed by distance xx, the additional buoyant force is F=ρAgxF = \rho Agx acting upward as restoring force.
Equation of motion: Md2xdt2=−ρAgxM\frac{d^2x}{dt^2} = -\rho Agx.
This gives ω2=ρAgM\omega^2 = \frac{\rho Ag}{M}.
Period of oscillation: T=2πω=2πMρAgT = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{M}{\rho Ag}}

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