Question

If at depth ‘$d$’ the gravitational force acting on a particle is 300 N, then what is the force on a particle at depth ‘$\frac d2$’ ?

Answer 1

The correct answer is 150 N.

We know that the gravitational force acting on a particle is directly proportional to the mass of the particle and the acceleration due to gravity.
$F = m \times g$, where F = Force, m = mass, g = acceleration due to gravity

Now, if we consider the situation where we have two particles, one at depth ‘d’ and another at depth ‘d/2’ we can use the formula above.
$F_d=m\times g$

Given: $F_d= 300N$
Similarly for the particle at $\frac{d}{2}$:
$F_{\frac{d}{2}}=m \times g _ {new}$

The acceleration due to gravity changes with depth because the gravitational field strength decreases as we move away from the center of the Earth.
Therefore,
$g'=G \times \frac{M}{r^2}$
where, g’ = gravitational field strength, G = gravitational Constant, M = Mass of the Earth and r = distance from the center of the earth.

As the gravitational field strength at depth 'd/2' is half of the gravitational field strength at depth 'd', the force on the particle at depth '$\frac{d}{2}$' is half of the force at depth 'd'.

$F_{\frac{d}{2}}= \frac{F_d}{2}$
$F_{\frac{d}{2}}= \frac{300}{2}$
$F_{\frac{d}{2}}= 150 N$

Therefore, the force on the particle at depth $‘\frac{d}{2}’ = 150 N$