A solid sphere of radius `R` is floating in a liquid of density `sigma` with half of its volume submerged. If the sphere is slightly pushed and released , it starts axecuting simple harmonic motion. Find the frequency of these oscillations.

Answer

`(1)/(2pi)sqrt((3g)/(2R))`

Explanation

A sphere of radius R is half sumerged of density `rho`
For equalibrium of sphere

Wieght of sphere = Upthrust of liquid on sphere.
`:. Vsigmag = (V)/(2) (rho)g`
where `sigma =` density of sphere
`rArr sigma = (rho)/(2)"........."(ii)`
From this position, the sphere is slightly pushed down. Upthrust of liquid on the sphere will increase and it will act the restroning force.
`:.` Restoring force
`F =` Upthrust due to extra-immession
`rArr F = -("extra volume of immersed") xx rhog`
`rArr ("mass of sphere m") xx ac c(la) = -piR^(2)rhogx`
`rArr F = -piR^(2)x xx rhog`
`rArr (4)/(3) piR^(3) xx sigma xx a = -R^(2)rhogx`
`rArr a = - (3grho)/(4Rsigma)x`
`rArr a = - (3g xx 2)/(4R)x`, [from (i)) `= - (3g)/(2R) x`
`rArr` a is proportional to x.
Hene the motion is simple harmonic.
`:.` Frequency of oscillation
`= (1)/(2pi) sqrt((a)/(x)) = (1)/(2pi) sqrt((3g)/(2R))`.