Question

A cylindrical block of density $\rho$ is partially immersed in a liquid of density $3\rho$. The plane surface of the block remains parallel to the surface of the liquid. The height of the block is $60cm$. The block performs SHM when displaced from its mean position. [Use $g=9.8m/s^2$]

• A. the maximum amplitude is $20cm$.
• B. the maximum amplitude is $40cm$
• C. the time period will be $\frac{2\pi }{7}$ seconds.
• D. $none$

Answer 1

Answer: (A), (C)

The correct options are
A the maximum amplitude is $20cm$.
C the time period will be $\frac{2\pi }{7}$ seconds.

When cylindrical block is partially immersed

$F_B=mg\Rightarrow 3\rho Ayg=\rho A(60\times {10}^{-2})g$

$y=20cm$

$\Rightarrow Maximumamplitude=20cm$

Restoring force when it is slightly depressed by an amount of x.

$F=-\left(\Delta V\sigma g\right)=-\left(A\sigma g\right)x$

$T$= $2\pi \sqrt{\frac{m}{A\sigma g}}$ = $2\pi \sqrt{\frac{\rho Ah}{3A\rho g}}$ = $2\pi \sqrt{\frac{h}{3g}}$

$=2\pi \sqrt{\frac{60\times {10}^{-2}}{3\times 9.8}}=\frac{2\pi }{7}$