Question

A right circular cylinder of specific gravity $\sigma$ floats in water with its axis vertical, one third volume being above the water. If ${}^{\prime }\rho {}^{\prime }$ is the specific gravity of air then select the correct option.

• A. $3\sigma =3+\rho$
• B. $3\sigma =2+\rho$
• C. $2\sigma =2+\rho$
• D. $3\sigma =1+2\rho$

Answer 1

The correct option is B $3\sigma =2+\rho$

From archimedes's principle
Weight of the cylinder is equal to the sum of weight of the fluids displaced( air and water)
Weight of the cylinder: $mg=\pi r^{2}h\rho g=\pi r^{2}h\sigma {\rho }_{w}g$
Weight of the water displaced: $m_{w}g=\pi r^2\frac{2}{3}h{\rho }_{w}g$
Weight of the air displaced: $m_{air}g=\pi r^2\frac{1}{3}h\rho {\rho }_{w}g$
$\therefore mg=m_{w}g+m_{air}g$
$\therefore \pi r^{2}h\sigma {\rho }_{w}g=\pi r^2\frac{2}{3}h{\rho }_{w}g+\pi r^2\frac{1}{3}h\rho {\rho }_{w}g$
$\therefore 3\sigma =2+\rho$