Question

What is the weight of a body placed at the center of the earth?

Answer 1

Hint: Use Newton's law of gravitation. Take the relation between acceleration due to gravity and force on the body into consideration. Find the relation between force and weight. Then using the relation between weight and acceleration due to gravity, find the weight of the body at the center of the earth.

Formula used:
F=mg
$g= \dfrac {G \rho r}{3}$

Complete step-by-step solution:
By definition, the weight of a body is defined as the force by which the body is attracted by the earth's gravitational pull towards the center of the earth.
According to Newton's Law of universal gravitation,
F=mg ……………….(1)
Where m is the mass of the body
             g is the acceleration due to gravity.
But, as the force and weight are the same.
$\therefore W=mg$ ……………...(2)
Where W is the weight of the body
We know, g at a distance is given by,
$g= \dfrac {G \rho r}{3}$..........(3)
At the center, r=0
Therefore, the equation. (3) becomes,
g=0
From the equation, (2), it can be inferred that the weight of the body is directly proportional to acceleration due to gravity and mass.
Therefore, the equation. (2) becomes,
W= 0
Thus, weight is zero where the acceleration due to gravity is zero.
Hence, the weight of the body placed at the center of the earth is zero.

Note: Acceleration due to gravity is maximum at the surface of the earth. Hence, the weight of a body placed at the surface of the earth is also maximum.
As the earth rotates, the body on the earth’s surface also rotates and thus, experiences a centrifugal force. Using this, we can also find a relation between angular velocity and acceleration due to gravity.