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
12π√g2R
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B
12π√2g3R
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C
12π√g3R
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D
12π√3g2R
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Solution
The correct option is D12π√3g2R 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)] ⇒a∝−x
Hence the motion is simple harmonic.
Comparing this with a=−ω2x we get, ω=√3g2R ∴ Frequency of oscillation f=ω2π=12π√3g2R
Thus, option (d) is the correct answer.