Question

What is the weight (or force of gravity) of a body of mass $m$ at the centre of the earth?
a. $mg$
b. $m/g$
c. $g$
d. $0$

Answer 1

Hint: At the centre of the earth there is no acceleration due to gravity because all the particles equally pull a body placed at the centre of the earth.

Complete step by step answer:
At the centre of the earth, the earth is distributed equally in all directions. Thus, all the mass particles equally pull the body in different directions, thereby cancelling each other out. As a result, there is zero net force acting on the body. Thus, acceleration due to gravity $g$ is zero.
We know that the weight or force of gravity of a body of mass $m$ is $mg$
Now as $g = 0$ the weight of the body will be, $mg = m \times 0 = 0$
Hence, the weight (or force of gravity) of a body of mass $m$ placed at the centre of the earth is zero.

Hence, the correct answer is option (D).

Additional information:
The centre of mass is the average position of all parts of the object or system, weighted according to their masses. The centre of gravity is the point through which the force of gravity acts on an object or system. Both these terms can be used interchangeably as they are often located at the same point. But the centre of mass is independent of gravity. If the object is not in a uniform gravitational field then the centre of gravity can be different from the centre of mass.

Note: Net force acting on the body at the centre of the earth is zero. This does not mean that there is no gravitational pull happening on the body by the earth particles. There is a gravitational pull on the body but it is distributed equally in all the different directions, thereby making the net pull zero.