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Body in Water

A plank of ar...

Question

A plank of area of cross section A is half immersed in liquid 1 of density and half in liquid 2 of density 2. What is the period of oscillation of the plank, if it is slightly depressed downwards

A

T = 2(m/3Ag)

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B

T = 2(m/Ag)

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C

T = 2(2m/3Ag)

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D

T = 2(3m/2Ag)

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Solution

The correct option is B T = 2(m/Ag)
The net buoyancy force acting on the block if it is given a downward displacement x is = $- 2 x A g - (- x A g) = - A g x$
Since buoyant force acts as restoring force, we can write $m a = - A g x$ or $a = - (A g / m) x = - w^{2} x$
Thus, acceleration is proportional to displacement and the motion is SHM.
The time period of oscillations $T = 2 = 2 / w = 2 (m / A g)$

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