Question

A cylindrical cork of radius R and length L is floating in a liquid such that its axis of symmetry is vertical and half of the length of the cork is below the fluid (the density of the fluid is p), as in Figure 2. For this problem, ignore any effects from friction, drag and the viscosity of the fluid (a) Show that the density of the cork, Pe is half that of the fluid (Pe ps) (b) If the cork is displaced vertically by a distance x (x < from its equilibrium (either pushed down or pulled up slightly compared to the depiction in the figure), show that the net force on the cork is given by: where the net force is in the opposite direction of the displacement, r (c) If you press down on the cork slightly (so that part of the cork is still above the fluid) and release it, the cork will oscillate up and down. What should the length of the cork be for it to oscillate with a frequency of 5 Hzi? liquid Figure 2: A cork floating in a liquid.

Answer 1

a)

Volume of cork $V={\pi R^2L}$,

Volume of displaced water $V'=\frac{V}{2}=\frac{\pi R^2L}{2}$

When half of the cork is in fluid,

Force of buoyancy $F_B=\rho_f V' g$ , Force due to gravitation $F_g=mg=\rho_c Vg$

$F_B=F_g$

$\rho_f\left ( \frac{\pi R^2 L}{2} \right )g=\rho_c(\pi R^2L)g$   $\Rightarrow \rho_c=\frac{\rho_f}{2}$

Density of cork is $\rho_c=\frac{\rho_f}{2}$

b)

When more than half part $\frac{L}{2}&plus;x$ of cork is in fluid,

$F_B=\rho_f\,\pi R^2\left ( \frac{L}{2}&plus;x \right ) g$

$F_g=\rho_c\,\pi R^2Lg$

Net force on the cork is $F=F_g-F_B=\rho_c\,\pi R^2Lg-\rho_f \,\pi R^2 \left ( \frac{L}{2}&plus;x \right )g$

$F=\frac{\rho_f}{2}\,\pi R^2Lg-\rho_f \,\pi R^2 \left ( \frac{L}{2}&plus;x \right )g$

That is $F=- \,\pi R^2 \rho_fgx$

c)

The cork is oscillating up and down,

Net force on the cork is $F=ma$ ( a is acceleration of cork)

$ma=- \,\pi R^2 \rho_fgx$

$(V\rho_c)\,a=- \,\pi R^2 \rho_fgx$

$(\pi R^2L\rho_c)\,a=- \,\pi R^2 \rho_fgx$

$(\pi R^2L\frac{\rho_f}{2})\,a=- \,\pi R^2 \rho_fgx$

$a=-\left ( \frac{2g}{L} \right )x$ comparing this equation with SHM equation $a=-\omega^2x$

$\omega=\frac{2g}{L}$

$2\pi f=\frac{2g}{L}$

$L=\frac{2g}{2\pi f}$

$L=\frac{2*9.81}{2\pi *5}=0.624\,m$

Length of cork $L=0.624\,m$