Formula & Calculator
Angular Momentum
Calculates the angular momentum of a rotating body from its moment of inertia and angular velocity.
Interpretation
Angular momentum is the rotational analogue of linear momentum, defined as the product of moment of inertia and angular velocity: L = I·ω. It is a vector quantity and is conserved in the absence of external torques.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| L | Angular momentum | kg.m2/s |
| I | Mass moment of inertia about the rotation axis | kg.m2 |
| omega | Angular velocity | rad/s |
What it means
Angular momentum (L) is a measure of the rotational motion of a body. For a rigid body rotating about a fixed axis, L = I ω, where I is the moment of inertia about that axis and ω is the angular velocity. Angular momentum is a vector, with direction given by the right‑hand rule. The law of conservation of angular momentum states that if no external torque acts on a system, its total angular momentum remains constant. This principle explains phenomena like the spinning of figure skaters (they spin faster when pulling in arms) and the stability of gyroscopes. In orbital mechanics, it governs the motion of planets. In engineering, angular momentum is important in designing flywheels, gyroscopes, and propulsion systems. The unit is kg·m²/s. The concept extends to quantum mechanics, where it is quantised. Understanding angular momentum is crucial for analyzing rotating machinery and for control systems in aerospace.
Worked example
Angular Momentum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| I | 5 kg·m² |
| ω | 10 rad/s |
| Parameter | Value |
|---|---|
| I | 2 kg·m² |
| ω | 20 rad/s |
Common mistakes
- Moment of inertia I: Use the correct moment of inertia about the axis of rotation.
- Angular velocity ω: In rad/s – not rpm.
- Units: I in kg·m², ω in rad/s → L in kg·m²/s.
- Vector nature: Angular momentum is a vector (axial) – direction matters.
- Conservation: Angular momentum is conserved only if no external torque acts.
Applications
Angular momentum is the rotational analogue of linear momentum, given by the product of moment of inertia and angular velocity. It is a conserved quantity in isolated systems, making it crucial in many engineering disciplines. In aerospace, angular momentum is used to control satellite attitude through reaction wheels. In mechanical systems, it governs the behaviour of gyroscopes, which are used in navigation and stabilisation. The concept also applies to the design of spinning objects like tops, wheels, and turbine rotors, where stability and control depend on angular momentum. In astrophysics, it explains the rotation of celestial bodies. Engineers use angular momentum to design systems that require precise rotational control or energy storage.
- Satellite attitude control systems
- Gyroscope design for navigation and stabilisation
- Flywheel energy storage systems
- Design of spinning machinery (turbines, compressors)
- Astronomical and planetary motion analysis
Frequently Asked Questions
Angular momentum is the rotational analog of linear momentum. For a rigid body rotating about a fixed axis, it is L = I·ω, where I is the moment of inertia about the axis and ω is the angular velocity. It is a vector quantity pointing along the axis of rotation (right‑hand rule).
In SI, angular momentum has units of kg·m²/s. It is conserved in a closed system with no external net torque (τ = 0). Conservation of angular momentum explains many phenomena: an ice skater spinning faster by pulling in arms, and a planet's orbital speed increasing as it approaches the sun.
- Using linear velocity instead of angular velocity – you must use ω, not v; if you have v, convert via v = r·ω.
- Using the wrong moment of inertia – I depends on the axis of rotation; use the appropriate I for that axis.
- Ignoring the vector nature – L is a vector; direction matters in torque calculations.
- Applying L = Iω to a system that is not a rigid body – for a system of particles, L = Σ rᵢ × pᵢ.
The net torque on an object equals the rate of change of its angular momentum: τ = dL/dt. This is the rotational analog of Newton's second law (F = dp/dt). If τ = 0, L is constant.
Linear momentum p = m·v describes translation; angular momentum L = I·ω describes rotation. They are independent in the sense that a translating object can have zero angular momentum (if it moves along a line through the reference point), and a rotating object can have zero linear momentum (if its centre of mass is stationary).
For the same ω, a larger I gives a larger L. This is why a flywheel with a large I stores a lot of angular momentum, which is used to smooth out torque fluctuations in engines.
For a particle of mass m moving with velocity v at a perpendicular distance r from a reference point, L = m·v·r·sinθ = m·v·r_perp. Even though it is moving linearly, it has angular momentum about the point. This is important in orbital mechanics (e.g., conservation of angular momentum in planetary orbits).
A gyroscope's spinning rotor has a large angular momentum. When a torque is applied perpendicular to the spin axis, the gyroscope precesses instead of toppling. This property is used in navigation (gyrocompasses), stabilisation of spacecraft, and in bicycle wheels (gyroscopic effect).
The total angular momentum is the vector sum of the angular momenta of each body: L_total = Σ L_i. If all bodies rotate about the same axis, L_total = (Σ I_i)·ω. If they have different axes, you must use vector addition.
In quantum mechanics, particles have spin (intrinsic angular momentum) independent of their orbital motion. Orbital angular momentum arises from the motion of a particle about a centre. Both are quantised. In classical mechanics, we only deal with orbital angular momentum of macroscopic bodies.