Question

Assertion: The difference in the value of acceleration due to gravity at pole and equator is proportional to square of angular velocity of earth.
Reason: The value of acceleration due to gravity is minimum at the equator and maximum at the pole.
A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C) Assertion is correct but Reason is incorrect
D) Both Assertion and Reason are incorrect

Answer 1

To solve this question, we have to understand the concept of acceleration due to gravity and the factors affecting the acceleration due to gravity at the pole and the equator. Here, the variation of the acceleration due to gravity as per the latitude is given by the expression –
$g = {g_0} - R{\omega ^2}{\cos ^2}\theta$
where ${g_0}$= acceleration due to gravity at the equator or zero degree latitude, R = radius of Earth, $\omega$ = angular velocity of Earth and $\theta$ = latitude of the place.

Complete answer:
The Earth exerts a force called gravitational force, based on Newton's law of gravitation, on all objects around it because of the phenomenon of gravitation.
Given that the force is equal to the product of mass and acceleration, the acceleration of any object that is attracted to the Earth due to gravitation is called the acceleration due to gravity.
The acceleration due to gravity depends on a variety of factors. One of them is latitude.
When the Earth is rotating about its own axis, the angular velocity of the entire Earth will not be the same. The central region of the Earth rotates at a very high angular velocity while the extreme ends of the Earth called the poles, rotate at almost zero angular velocity.
Due to the change in the angular velocity, the centripetal force component affecting the acceleration due to gravity results in different acceleration due to gravity across different latitudes of Earth.
The mathematical expression is given by –
$g = {g_0} - R{\omega ^2}{\cos ^2}\theta$
where ${g_0}$= acceleration due to gravity at the equator or zero degree latitude, R = radius of Earth, $\omega$ = angular velocity of Earth and $\theta$ = latitude of the place.
The acceleration due to gravity,
At poles: $\theta = {90^ \circ } \Rightarrow {\cos ^2}\theta = 0$
${g_p} = {g_0}$
At equator: $\theta = {0^ \circ } \Rightarrow {\cos ^2}\theta = 1$
${g_e} = {g_0} - R{\omega ^2}$
The difference,
${g_p} - {g_e} = {g_0} - \left( {{g_0} - R{\omega ^2}} \right)$
$\Rightarrow {g_p} - {g_e} = {g_0} - {g_0} + R{\omega ^2}$
$\Rightarrow {g_p} - {g_e} = R{\omega ^2}$
$\Rightarrow {g_p} - {g_e} \propto {\omega ^2}$
This proves that the difference between the acceleration due to gravity at pole and equator is proportional to the angular velocity of the Earth.
Thus, the assertion statement is correct.
We have seen here that, the value of acceleration due to gravity at pole, ${g_p} = {g_0}$ is maximum and acceleration due to gravity at equator, ${g_e} = {g_0} - R{\omega ^2}$ is minimum.
Hence, the Reason statement is correct.
However, the difference between the accelerations at pole and equator being proportional to angular velocity cannot be reasoned or explained by the fact that their values are maximum or minimum, but only derived by the mathematical definition as explained here, in this problem.
Thus, Reason is not the correct explanation of the Assertion.

Hence, the correct option is Option B.

Note: Based on the fact that the acceleration due to gravity is minimum at equator and maximum at equator, it can be said that a body of same mass will have 0.5% higher weight in the poles compared to the equator. This is because, the weight is force equal to the product of mass and acceleration due to gravity.