Question

at what depth from the surface of the earth will a body of mass 5 kg experience the same force as at a height of R/3 above the earths surface? assume the earth to be a uniform sphere of radius R.

Answer 1

Dear student

The force on a body of mass m is given by mg. where g is the acceleration due to gravity. the value of g changes with the height and depth.
$$\begin{gathered} Thevalueofgatadepthx=g\left(1-\frac{x}{R}\right) \\ andatheighththevalueofg,=\frac{g}{{\left(1+{\displaystyle \frac{h}{R}}\right)}^2} \\ forh<<Ritis=g\left(1-\frac{2h}{R}\right) \\ soaccordingtothequestionh=\frac{R}{3} \\ forceonbodyatheighth=forceonbodyatadepthx \\ m\times gatheighth=m\times gatdepthx \\ g\left(1-\frac{2h}{R}\right)=g\left(1-\frac{x}{R}\right) \\ 1-\frac{2}{3}=1-\frac{x}{R} \\ x=\frac{2}{3}R \end{gathered}$$