JEE Advanced 2022 Paper 1 · Physics · Fluid Mechanics, Elasticity and Surface Tension · medium

A uniform solid cylindrical block of density 600 kg m^-3 and height 0.2 m floats upright in water (density 1000 kg m^-3) inside a lift. The lift…

Numerical answer

A uniform solid cylindrical block of density

600\ \mathrm{kg\,m^{-3}}

and height

0.2\ \mathrm{m}

floats upright in water (density

1000\ \mathrm{kg\,m^{-3}}

) inside a lift. The lift accelerates upward at

5\ \mathrm{m\,s^{-2}}

and the block is slightly pressed down and released. Take

g=10\ \mathrm{m\,s^{-2}}

. The time period (in seconds) of the resulting small vertical oscillations, as observed in the lift, is ______ . Give the answer correct to two decimal places.

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Answer: 0.56

In the lift frame the effective gravity is

g'=g+a=15\ \mathrm{m\,s^{-2}}

. For a displacement

x

the extra buoyant force is

\rho_w A x g'

, so the restoring constant is

k=\rho_w A g'

. The mass is

\rho A h

with

h=0.2\ \mathrm{m}

. Thus

T=2\pi\sqrt{\dfrac{\rho h}{\rho_w g'}}=2\pi\sqrt{\dfrac{600\times0.2}{1000\times15}}=2\pi\sqrt{0.008}=0.56\ \mathrm{s}

. Water's inertia is neglected as the container is large.

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