question_answer

If the radius of earth of R then the height h at which the value of g becomes one fourth, will be

A)  \[~~\frac{R}{8}~~~\]                    

B)  \[~~\frac{3R}{8}~~~\]  

C)         \[~~\frac{3R}{4}~~~\]  

D)         \[\frac{R}{2}\]

Correct Answer: B

Solution :

The value of acceleration due to gravity at a height h above the surface of the earth is given by        \[g=\frac{g}{{{\left( 1+\frac{h}{R} \right)}^{2}}}\] where R is radius of earth. When h is negligible compared to R, we have                 \[g=g{{\left( 1+\frac{h}{R} \right)}^{-2}}=g\left( 1-\frac{2h}{R} \right)\]                 Given. \[g=\frac{g}{4}\]                                 \[\frac{g}{4}=g\left( 1-\frac{2h}{R} \right)\]                 \[\Rightarrow \]               \[\frac{g}{4}=g\left( 1-\frac{2h}{R} \right)\] \[\Rightarrow \]               \[\frac{1}{4}-1-\frac{2h}{R}\] \[\Rightarrow \]               \[\frac{2h}{R}=\frac{3}{4}\] \[\Rightarrow \]               \[h=\frac{3R}{8}\] Note: The value of acceleration due to gravity decreases on going above or below the surface of earth.