Two identical blocks of mass $\mathrm{m}$ are suspended from a beam balance whose scale pans differ in vertical height by $\mathrm{h}\left(\ll \mathrm{R}\right)$, If $\mathrm{R}$ and $\mathrm{\rho }$ are the radius and density of the earth, then the error in weighing is

• A

  $\frac{2}{3}{\mathrm{\pi \rho R}}^3\mathrm{Gm}$
• Correct answer
  B

  $\frac{8}{3}\mathrm{\pi \rho Gmh}$
• C

  $\frac{8}{3}{\mathrm{\pi \rho R}}^3\mathrm{Gm}$
• D

  $\frac{4}{3}{\mathrm{\pi \rho Gm}}^2\mathrm{h}$

Solution

As with height $\mathrm{g}$ varies as



and in accord with the figure ${\mathrm{h}}_1>{\mathrm{h}}_2$ so ${\mathrm{W}}_1$ will be lesser than ${\mathrm{W}}_2$ and

${\mathrm{W}}_2-{\mathrm{W}}_1={\mathrm{mg}}_2-{\mathrm{mg}}_1=2\mathrm{mg}\left[\frac{{\mathrm{h}}_1}{\mathrm{R}}-\frac{{\mathrm{h}}_2}{\mathrm{R}}\right]$

or ${\mathrm{W}}_2-{\mathrm{W}}_1=2\mathrm{m}\frac{\mathrm{GM}}{{\mathrm{R}}^2}\frac{\mathrm{h}}{\mathrm{R}}$

$\left[\mathrm{as}\mathrm{ }\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}\mathrm{ }\mathrm{and}\mathrm{ }\left({\mathrm{h}}_1-{\mathrm{h}}_2\right)=\mathrm{h}\right]$

or ${\mathrm{W}}_2-{\mathrm{W}}_1=\frac{2\mathrm{mhG}}{{\mathrm{R}}^3}\left(\frac{4}{3}{\mathrm{\pi R}}^3\mathrm{\rho }\right)$

$=\frac{8}{3}\mathrm{ }\mathrm{\pi \rho Gmh}\mathrm{ }\left[\mathrm{as}\mathrm{ }\mathrm{M}=\frac{4}{3}{\mathrm{\pi R}}^3\mathrm{\rho }\right]$

Hence, the correct answer is option: 'B'.