Question

Consider a solid cylinder of density ${\rho }_s$ cross – sectional area A and height h floating in a liquid of density ${\rho }_l$, as shown in the adjacent figure $({\rho }_l>{\rho }_s)$. It is depressed slightly and allowed to oscillate vertically.

The frequency of small oscillations is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

At equilibrium the net force on the cylinder is zero in the vertical direction; $F_{net}=B-W=0;B\equiv$ the buoyancy and $W\equiv$ the weight of the cylinder.
when the cylinder is depressed slightly by x, the buoyancy increases from $BtoB+\delta B$ where : $\delta B=x{\rho }_{l}Ag$
While the weight W remains the same.
$\therefore$ The force, $F_{net}^{\prime }=B+\delta B-W=\delta B=x|{\rho }_{l}Ag$
The equation of motion is, the therefore,
${\rho }_{s}Ah\frac{d^{2}x}{d{t}^2}=-x{\rho }_{l}Ag$
The minus sign takes into account the fact that x and restoring force are in opposite directions.
$\therefore \frac{d^{s}x}{d{t}^2}=-x\frac{\rho lg}{{\rho }_{s}h}$ and the angular frequency, $\omega =\sqrt{\frac{g\rho l}{h{\rho }_s}}$