Question

A homogeneous circular cylinder of length h, radius r and specific gravity S, floats in water. It is noted that r = 2/3h Under which one of the followingconditions willthe flotation be unstable?

• A. $0.11\le S<0.22$
• B. $0.22\le S<0.33$
• C. $0.33\le S<0.66$
• D. $0.66\le S\le 0.99$

Answer 1

The correct option is C $0.33\le S<0.66$

For unstable floatation GM < O i.e., metacentric height will be less than zero.
$h^{\prime }\pi r^2=Sh\pi r^2$
$h=h^{\prime }/S$
$OB=h^{\prime }/2$
$OG=h/2$
$BM=I_G/V$
$r=\frac{2}{3}h=\frac{\pi r^4/4}{\pi r^2{h}^{\prime }}=\frac{r^2}{4h^{\prime }}$
$=\frac{r^2}{4Sh}=\frac{{\left(\frac{2}{3}h\right)}^2}{4Sh}=\frac{h}{9S}$
$BG=\frac{h}{2}(1-S)$
$GM=BM-BG$
$=\frac{h}{9S}-\frac{h}{2}(1-S)<0$
$=2-9S+9S^2<0$
$=2-6S-3S+9S^2<0$
$=2(1-3S)-3S(1-3S)<0$
$(1-3S)(2-3S)<0$
$S>\frac{1}{3}andS<\frac{2}{3}$
$So\frac{1}{3}<S<\frac{2}{3}$