Question

Assume gravitational potential energy $‘U^{\prime }$ at ground level to be zero. All objects are made up of same material.
$U_P=$ Gravitational potential energy of solid sphere
$U_Q=$ Gravitational potential energy of solid cube
$U_S=$ Gravitational potential energy of solid cylinder

• A. $U_S>U_P$
• B. $U_Q>U_S$
• C. $U_P>U_Q$
• D. $U_P>U_S$

Answer 1

Answer: (A), (B)

The correct options are
A $U_S>U_P$
B $U_Q>U_S$

$U_{\text{solid-sphere}}=mg{h}_{\text{com}}$
$U_P=\rho .\left[\frac{4}{3}\pi R^3\right]\frac{D}{2}g$ (As $R=\frac{D}{2}$)
$=\rho \left(\frac{\pi }{6}{D}^3\right).\frac{D}{2}g$
$=\frac{\rho \pi D^{4}g}{12}=0.26\rho D^{4}g$
$U_{\text{solid-cube}}=mg{h}_{\text{com}}$
$U_Q=[\rho .D^{3}g]\frac{D}{2}$
$=\frac{\rho D^{4}g}{2}=0.5\rho D^{4}g$
$U_{\text{solid-cylinder}}=mg{h}_{\text{com}}$
$U_S=\rho \left[\frac{\pi }{4}{D}^2\right].D\times \frac{D}{2}$
$=\frac{\rho \pi D^{4}g}{8}=0.39\rho D^{4}g$
Hence, $U_S>U_P$
$U_Q>U_S$
Option A and B are correct.