Question

A cylindrical object of outer diameter 20cm and mass 2kg floats in water with its axis vertical. If it is slightly depressed and then released find time period of the resulting SHM of the object.

Answer 1

Let, $\rho$ = density of the water
A = cross section of the cylinder
m = mass of the cylinder
Initially
Let $x_o$ is the length of cylinder immersed in water, as the cylinder is floating.
Bouyancy force = Gravitational force
$\Rightarrow \rho A{x}_{0}g=mg$
$\Rightarrow \rho A{x}_0=m$ (i)
Now after it is displaced and released,
Let it be x away from its initial positin at any instant.

Now, there will be an upwards
Net force $=\rho A(x+x_0)g-mg=ma$ (ii)
From (i) and (ii)
$\rho A\times g=ma$
Now comparing with SHM equation, $a=w^{2}x$
$w^2=\frac{\rho Ag}{m}=\frac{{10}^3\times 3.14\times {10}^{-2}\times 9.8}{2}$
$\Rightarrow w=12.4$
$\Rightarrow \tau =\frac{2\tau }{w}=\frac{2\times 3.14}{12.4}=0.5065$.