Question

There is a cylindrical bottle of negligible mass and radius $2.5 \, cm$ floating in the water of density $10^{3} \, kg \, m^{- 3}$ filled with $310 \, ml$ of water inside it. Now the bottle is slightly dipped into water and released. The frequency of oscillation is

• A. $3.75 \, s^{- 1} \, $
• B. $1.25 \, s^{- 1} \, $
• C. $2.75 \, s^{- 1} \, $
• D. $3.00 \, s^{- 1} \, $

Answer 1

Answer: (B)

Let at equilibrium, $y$ length of the bottle is immersed in water:-
So, $mg=f_{b u o}$
$mg=\rho \left(A y\right)g$ ..... $\left(\right.1\left.\right)$ [Equilibrium Condition]
Now it is further pushed by distance $′x′$
$F_{r e s t o r i n g}=\rho \left[\right.Ax + y \left.\right]g-mg$
$m\omega ^{2}x=\rho Axg+\rho Ayg-mg$
$\omega =\sqrt{\frac{\rho A g}{m}}=\sqrt{\frac{\rho A g}{\rho V}}=\sqrt{\frac{\pi r^{2} g}{V}}=2\pi f$
$\therefore f=\frac{1}{2 \pi }\sqrt{\frac{\pi r^{2} g}{V}}=1.25 \, s^{- 1} \, $