# Definition:Momentum

## Definition

### Linear Momentum

The linear momentum of a body is its mass multiplied by its velocity.

$\mathbf p = m \mathbf v$

### Particle

Let $P$ be a particle of mass $m$ moving with velocity $\mathbf v$ relative to a point $O$.

The angular momentum of $P$ relative to (or about) $O$ is defined as:

$\mathbf L = \mathbf r \times \mathbf p = m \paren {\mathbf r \times \mathbf v}$

where:

$\mathbf p$ denotes the (linear) momentum of $P$
$\mathbf r$ denotes the position vector of $P$ with respect to $O$
$\times$ denotes the vector cross product.

### Aggregation of Particles

Let $\PP = \set {P_i: i \in I}$ be an aggregation of particles, indexed by $I$, all in motion relative to a point $O$.

For all $i \in I$, let:

the mass of particle $P_i$ be $m_i$
the velocity of particle $P_i$ relative to $O$ be $\mathbf v_i$.

The angular momentum of $\PP$ relative to (or about) $O$ is defined as the sum of the angular momenta of each of the particles in $\PP$:

$\ds \mathbf L = \sum_{i \mathop \in I} \mathbf r_i \times \mathbf p_i = m \paren {\mathbf r_i \times \mathbf v_i}$

where:

$\mathbf p_i$ denotes the (linear) momentum of particle $P_i$ for $i \in I$
$\mathbf r_i$ denotes the position vector of particle $P_i$ for $i \in I$ with respect to $O$
$\times$ denotes the vector cross product.

### Rigid Body

The angular momentum of a rigid body about a point $O$ is the vector cross product of its moment of inertia $\mathbf I$ about $O$ by its angular velocity $\omega$ about $O$:

$\mathbf L = \mathbf I \times \omega$

## Linguistic Note

The plural of momentum is momenta.