Center of Gravity equals Center of Mass if it exists

Theorem

Let $B$ be a body in a gravitational field $\mathbf G$.

Let $B$ have a center of gravity $P$.

Then the center of mass of $B$ is also $P$.

Proof

Let $Q$ denote the center of mass of $B$

First suppose that $\mathbf G$ is uniform.

From Center of Gravity in Uniform Gravitational Field is Center of Mass:

$P = Q$

$\Box$

Now suppose that $\mathbf G$ is non-uniform.

There are two possibilities:

$(1): \quad$ $B$ is barycentric

$(2): \quad$ $B$ is not barycentric.

If $(1)$, then from Center of Gravity of Barycentric Body is Center of Mass:

$P = Q$

If $(2)$, then from Center of Gravity in Non-Uniform Gravitational Field, the forces on $B$ induced by $\mathbf G$ consist of:

a single force $\mathbf F$

a couple $C$ whose plane is perpendicular to the line of action of $\mathbf F$.

The line of action of $\mathbf F$ does not necessarily pass through some fixed point as $B$ rotates in $\mathbf G$.

Hence $B$ has no center of gravity.

All cases are covered, and the result follows.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): centre of mass
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): centre of mass