Chapter 3
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)} }
Step-by-Step
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)} }