A cylindrical block of wood of mass M is floating in water with its axis vertical. It is depressed a little and then released. Show that the motion of the block is simple harmonic and find its frequency.

Explanation

Suppose a height h of the block is dipped in the water in equilibrium position. If r be the radius of the cylindrical block, the volume of the water displasced `=pir^2h`. For floating in equilibrium
`pir^2hrhog=W`…….i
whre rho is the densilty of water and W the weight of the block.
Now suppose during the vertical motion, the block is further diped through a distance x at some instasnt. The volume of the displascement waster is `pir^2(h+x)`. The forces acting on the block are the weight W vertically downward and the buoyance `pir^2(h+x)rhog` verticaly upward.
Net force on the block at displacement x from the equilibrium position is
`F=W-pir^2(h+x)rho g`
`=W-pir^2hrhog-pir^2rhoxg`
using i
`F=-pir^2rhogx=-kx, where k=pir^2rhog`
Thus the block executes SHM with frequency
`v=1/(2pi) sqrt(k/M)=1/(2pi) sqrt((pir^2rhog)/M)`