Question

Why is the value of acceleration due to gravity zero at the center of the Earth? Prove with mathematical calculations.

Answer 1

Step 1: Description of acceleration due to gravity

1. The acceleration due to gravity is the measure of gravitational attraction that Earth exerts on an object.
2. This attraction is directed towards the center of the Earth, i.e., the Earth tries to pull everything towards its center.
3. Hence, when the body is at the center there is no need to pull it towards the center further. So, gravitational acceleration is zero at the center of the Earth.

Step 2: Proof

Consider a test mass ($\mathrm{m}$) taken at a depth ($d$) below the Earth’s surface,

On the Earth's surface, the value of gravitational acceleration $g$ in terms of density $\rho$ is given by:

$g=\frac{4}{3}\pi \rho RG$

Where, $R$ is the radius of the Earth,

At a distance ($d$) below the surface of Earth, the gravitational acceleration is given by:

$g_d=\frac{g(R–d)}{R}$

Where, the acceleration due to gravity at that point ($g_d$) is obtained by taking the value of $g$ in terms of density.

At the Earth's center,
$R=d$

Substitute in the above equation.

$g_d=\frac{g(d–d)}{R}$

$g_d=\frac{g\left(0\right)}{R}$

$g_d=0$

Hence, it is proved that at the center of Earth the acceleration due to gravity is zero.