Question

A uniform spherical planet (Radius R) has acceleration due to gravity at its surface \[g\]. Points P and Q located inside and outside the planet have acceleration due to gravity \[\dfrac{g}{4}\]. Maximum possible separation between P and Q is
A. \[\dfrac{{7R}}{4}\]
B. \[\dfrac{{3R}}{2}\]
C. \[\dfrac{{9R}}{4}\]
D. None

Answer 1

Hint:Use the formulae for the variation of the acceleration due to gravity with depth and height from the surface of a planet. Calculate the depth of point P and height of point Q from the surface of the planet and take addition of these distances to determine the maximum possible distance between the points P and Q.

Formulae used:
The variation of acceleration due to gravity \[{g_d}\] with depth is given by
\[{g_d} = g\left( {1 - \dfrac{d}{R}} \right)\] …… (1)
Here, \[g\] is acceleration due to gravity on the surface of the planet, \[d\] is depth from the surface of the plane and \[R\] is radius of the planet.
The variation of acceleration due to gravity \[{g_h}\] with height is given by
\[{g_h} = g\left( {1 - \dfrac{{2h}}{R}} \right)\] …… (2)
Here, \[g\] is acceleration due to gravity on the surface of the planet, \[h\] is height from the surface of the plane and \[R\] is the radius of the planet.

Complete step by step answer:
We have given that the radius of the spherical planet is \[R\] and acceleration due to gravity at its surface is \[g\].We have also given that at the points P and Q which are inside and outside the planet, the value of acceleration due to gravity is \[\dfrac{g}{4}\].We have asked to determine the maximum possible distance between the points P and Q for having the same value of acceleration due to gravity.

Let point P is inside the planet at depth \[d\] and point Q is outside the planet at height \[h\]. Hence, the maximum possible distance between these two points is
\[r = d + h\] …… (3)
Let us first determine the depth of point P from the surface of the planet.Substitute \[\dfrac{g}{4}\] for \[{g_d}\] in equation (1).
\[\dfrac{g}{4} = g\left( {1 - \dfrac{d}{R}} \right)\]
\[ \Rightarrow 1 - \dfrac{d}{R} = \dfrac{1}{4}\]
\[ \Rightarrow \dfrac{d}{R} = 1 - \dfrac{1}{4}\]
\[ \Rightarrow d = \dfrac{{3R}}{4}\]
This is the depth of point P from the surface of the planet.

Let us first determine the height of point Q from the surface of the planet.Substitute \[\dfrac{g}{4}\] for \[{g_h}\] in equation (1).
\[\dfrac{g}{4} = g\left( {1 - \dfrac{{2h}}{R}} \right)\]
\[ \Rightarrow 1 - \dfrac{{2h}}{R} = \dfrac{1}{4}\]
\[ \Rightarrow \dfrac{{2h}}{R} = 1 - \dfrac{1}{4}\]
\[ \Rightarrow h = \dfrac{{3R}}{8}\]
This is the height of point Q from the surface of the planet.
Substitute \[\dfrac{{3R}}{4}\] for \[d\] and \[\dfrac{{3R}}{8}\] for \[h\] in equation (3).
\[r = \dfrac{{3R}}{4} + \dfrac{{3R}}{8}\]
\[ \therefore r = \dfrac{{9R}}{8}\]
Therefore, the maximum possible distance between the points P and Q is \[\dfrac{{9R}}{8}\].

Hence, the correct option is D.

Note:The students should correctly use the formulae for the variation of acceleration due to gravity from the surface of a planet with the depth and height. If these formulae are not used correctly then the final answer for the maximum possible distance between the points P and Q will be incorrect. Also the students should not forget to take addition of these two distances from the surface of the planet.