Question

Two equal masses $m$ and $m$ are hung from a balance whose scale pans differ in vertical height by 'h'. The error in weighing in terms of density of the earth $\rho $ is :-

• A. $\pi G\rho mh$
• B. $\frac{1}{3}\pi G\rho mh$
• C. $\frac{8}{3}\pi G\rho mh$
• D. $\frac{4}{3}\pi G\rho mh$

Answer 1

Answer: (C)

$g_{1}=\frac{g}{\left[1 + \frac{h}{R}\right]^{2}}=g\left[1 - \frac{2 h}{R}\right]$
$W_{2}-W_{1}=mg_{2}-mg_{1}$
$=2mg\left[\frac{h_{1}}{R} - \frac{h_{2}}{R}\right]=2m\frac{GM}{R^{2}}\frac{h}{R}$
$\left[\because g = \frac{G M}{R^{2}} \text{ and } h_{1} - h_{2} = h\right]$
$\therefore W_{2}-W_{1}=$ error in weighing
$=2mG\cdot \frac{4}{3}\pi R^{3}\rho \frac{h}{R^{3}}=\frac{8 \pi }{3}Gm\rho h$