Definition

Centre of mass

r_{c m} = \frac{1}{M} i \sum m_{i} r_{i}

Statement

The mass-weighted average position of the parts of a system.

Symbols and units

$r_{c m}$	centre-of-mass position	m
$m_{i}, r_{i}$	mass and position of each part	kg, m
$M$	total mass of the system	kg

When it applies

Any system of discrete masses or, in integral form, any continuous mass distribution.

Derivation

Weight each particle's position by its share of the total mass, and sum:

r_{c m} = \frac{\sum _{i} m _{i} r _{i}}{\sum _{i} m _{i}} = \frac{1}{M} i \sum m_{i} r_{i}

Worked examples

Centre of mass of two objects

m₁ = 2.0 kg at x = 0, m₂ = 3.0 kg at x = 5.0 m:

x_{c m} = \frac{( 2.0 ) ( 0 ) + ( 3.0 ) ( 5.0 )}{5.0} = 3.0 m

Closer to the heavier mass, as expected of a weighted average.

Watch out for

The centre of mass need not lie on the object itself — for a ring or a horseshoe, it sits in empty space at the centre.