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

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

Explanation

Weight = Buoyancy
`(4)/(3)piR^(3)sigmag=(2)/(3)piR^(3)rhog" "implies" "sigma=(rho)/(2)" "," "sigma`=density of shpere
when sphere is slightly pushed down by x and released. Restoring force will be provioded by extra buoyancy.
`F_(R)=-(piR^(2)x)rhog" "implies" "F_(R)=-piR^(2)rhogx`
`k=piR^(2)rhog`
`n=(1)/(2)pisqrt((k)/(m))" "implies" "n=(1)/(2)pisqrt((piR^(2)rhog)/((4)/(3)piR^(3)sigma))" "impliesn=(1)/(2)pisqrt((3)/(4)(rho)/(sigma)(g)/(R))`
by putting the value of `sigma`, we get, `n =(1)/(2)pisqrt((3g )/(2R))`