Center of mass

The average position of mass

The center of mass is a weighted average position of a system's mass. For particles along one dimension,

x_{cm}=rac{sum_i m_i x_i}{sum_i m_i}

In vector form,

ec{r}_{cm}=rac{sum_i m_iec{r}_i}{M}

where

$M=sum_i m_i$

The center of mass may be inside an object, outside it, or between separate objects. For example, the center of mass of a ring is at its center, where there may be no material.

Two-particle example

For two masses on a line,

x_{cm}=rac{m_1x_1+m_2x_2}{m_1+m_2}

The center of mass lies closer to the larger mass. If the masses are equal, it lies halfway between them.

This idea generalizes to continuous objects using integrals:

ec{r}_{cm}=rac{1}{M}int ec{r},dm

Velocity of the center of mass

Differentiate the center-of-mass position:

ec{v}*{cm}=rac{dec{r}*{cm}}{dt}=rac{sum_i m_iec{v}_i}{M}

The total momentum of the system is

ec{P}=sum_i m_iec{v}_i

Therefore

ec{P}=Mec{v}_{cm}

This equation gives center of mass deep physical meaning: total momentum is the mass of the system times the velocity of its center of mass.

Acceleration of the center of mass

Differentiating again,

ec{F}*{ext,net}=Mec{a}*{cm}

This says the center of mass moves as if all external force acted on a single particle of mass $M$ located at the center of mass.

Internal forces can cause rotation, vibration, or rearrangement within the system, but they do not affect the acceleration of the center of mass.

Projectile motion of extended objects

If a thrown object rotates in the air, its parts may follow complicated paths. However, if air resistance is negligible, the center of mass follows the same parabolic path as a point projectile. A flipping wrench or tumbling baton has a center of mass that moves simply under gravity.

This is a powerful simplification: complicated internal motion can be separated from overall motion.

Explosions and center of mass

Suppose an object at rest explodes into fragments. If external impulse is negligible, the center of mass remains at rest even though fragments fly apart. The vector momenta of the fragments sum to zero.

If the object was moving before the explosion, the center of mass continues with the same velocity after the explosion.

Stability and balance

An object balances when its center of mass is supported. If the vertical line through the center of mass falls outside the base of support, the object tends to tip. This explains why widening your stance improves stability and why tall, narrow objects tip easily.

Continuous bodies

For a uniform symmetric object, the center of mass lies at the geometric center. For nonuniform objects, more mass on one side shifts the center of mass toward that side. Calculating center of mass with integrals is important for rods, plates, and solids with varying density.

The big idea

The center of mass is the mass-weighted average position of a system. Its motion is governed only by net external force, according to ec{F}*{ext}=Mec{a}*{cm}. This lets us separate overall translational motion from internal motion, making complex systems much easier to analyze.