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The Potential Energy of a Mountain

CLMECH-VKYI4L

Mountains on Earth are limited in size by gravitational forces (this is why the largest mountain in the solar system is on the much less massive planet Mars). Derive an equation for the total potential energy of a mountain along a planet, in terms of the mountain's base radius $R$, it's total height $H$, its uniform density $\rho_0$, and the local gravitational acceleration $g$.

For simplicity, we will assume that the mountain is a perfect circular cone and that the mountain is short enough that the gravitational acceleration is constant from base to peak.

A

$\pi R H^3 \rho_0 g$

B

$\pi R^2 H \rho_0 g$

C

$\frac{1}{3}\pi R^2 H^2 \rho_0 g$

D

$\frac{1}{4} \pi R^2 H^2\rho_0 g$

E

$\frac{1}{12} \pi R^2 H^2\rho_0 g$