Consider a cylinder, of length L, area of cross section A and density ρ, floating in a liquid of density σ. If the cylinder floats up to depth d in the liquid, then by the law of floatation, the weight of the cylinder equals the weight of the liquid displaced, i.e.,
ALρg = Adσg
∴ L = dσ/p … (1)
Let the cylinder be pushed down by a distance y. Then, the weight of the liquid displaced by the cylinder of length y will exert a net upward force on the cylinder :
F = Ayσg,
which produces an acceleration,
a = \(\frac Fm\) = \(\frac {Ay\sigma g}{AL\rho}\) = \(\frac {\sigma g}{\rho L}\)y = - \(\frac {\sigma g}{\rho (d\sigma/\rho)}\) = - \(\frac gd\)y [from Eq. (1)]
\(\therefore\) Acceleration per unit displacement, |a/y| = \(\frac gd\)
\(\therefore\) period of SHM of the floating cylinder ,
T = \(\frac{2\pi}{ Acceleration \,per \,unit \,displacement,}\) = 2\(\pi\)\(\sqrt{\frac dg}\)
