A plank of mass 'm' and area of cross - section `A` is floating in a non - viscous liquid of desity `rho`. When displaced slightly from the mean position, it starts oscillating. Prove that oscillations are simple harmonic and find its time period.

Answer

A, B

Explanation

Two forces are action on the plank: weight and upthrust. Weight is a constant force. So, forget it. Upthrust (if partially immersed ) is a variable force. When displaced downwards by a distance `x`, then
Net restoring force`F_(net) = - ["change or increase in variable force upthrust"]`
= - [(extra immersed volume) (density of liquid) (g)]
` = -[(Ax)(rho)(g)]`
This force is of type `F = - kx`
So, motion is simple harmonic, where `k = rho Ag`
`:. T = 2pi sqrt((m)/(k))` or `T = 2pi sqrt((m)/(rho Ag))`.