Gravitational potential energy of the CM

There is a very useful result linking the centre of mass to gravitational potential energy: to compute the total $E_\text{pot,g}$ of a system — an extended object or a set of several bodies — you just need the height of the centre of mass.

Principle — Potential energy and centre of mass

$\ev{E_{\text{pot,g}} = M_{\text{tot}}\,g\,h_{\text{CM}}}$

The proof is immediate: the total potential energy is $\sum_i m_i g h_i = g \sum_i m_i h_i$. But $\sum_i m_i h_i = M_\text{tot}\, h_\text{CM}$ by the very definition of the centre of mass (it is its height, multiplied by the total mass). So all the information about the vertical distribution of the masses condenses into a single number, $h_\text{CM}$.

This enormously simplifies calculations in problems with extended objects or systems of several bodies: instead of summing many $m_i g h_i$ terms, you find the height of the centre of gravity and multiply by the total weight. It is particularly handy when a body changes shape — a chain sliding, a rope unrolling — because you only need to track how the centre of gravity drops.

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Topics: Centre of mass Concepts: Centre of mass · Gravitational potential energy

Related exercises: Challenge: chain on a table · Worked exercise — The cat on the trolley · Rocket exploding mid-trajectory