Question

A hole is drilled along the earth’s diameter and a stone is dropped into it. When the stone is at the centre of the earth, it has
(A) Acceleration
(B) Weight
(C) Mass
(D) Potential energy

Answer 1

Hint:Use the formula for acceleration due to gravity at depth. Weight is the gravitational force acting on the stone. Refer to the formula for potential energy to check whether the stone will have potential energy or not.

Complete step by step answer:
We know the variation of acceleration due to gravity at depth d below the earth’s surface.
\[\Rightarrow{g_d} = g\left( {1 - \dfrac{d}{R}} \right)\]
Here, g is the acceleration due to gravity on the surface of earth and R is the radius of earth.
At the centre of earth, the depth equals the radius of the earth. Therefore, above equation becomes,
\[\Rightarrow{g_d} = g\left( {1 - \dfrac{R}{R}} \right)\]
\[ \Rightarrow {g_d} = 0\,m/{s^2}\]
Therefore, the acceleration due to gravity at the centre of the earth is zero.
To answer this question, we need to verify every option.
(A) Acceleration:
The velocity of the stone is only due to the acceleration due to gravity. The stone experiences no other net acceleration on it. Therefore, it has no acceleration at the centre of the earth as acceleration due to gravity is zero.
(B) Weight:
We know that the weight of the stone is its mass times the acceleration.
\[\Rightarrow W = mg\]
Since the acceleration due to gravity is zero at the centre of the earth, the stone is weightless.Therefore, the option (B) is incorrect.
(C) Mass:
The mass of the stone is the amount of matter present in the stone. It will be constant at every position even on the other planet.
(D) Potential energy:
The potential energy of the object is given as,
\[\Rightarrow U = mgh\]
Since the acceleration due to gravity is zero at the centre of the earth, the stone will have zero potential energy.Therefore, the option (D) is incorrect.

So, the correct option is (C).

Note: You may think without acceleration due to gravity, the velocity of the stone will also be zero at the centre of the earth. But the stone already has gained velocity on the way from the surface to the centre, therefore, due to the inertia, the stone will not stop at the centre of the earth.