A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period where m is mass of the body and ρ is the density of the liquid.

contexto.120

3 years ago

Answer:

Let us consider that the verticle displacement at the equilibrium position is $ {{x}_{0}} $

At equilibrium

$m g = B u o y a n t F o r c e =$ $ A{{x}_{0}}\rho g $

When it is displaced further by a displacement x, the buoyant force is $ A({{x}_{0}}+x)\rho g $

The net restoring force then becomes,

F = B u o y a n t F o r c e - w e i g h t

$ F=A({{x}_{0}}+x)\rho g-mg=A\rho gx $

F \propto x

[We have taken mod value]

So, we get

$ T=2\pi \sqrt{\frac{m}{A\rho g}} $