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)} }

Step-by-Step

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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)} }

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