Q. Two equal masses $m$ and $m$ are hung from a balance whose scale pan differs in vertical height by $h/2$. The error in weighing in terms of density of the earth $\rho$ is
A) $\frac{1}{3}\pi G\rho mh$
B) $\pi G\rho mh$
C) $\frac{4}{3}\pi G\rho mh$
D) $\frac{8}{3}G\rho mh$
Explanation
(c) Error in weight = difference in weight at two different heights
$= mg\left\lbrack 1 - \frac{2h_{1}}{R} \right\rbrack - mg\left\lbrack 1 - \frac{2h_{2}}{R} \right\rbrack$
$= \frac{2mg}{R}\left( h_{2} - h_{1} \right) = \frac{2m}{R} \times \frac{GM}{R^{2}} \times \frac{h}{2}$
[where, $h_{2} - h_{1} = h_{2}$]
$= \frac{2m}{R^{3}} \times G \times \frac{4}{3}\pi R^{2}\rho \times \frac{h}{2} = \frac{4}{3}\pi Gm\rho h$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.