A man in a coal mine experience a loss in his weight from 800 to 777 Newton . Find the depth of coal mine , given that radius of Earth is 6400km

Answer 1

I'm not sure how your class wants you to handle this problem. If you assume a constant density for the earth, then the depth of the coal mine is about 184 km.

There is something called Gauss' Law, which essentially states that the gravity felt would be due to the matter UNDER you. This means that the mass pulling on the miner decreases as they dig down. You would have to find the mass by multiplying density by the volume. Mass is then ρ*(⁴/₃)πr³.

Your gravitational force is then (GM_{earth_underneath}M_miner) / r² = (Gρ[⁴/₃]πr³) / r² = (⁴/₃)Gρπr

You can then just use the ratio of the forces:

F₁ / F₂ = 800 / 777 = {(⁴/₃)Gρπr₁} / {(⁴/₃)Gρπr₂} = r₁ / r₂ = 6400 / r₂.

==> 800 / 777 = 6400 / r₂

==> r₂ = 6400 * (777 / 800) = 6216 km.

==> Depth of the mine = 6400 km - 6216 km = 184 km.