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Question

A sphere of radius R is half submerged in liquid of density ρ. If the sphere is slightly pushed down and released, it oscillates, then what is the frequency of its oscillation ?

A
12πg2R
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B
12π2g3R
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C
12πg3R
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D
12π3g2R
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Solution

The correct option is D 12π3g2R
A sphere of radius R is half submerged in a liquid of density ρ.
At equilibrium ,
Weight of sphere = Upthrust of liquid on sphere.


Vσg=V2ρg,
where σ = density of sphere.
σ=ρ2 ......(1)
From this position, the sphere is slightly pushed down. Upthrust of liquid on the sphere will increase and it will act as restoring force.
Restoring force = Upthrust due to extra immersion
F = - (extra volume immersed) ×ρg
43πR3×σ×a=πR2ρgx
a=3gρ4Rσx
or a=3g2Rx, [From (1)]
ax
Hence the motion is simple harmonic.

Comparing this with a=ω2x we get,
ω=3g2R
Frequency of oscillation f=ω2π=12π3g2R
Thus, option (d) is the correct answer.

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