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.

Options

A. `(1)/(2pi) sqrt((3g)/(2R))`
B. `(1)/(2pi) sqrt(2R)/(3g)`
C. `(1)/(2pi)sqrt((R )/(g))`
D. `(1)/(2pi)sqrt((5g)/(3R))`

Answer

A

Explanation

Half of the volume of sphere is submerged. For equilibrium of sphere, Weight =upthrust
`:. Vr_(x)g =(V)/(2) (r_(L)) (g), r_(s) =(rho_(L))/(2) ..(1)`
When slightly pushed downwards by `x`, weigth 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") (r_(L)) (g)`
or `ma =- (piR^(2)) xr_(L) (a= "acceleration")`
`:. (4)/(3)piR^(3) ((rho)/(2))a =- (piR^(2)) xr_(L)g (a= "acceleraiton")`
`:. a =- ((3g)/(2R))x` as the given motion is simple harmonic
Frequency of oscillation, `f = (1)/(2pi) sqrt(|(a)/(x)|) =(1)/(2)sqrt((3g)/(2R))`