# Derivation of the Center of Mass Formula

## Torque

To derive a formula in coordinates for the center of mass, we
introduce the concept of **torque**. To define torque, imagine a
single particle at the end of a beam connected to a pivot.

The tendency of the particle to rotate around the pivot
depends on the *force
**F * perpendicular to the connecting rod applied to
the particle, and the
*distance r to the pivot *. We define the torque
*T * by the formula

*T * = *F * *r *

The greater the force or the greater the distance from the pivot, the
larger the torque, and hence the greater the tendency to rotate.

### Group Discussion

If you apply a force which is not perpendicular to the rod, but instead
is at an angle, *t *, to the rod, what torque is being applied?

## Center of Mass

Now consider the problem of finding the center of mass for the two
particle system shown below.

If we try to balance the system on a pivot at a point *x_p * in
between *x_1 * and *x_2 *, both particles exert
torques that tend to tip the beam, rotating it around the pivot. In
order to balance the system on the pivot, we want the torques caused
by each of the particles to cancel each other. Since the force
exerted by each particle is given by its mass times the acceleration
of gravity, *x_p * must be chosen so that

*m_1 g * (*x_1 - x_p *) = *m_2 g * (*x_2 - x_p *)

Rearranging this equation gives

*x_p *(*m_1 * + *m_2 *) = (*x_1 m_1 * + *x_2
m_2 *)

This line of reasoning easily generalizes. If a rigid body is
composed of n particles connected in a straight line, then the
location of the center of mass for the system is *x_cm * where

*(m1 + m2 + . . . + m_n)x_cm = m1 x1 + . . . + m_n x_n *

**Return to: **Centers of Mass and Centroids

The Geometry Center Calculus Development Team
Copyright © 1996 by The Geometry Center.
Last modified: Fri Apr 12 15:46:55 1996