Question

The acceleration due to gravity at the poles and the equator is ${g_p}$​ and ${g_e}$​ respectively. If the earth is a sphere of radius RE​ and rotating about its axis with angular speed ω, then ${g_p} - {g_e}$ is then given by
(A) \[\dfrac{{{\omega ^2}}}{{{R_E}}}\]
(B) \[\dfrac{{{\omega ^2}}}{{{R_E}^2}}\]
(C) ${R^2}_E{\omega ^2}$
(D) ${R_E}{\omega ^2}$

Answer 1

Hint
In order to solve the problem, we have to take into the account of acceleration due to the gravity at the pole and the equator at a certain latitude. We also have to take in the account of the rotation of the earth. The acceleration due to gravity at a place of latitude $\lambda $ due to the earth’s rotation is;
$g' = g - R{\omega ^2}{\cos ^2}\lambda $
Where $g$ denotes the acceleration due to gravity, $R$ denotes the radius of the earth, $\omega $ denotes the angular velocity,
$\lambda $ denotes the latitude.

Complete step by step solution
Given data:
Acceleration due to gravity at pole is ${g_p}$,
Acceleration due to gravity at the equator is ${g_e}$ .
The acceleration due to gravity at a place of latitude $\lambda $ due to the earth’s rotation is;
$g' = g - R{\omega ^2}{\cos ^2}\lambda $
At the equator
The acceleration due to gravity;
$g' = g - {R_E}{\omega ^2}{\cos ^2}\lambda $
Since at equator the latitude $\lambda = {0^ \circ }$ , $\cos \,{0^ \circ } = 1$;
By substituting we get;
$g' = g - {R_E}{\omega ^2}$
Since $g' = {g_e}$ ;
${g_e} = g - {R_E}{\omega ^2}$
At the pole:
The acceleration due to gravity;
$g' = g - {R_P}{\omega ^2}{\cos ^2}\lambda $
Since at the pole the latitude $\lambda = {90^ \circ }$, $\cos \,{90^ \circ } = {0^ \circ }$;
By substituting we get;
$g' = g$
Since $g' = {g_p}$ we get;
${g_p} = g$
Since we need ${g_p} - {g_e}$
${g_p} - {g_e} = g - g - {R_E}{\omega ^2}$z
By cancelling the like terms, we get;
${g_p} - {g_e} = {R_E}{\omega ^2}$
Therefore, the answer is ${g_p} - {g_e} = {R_E}{\omega ^2}$ .
Hence, the option (D) ${g_p} - {g_e} = {R_E}{\omega ^2}$ is the correct answer.

Note
The earth shape is not circular, it is elongated in shape. Since the shape of the earth is not round the radius of the earth differs at the poles and at the equator. Due to this the gravitational force gets affected. In light of this reason the mass placed in the equator and pole gets affected.