Question

At the centre of the earth acceleration due to gravity is:
(A) zero
(B) infinity
(C) $9.8\;$
(D) $98\;$

Answer 1

Gravitational force is the mass of the body times the acceleration due to gravity times at that position. Since the acceleration due to gravity at the centre of the earth is $0$ , the mass times this acceleration is also $0$ and therefore no force will act on any object which may be placed at the centre of the earth.

Complete Answer:
The acceleration due to the gravity of the earth is the acceleration with which anybody close to the surface of the earth will fall towards the centre of the earth when it is in free fall. This value of the acceleration due to gravity can be computed using the law of gravitation.
We can compute the force a body experiences on the surface of the earth and divide that force by the mass of the concerned body to get the approximate value of the acceleration with which it would accelerate when acted upon by this gravitational force. Since this acceleration of about $9.8\;m/{s^2}$ is caused due to the force of gravitation, it is termed as acceleration due to gravity.
The gravitational force of attraction varies depending on whether the body is inside the surface of the earth, say in mines or above the surface of the earth, say in orbit. Now when the body is inside the surface of the earth, the portion of the earth which will be pulling the body will be only the amount of earth inside the Gaussian sphere with a radius equal to the distance of the body from the centre.
Let $R$ be the radius of the earth and $r$ be the distance from the centre of the earth at which the body is present, such that $R > r$ . In such a case the gravitational pull on the body will be only due to the mass of the earth inside the radius of $r$ , i.e. a Gaussian sphere of radius $r$ .
Thus when an object is at the centre of the earth, the value of $r$ becomes $0$ . Thus the gaussian sphere will have a radius of $0$ , i.e. effectively the gaussian sphere will not exist. Due to this no mass of the earth will be acting on the object and so the force on the object will be $0$ .
Thus since the gravitational force on the object is $0$ at the centre of the earth, the acceleration due to gravity will also be $0$ .
The correct answer is option (A) zero.

Note:
The centre of any planet will have the acceleration due to gravity as $0$ . The gravitational forces are attractive on nature only and not repulsive since such forces have not been discovered so far. The acceleration due to gravity and in-turn the force of attraction on a body decreases as we rise above the surface of the earth or go below the surface of the earth, and is the maximum at the poles.