Angular momentum
This gyroscope remains upright while spinning owing to the conservation of its angular momentum.
Common symbols	L
In SI base units	kg⋅m2⋅s−1
Conserved?	yes
Derivations from other quantities	L = Iω = r × p
Dimension	${\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}$

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In physics, angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs,^[1] rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes^[2] form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

The three-dimensional angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it.

Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body or a fluid, the total angular momentum is the volume integral of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.

Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque. Torque can be defined as the rate of change of angular momentum, analogous to force. The net external torque on any system is always equal to the total torque on the system; in other words, the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's third law of motion). Therefore, for a closed system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant.

The change in angular momentum for a particular interaction is called angular impulse, sometimes twirl.^[3] Angular impulse is the angular analog of (linear) impulse.

Examples

The trivial case of the angular momentum ${\displaystyle L}$ of a body in an orbit is given by

$${\displaystyle L=2\pi Mfr^{2}}$$

${\displaystyle M}$

${\displaystyle f}$

${\displaystyle r}$

The angular momentum ${\displaystyle L}$ of a uniform rigid sphere rotating around its axis, instead, is given by

$${\displaystyle L={\frac {4}{5}}\pi Mfr^{2}}$$

where ${\displaystyle M}$ is the sphere's mass, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the sphere's radius.

Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 10⁴⁰ kg⋅m²⋅s⁻¹, while its rotational angular momentum is about 7.05 × 10³³ kg⋅m²⋅s⁻¹.

In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density is known, the angular momentum ${\displaystyle L}$ is given by

$${\displaystyle L={\frac {16}{15}}\pi ^{2}\rho fr^{5}}$$

where ${\displaystyle \rho }$ is the sphere's density, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the sphere's radius.

In the simplest case of a spinning disk, the angular momentum ${\displaystyle L}$ is given by^[4]

$${\displaystyle L=\pi Mfr^{2}}$$

where ${\displaystyle M}$ is the disk's mass, ${\displaystyle f}$ is the frequency of rotation and ${\displaystyle r}$ is the disk's radius.

If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum ${\displaystyle L}$ is given by^[4]

$${\displaystyle L={\frac {1}{2}}\pi Mfr^{2}}$$

Definition in classical mechanics

Just as for angular velocity, there are two special types of angular momentum of an object: the spin angular momentum is the angular momentum about the object's centre of mass, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.

Orbital angular momentum in two dimensions

Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar).^[5] Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum p is proportional to mass m and linear speed v,