Two equal masses m and m are hung from a balance whose scale pans differ in vertical height by h. Then the error in weighing, if any, in terms of density of earth $ρ$

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$3 ρπGmh$

$\frac{1}{3} ρπGmh$

$\frac{8}{3} ρπGmh$

$\frac{4}{3} ρπGmh$

answer is A.

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Detailed Solution

for small heights, $g^{1} = g (1 - \frac{2 h}{R}); g - g^{1} = \frac{2 h}{R} g$
Error in weighing
$\begin{array}{l} ΔW = m (g - g^{1}) = \frac{2 hg}{R} m \\ g = \frac{GM}{R^{2}} = \frac{G}{R^{2}} \cdot \frac{4}{3} {πR}^{3} ρ \\ ΔW = \frac{2 hm}{R} \cdot \frac{G}{R^{2}} \cdot \frac{4}{3} {πR}^{3} ρ \end{array}$
$ΔW = \frac{2 hm}{R} \cdot \frac{G}{R^{2}} \cdot \frac{4}{3} {πR}^{3} ρ = \frac{8 hGπρm}{3}$