A solid sphere of radius R is half submerged in a liquid of density ρ . The sphere is slightly pushed down and released, If the frequency of small oscillations is 
, then find n. Take π = 
.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(8)
Half of the volume of sphere is submerged.
For equilibrium of sphere,
weight = upthrust
∴ V ρ s g = 
( ρ L ) (g)
⇒ ρ s = 
When slightly pushed down by x weight will remain as it is while upthrust will increase. The increased upthrust will become the net restoring force (upwards).
F = – (extra upthrust)
= (extra volume immersed) ( ρ L ) (g)
or ma = –( π R 2 ) x ρ L g (a = acceleration)
∴ 
π R 3 
a = – ( π R 2 ρ L g) x
∴ a = 
x
as a ∝ – x motion is simple harmonic
Frequency of oscillation,
f = 
= 
.
f = 
∴ N = 8
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