Question

A solid cylinder of circular section of diameter d is of material with specific gravity $S_s$ . This floats in a liquid of specific gravity $S_l$. What is the maximum length of the cylinder if equilibrium is to be stable with the cylinder axis vertical?

• A. $\frac{d{S}_l}{2\sqrt{S_s(S_l-S_s)}}$
• B. $\frac{d{S}_l}{\sqrt{8S_s(S_l-S_s)}}$
• C. $\frac{d{S}_l}{\sqrt{2S_s(S_l-S_s)}}$
• D. $\frac{d}{\sqrt{8(S_l-S_s)}}$

Answer 1

The correct option is B $\frac{d{S}_l}{\sqrt{8S_s(S_l-S_s)}}$

$M_{b}g=F_B$

${\rho }_s\left[\frac{\pi d^2}{4}\right]Lg={\rho }_l\left[\frac{\pi d^2}{4}\right]hg$

${\rho }_{s}L={\rho }_{l}h$

$h=\frac{{\rho }_s}{{\rho }_l}\times L$

$orh=\frac{S_s}{S_l}\times L$ ... (i)
Take GM = 0
$\frac{l}{\forall }-BG=0$

$\frac{\frac{\pi d^4}{64}}{\frac{\pi d^2}{4}\times h}-\frac{1}{2}(L-h)=0$

$\frac{d^2}{16h}-\frac{1}{2}[L-h]=0$

$\frac{d^2}{8h}=L-h$

$d=\sqrt{8h(L-h)}$

$=\sqrt{8\frac{S_s}{S_l}L(L-\frac{S_s}{S_l})}$

$=L\sqrt{8\frac{S_s}{S_l}[1-\frac{S_s}{S_l}]}$

$=\frac{L}{S_l}\sqrt{8S_s[S_l-S_s]}$

$L=\frac{d\times S_l}{\sqrt{8S_s(S_l-S_s)}}$