The height \({ }^{\prime} h ^{\prime}\) at which the weight of a body will be the same as that at the same depth \('h'\) from the surface of the earth is (Radius of the earth is \(R\) and effect of the rotation of the earth is neglected):

\(M _{1}= mass\) of shaded portion

\(R =\) Radius of earth

\(M_{1}=\frac{M}{\frac{4}{3} \pi R^{3}} \cdot \frac{4}{3} \pi(R-h)^{3}\)

\(=\frac{M(R-h)^{3}}{R}\)

Weight of body is same at \(P\) and \(Q\)

i.e. \(mg _{ P }= mg _{ Q }\)

\(g _{ P }= g _{ Q }\)

\(\frac{G M_{1}}{(R-h)^{2}}=\frac{G M}{(R+h)^{2}}\)

\(\frac{G M(R-h)^{3}}{(R-h)^{2} R^{3}}=\frac{G M}{(R+h)^{2}}\)

\(( R - h )( R + h )^{2}= R ^{3}\)

\(R^{3}-h R^{2}-h^{2} R-h^{3}+2 R^{2} h-2 R h^{2}=R^{3}\)

\(R^{2}-R h^{2}-h^{3}=0\)

\(R^{2}-R h-h^{2}=0\)

\(h ^{2}+ Rh - R ^{2}=0 \Rightarrow h =\frac{- R \pm \sqrt{ R ^{2}+4 R ^{2}}}{2}\)

ie \(h =\frac{- R +\sqrt{5} R }{2}=\left(\frac{\sqrt{5}-1}{2}\right) R\)