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 lon would execute SHM with a time period. `T=2pisqrt((m)/(Apg))` where, m is mass of the body and p is density of the liquid.

Explanation

Consider the diagram.

Let the log be pressed let the vertical displacement at the equilibrium position be `x^(0).` At equilibrium,
`mg="buoyant force"=(qAx_(0))g" " [:.m=v_(p)=(Ax_(0))p]`
When it is displaced by a further displacement x, the buoyant force is `A(x_(0)+x)pg`
`"Net restoring force"="Buoyant force"-"Weight"`
`=A(x_(0)+x)pg-mg`
`=(Apg)x" " [:.mg=pAx_(0)g]`
As displacement x is downward and restoring force is upward, we can write
`F_("restoring")=-(Apg)x`
where `k="constant"=Apg`
So, the motion is SHM `(":.Fprop-x)`
Now, `"Acceleration a"=(F_("restoring"))/(m)=-(k)/(m)x`
Comparing with `a=-omega^(2)x`
`rArromega^(2)=(k)/(m)rArromega=sqrt((k)/(m))`
`rArr(2pi)/(T)=sqrt((k)/(m))rArrT=2pisqrt((m)/(k))`
`rArrT=2pisqrt((m)/(Apg))`