Question

A cylindrical cork (area A, length L, density pc) is floating in a liquid of density Pw with only a part of its length L submerged in the liquid. If the cork is pushed down by a small distance Xm and then let go, what is the frequency f of the subsequent bobbing simple harmonic motion? Express your answer in terms of the quantities Pc, Pw, A, L, and g.

Answer 1

Answer:

Explanation:

Given

$\rho _c=$density of cork

$\rho _w=$density of water

L=Length of cylinder

If initially x length is under water

At equilibrium

$\rho _wAxg-\rho _cALg=0$

After giving $X_m$ push

$\rho _wAg(x+X_m)-\rho _cALg=\rho _cALa$

where a  is acceleration of system

and $a=\frac{\mathrm{d^2} X_m}{\mathrm{d} t^2}$

$\rho _wAgX_m=\rho _cAL\frac{\mathrm{d^2} X_m}{\mathrm{d} t^2}$

$\frac{\mathrm{d^2} X_m}{\mathrm{d} t^2}=\frac{\rho _wAg}{\rho _cAL}$

thus $\omega ^2=\frac{\rho _wg}{\rho _cL}$

thus $\omega =\sqrt{\frac{\rho _wg}{\rho _cL}}$

and $2\pi f=\omega$

$f=\frac{\sqrt{\frac{\rho _wg}{\rho _cL}}}{2\pi }$