SHMMedium
Question
A cylindrical block of mass M and area of cross section A is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is $\_\_\_\_$ .
Options
A.$2\pi\sqrt{\frac{M}{\rho Ag}}$
B.$\pi\sqrt{\frac{2M}{\rho Ag}}$
C.$\pi\sqrt{\frac{\rho A}{Mg}}$
D.$2\pi\sqrt{\frac{\rho\text{ }A}{Mg}}$
Solution
Sol.
At equilibrium
$$\rho Ahg = Mg $$After displacing by x ,
$${Ma = - \rho A(h + x)g + Mg }{Ma = - \rho Ahg - \rho Axg + Mg }{Ma = - \rho Axg }{a = \left( \frac{- \rho Ag}{M} \right)x }$$on comparing with,
$${a = - \omega^{2}x }{\omega = \sqrt{\frac{\rho Ag}{M}} }{T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{M}{\rho Ag}}}$$
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