Question

A cylindrical plastic bottle of negligible mass is filled with $310\text{ml}$ of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega$. If the radius of the bottle is $2.5\text{cm}$ then $\omega$ is close to: (density of water ${10}^3{\text{kg/m}}^3$)

• A. $3.75{\text{rad s}}^{-1}$
• B. $1.25{\text{rad s}}^{-1}$
• C. $2.50{\text{rad s}}^{-1}$
• D. $2.50\sqrt{10}{\text{rad s}}^{-1}$

Answer 1

The correct option is D $2.50\sqrt{10}{\text{rad s}}^{-1}$

At equilibrium
$B_0=mg$
Extra boyant force $=\rho Axg$
From NLM,
$mg-B_0-B=ma$
$\therefore B=ma=-\rho Axg=-\left(\pi r^2\rho g\right)x$
$a=-\frac{\left(\pi r^2\rho g\right)x}{m}$

Using,
$a=-{\omega }^{2}x$
$\Rightarrow \omega =\sqrt{\frac{\pi r^2\rho g}{m}}$
$\Rightarrow \omega \approx 7.9{\text{rad s}}^{-1}$.