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

Step-by-step 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}=(\rho _wAg)/(\rho _cAL)`

thus
`\omega ^2=(\rho _wg)/(\rho _cL)`

thus
`\omega =\sqrt{(\rho _wg)/(\rho _cL)}`

and
`2\pi f=\omega`

`f=\frac{\sqrt{(\rho _wg)/(\rho _cL)}}{2\pi }`