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 S.H.M. with a time period.
T = 2π √

where m is mass of the body and ρ is density of the liquid.

Question

Philoid · Accepted Answer

Here we apply the concept of buoyant force

Let the wood part of the height of the wood be immersed into the water

Here the wood log experiences a buoyant force which is given as,

,

where ρ =density of wood log, V=volume of the water displaced by wood now at equilibrium, the buoyant force is balanced by the weight, i.e.,

------(1)

Now let the wood be displaced by depth into the water,

∴ the volume of the water displaced by the wood is,

∴ the buoyant force

∴ there acts net restoring force on the wood log,

Putting equation (1) in the above we get,

-------(2)

∵ is proportional to

∴ wood performs a simple harmonic motion

Now the in S.H.M., we have

Comparing this equation with equation (2) we get,

For S.H.M, Time period

Substituting the value of k, we get,

Hence proved.

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 S.H.M. with a time period.T = 2π √where m is mass of the body and ρ is density of the liquid.