Physics · Engineering Physics · A2 Level

Angular Momentum

Momentum in a straight line is easy to picture: a heavier object, or a faster one, is harder to stop. Rotating objects carry an equivalent quantity — angular momentum — and it turns out to be one of the most powerful conserved quantities in the whole of mechanics, underpinning everything from a spinning skater to a satellite's orbit.

7 min read

The rotational analogue of momentum

Linear momentum is defined as $p = mv$: the product of an object's mass and its velocity. Angular momentum, symbol $L$, is defined in the exactly parallel way, as the product of moment of inertia and angular velocity:

$$L = I\omega$$

Everything you already know about how $I$ stands in for $m$, and how rotational quantities mirror linear ones, applies here directly. An object with a large moment of inertia spinning slowly can carry the same angular momentum as an object with a small moment of inertia spinning fast — exactly as a heavy, slow-moving object and a light, fast-moving object can carry the same linear momentum.

Worked exampleAngular momentum of a rotating disc

A uniform disc has moment of inertia $I = 0.80\text{ kg m}^2$ and rotates at $\omega = 15\text{ rad s}^{-1}$.

Find its angular momentum.

Apply $L = I\omega$ directly.

$L = 0.80 \times 15$

Evaluate, including correct units.

$L = 12\text{ kg m}^2\text{ s}^{-1}$

L = 12 kg m² s⁻¹

Angular impulse: changing angular momentum

In linear dynamics, a constant force $F$ applied for a time $\Delta t$ delivers a linear impulse, producing a change in momentum: $F\Delta t = \Delta p$. Angular motion has a direct parallel. A constant torque $T$ applied for a time $\Delta t$ delivers an angular impulse, producing a change in angular momentum:

$$T\Delta t = \Delta(I\omega)$$

This equation is genuinely useful whenever a torque acts only briefly — a sudden twist, a sharp kick applied off-centre, a sports action — and you want to relate that brief interaction to the resulting change in spin, without needing to know exactly how the angular velocity varied moment-to-moment during the interaction itself.

Real torques aren't always constant. When a torque varies with time, the angular impulse it delivers is found not by simple multiplication but by finding the area under a graph of torque against time — exactly as the area under a force–time graph gives linear impulse. A large brief torque spike and a smaller torque sustained for longer can deliver the same angular impulse, provided the areas under their respective torque–time graphs are equal.

Worked exampleReading angular impulse from a torque–time graph

A machine component experiences a torque that rises linearly from zero to a peak of $18\text{ N m}$ over the first $0.40\text{ s}$, then falls linearly back to zero over the following $0.20\text{ s}$ — a triangular torque–time graph.

Find the angular impulse delivered over the full $0.60\text{ s}$.

The angular impulse equals the area under the torque–time graph — here, the area of a triangle.

Area of a triangle $= \tfrac{1}{2} \times \text{base} \times \text{height}$

Substitute the base (total time) and height (peak torque).

Angular impulse $= \tfrac{1}{2} \times 0.60 \times 18 = 5.4\text{ N m s}$

Angular impulse = 5.4 N m s

Since $T\Delta t = \Delta(I\omega)$, this angular impulse of $5.4\text{ N m s}$ tells you directly how much the component's angular momentum changed over that interval — without ever needing to know the angular velocity at every instant in between.

Looking ahead

The impulse equation $T\Delta t = \Delta(I\omega)$ hints at something important: if no external torque acts on a system at all ($T = 0$), then $\Delta(I\omega) = 0$, meaning the system's angular momentum cannot change. This is the principle of conservation of angular momentum, and it's powerful enough to deserve a lesson of its own — covered next.

Key takeaways

Angular momentum is defined as $L = I\omega$, the rotational analogue of linear momentum $p = mv$.
A constant torque applied for a time $\Delta t$ delivers an angular impulse, producing a change in angular momentum: $T\Delta t = \Delta(I\omega)$.
This directly parallels linear impulse, $F\Delta t = \Delta p$.
For a non-constant torque, the angular impulse delivered equals the area under a torque-against-time graph.
Different torque–time profiles can deliver the same angular impulse provided the areas under their graphs are equal.
When no external torque acts on a system, its angular momentum cannot change — the basis of conservation of angular momentum, covered in the next lesson.

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Rotational Kinetic Energy

Conservation of Angular Momentum