Angular Momentum Formula: L = Iω and L = mvr with Examples

The angular momentum formula for a rigid body turning about a fixed axis is $L = I\omega$ , the moment of inertia multiplied by the angular velocity. For a single particle the same quantity is $\vec{L} = \vec{r} \times \vec{p}$ , which has size $L = mvr\sin\theta$ and reduces to $L = mvr$ when the velocity is at right angles to the radius.

Angular momentum is the turning version of linear momentum. You measure it in kg m²/s, a unit identical to the joule second (J s). It stays conserved whenever no external torque acts, and that one rule explains spinning skaters, collapsing stars and the steady axis of a top. The dimensional formula of angular momentum page derives [M L² T⁻¹] step by step.

Quick facts

Rigid body
$L = I\omega$
Particle
$L = mvr\sin\theta$ , vector form $\vec{L} = \vec{r} \times \vec{p}$
SI unit
kg m²/s, equal to J s
Dimensional formula
[M¹ L² T⁻¹]
Type
Vector along the axis, direction by the right hand rule
Torque link
$\tau = \frac{dL}{dt}$
Conservation
$I_1\omega_1 = I_2\omega_2$ when net external torque is zero

Symbols in the angular momentum formula

Letters in L = Iω and L = mvr
Letter	What it measures	SI unit
$L$	Angular momentum	kg m²/s
$I$	Moment of inertia about the same axis	kg m²
$\omega$	Angular velocity	rad/s
$m, v$	Mass and speed of a particle	kg, m/s
$r$	Position vector from the reference point to the particle	m
$\theta$	Angle between $\vec{r}$ and $\vec{v}$	degree or rad

Both forms describe the same thing. Break a wheel into tiny particles, give each one $L = mvr$ with $v = \omega r$ , and the total is $\sum m r^2 \omega = I\omega$ . The moment of inertia formula and the angular velocity formula are therefore the two inputs you need.

Steps to solve an angular momentum problem

Choose the axis or reference point. L depends on it, just as I does.
Pick the right form. Spinning body: $L = I\omega$ . Particle moving past a point: $L = mvr\sin\theta$ , where $r\sin\theta$ is the perpendicular distance from the point to the line of motion.
Convert to SI. Revolutions per minute become rad/s by multiplying by 2π and dividing by 60. Grams become kg.
Check for external torque. If there is none, write $L_{before} = L_{after}$ and solve for the unknown.
State direction. Wrap your right hand around the axis with the fingers following the spin; the thumb then gives the direction of L.

Solved examples

Example 1: a spinning flywheel

A flywheel with I = 0.8 kg m² spins at 15 rad/s. Find its angular momentum.

$L = I\omega = 0.8 \times 15 = 12\ \text{kg m}^2/\text{s}$

Example 2: a ball passing a point

A 150 g cricket ball travels at 20 m/s along a straight line that passes 40 cm from a stump. Find its angular momentum about the stump.

Convert: m = 0.15 kg, perpendicular distance = 0.40 m.

$L = mvr_{\perp} = 0.15 \times 20 \times 0.40 = 1.2\ \text{kg m}^2/\text{s}$

A particle moving in a straight line still has angular momentum about any point that is not on its line.

Example 3: skater pulling in her arms

A skater spins at 2.0 rad/s with I = 4.0 kg m². She pulls her arms in and I drops to 1.6 kg m². Find the new angular velocity.

No external torque about the vertical axis, so $I_1\omega_1 = I_2\omega_2$ .

$\omega_2 = \frac{4.0 \times 2.0}{1.6} = 5.0\ \text{rad/s}$

Her kinetic energy rises from $\frac{1}{2}(4.0)(2.0)^2 = 8\ \text{J}$ to $\frac{1}{2}(1.6)(5.0)^2 = 20\ \text{J}$ . The extra 12 J comes from the work her muscles do pulling the arms in.

Conservation of angular momentum

Newton's second law for rotation can be written $\tau_{ext} = \frac{dL}{dt}$ . Set the external torque to zero and L cannot change. Internal forces, such as a skater's muscles, can move mass inward or outward, which changes I, but ω must adjust so that the product stays fixed.

The idea works for planets too. A planet's path brings it closer to the Sun and then farther away, yet gravity points along the line to the Sun and exerts no torque about it. So the planet moves faster near the Sun, which is Kepler's second law in another form. In the Bohr model of hydrogen, the electron's orbital angular momentum is limited to $\frac{nh}{2\pi}$ .

Angular momentum is conserved only when the net external torque is zero. Friction at an axle is an external torque.

Units and dimensions

Start from $L = mvr$ . Multiply kilograms by metres per second and then by metres, and you get kg m²/s. One joule equals one kg m²/s², and one more factor of seconds turns it back into kg m²/s, so Planck's constant h shares the unit of angular momentum. Mass times length per time times length gives [M¹ L² T⁻¹].

Do not confuse the two momenta. Linear momentum p is kg m/s; angular momentum has an extra metre. If your answer for L comes out in kg m/s, you have dropped the radius.

Mistakes that cost marks

Taking r as the distance along the path instead of the perpendicular distance to the line of motion.
Using rpm directly in L = Iω. Always convert to rad/s first.
Claiming kinetic energy is conserved when a body changes shape. Only L is conserved; energy can rise or fall.
Mixing axes: I and ω must both refer to the same axis.
Forgetting the vector nature. Two discs spinning in opposite senses have angular momenta that subtract.

For the rest of the rotation toolkit, from torque to rolling, open the rotation and circles hub and work through its pages in order.

Frequently asked questions

What is the formula of angular momentum?

For a rigid body it is L = Iω. For a single particle it is L = mvr sin θ, the cross product of position and linear momentum. Both give kg m²/s.

Which SI unit does angular momentum carry?

Kilogram metre squared per second, kg m²/s. A joule second (J s) works out to exactly the same combination of base units.

Does angular momentum have a direction?

Yes, it is a vector. Its direction lies along the axis of rotation and is found with the right hand rule.

When is angular momentum conserved?

It is conserved whenever the net external torque on the system is zero. Then I₁ω₁ = I₂ω₂ for a body that changes shape.

What is the difference between linear and angular momentum?

Linear momentum p = mv measures motion along a line in kg m/s. Angular momentum L = Iω measures rotation about an axis in kg m²/s, and it depends on the chosen axis.