## Q. 3.13

A cylinder of diameter d with specific gravity s floats on water as shown in Figure 3.31. Show that the permissible length 1 of the cylinder to float in stable equilibrium with axis vertical is

$l\lt \frac{d}{\sqrt{8s(1-s)} }$

## Verified Solution

From Figure 3.31,

OG = $\frac{l}{2}$

Let y be the depth of immersion. Then

Weight of cylinder = Weight of water displaced

$(\frac{\pi }{4} )d^{2}lws=(\frac{\pi }{4} )d^{2}wy$

Therefore,                                        $y=sl$      or       OB = $\frac{sl}{2}$

Now,

OM = OB + BM = $\frac{sl}{2}+\frac{({\pi d^{4} }/{64})}{({\pi d^{2} }/{4})sl{}}=\frac{sl}{2}+\frac{d^{2} }{16sl}$

For stable equilibrium, we have

OM > OG

$\frac{sl}{2}+\frac{d^{2} }{16sl}\gt\frac{l}{2}$

Solving for 1, we get

$l\lt \frac{d}{\sqrt{8s(1-s)} }$