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                 \[g={{\left( 1+\frac{h}{R} \right)}^{2}}=g\,\left( 1-\frac{2\,h}{R} \right)\] Given,   \[g=\frac{g}{4}\]                 \[\frac{g}{4}=g\,\left( 1-\frac{2h}{R} \right)\] \[\Rightarrow \] \[\frac{1}{4}=1-\frac{2\,h}{R}\] \[\Rightarrow \] \[h=\frac{3\,R}{8}\] Note: The value of acceleration due to gravity decreases on going above or below the surface of earth.