Question

The figure shows a uniform metal ball suspended by a thread of negligible mass from an upright cylinder that is partially submerged in water. The cylinder has height $6cm$, face area $11c{m}^2$ on the top and bottom, and density $0.5g/c{m}^3$. $4cm$ of the cylinder's height is inside the water surface and density of the metal ball is $8g/c{m}^3$. If R is the radius of the ball such that $R^3=\frac{3}{\alpha }c{m}^3$, where $\alpha$ is an integer, then $\alpha$ is (${\rho }_w=1g/c{m}^3$ and the system is in equilibrium). Write upto two digits after the decimal point.

Answer 1

Taking cylinder and the ball as a system,
Weight of the system = buoyant force
$\frac{4}{3}\pi R^3.{\rho }_2.g+Ah.{\rho }_{1}g=\frac{4}{3}\pi R^3.{\rho }_w.g+A{h}_1.{\rho }_{w}g$
$\Rightarrow R^3=\left[\frac{3A(h_1{\rho }_w-h{\rho }_1)}{4\pi ({\rho }_2-{\rho }_w)}\right]$
using values
$A=11c{m}^2$; $h_1=4cm$; ${\rho }_w=1g/c{m}^3$; ${\rho }_1=0.5gm/c{m}^3$; ${\rho }_2=8gm/c{m}^3$
$R^3=\left[\frac{3\times 11(4\times 1-6\times 0.5)}{4\times \left(\frac{22}{7}\right)\times (8-1)}\right]=\left(\frac{3}{8}\right)c{m}^3$