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)

$g_1=\frac{g}{{\left[1+\frac{h}{R}\right]}^2}=g\left[1-\frac{2h}{R}\right]$
$W_2-W_1=m{g}_2-m{g}_1$
$=2mg\left[\frac{h_1}{R}-\frac{h_2}{R}\right]=2m\frac{GM}{R^2}\frac{h}{R}$
$\left[\because g=\frac{GM}{R^2}\text{ and }{h}_1-h_2=h\right]$
$\therefore W_2-W_1=$ error in weighing
$=2mG\cdot \frac{4}{3}\pi R^3\rho \frac{h}{R^3}=\frac{8\pi }{3}Gm\rho h$