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Angular Momentum Formula – Definition, Derivation & Examples

The angular momentum formula is L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity. For a single particle, it can also be written as L = r × p (the cross product of the position vector and linear momentum). Angular momentum is a vector quantity measured in kg·m²/s, and it is the rotational analogue of linear momentum, playing the same role in rotational motion that momentum plays in straight-line motion.
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Angular Momentum Formula – Definition, Derivation & Examples

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What Is the Angular Momentum Formula?

The angular momentum formula depends on whether you're analysing a single particle or a rigid rotating body:

  • For a single particle: L = r × p, where r is the position vector from the axis of rotation and p is linear momentum. In magnitude form, L = rp sin θ, where θ is the angle between r and p.
  • For a rigid body rotating about a fixed axis: L = Iω, where I is the moment of inertia of the body about that axis and ω is its angular velocity.

Angular momentum is the rotational analogue of linear momentum — just as p = mv describes the "quantity of motion" in a straight line, the angular momentum equation L = Iω describes the "quantity of rotational motion" of a spinning or orbiting body.

What Is the SI Unit and Dimensional Formula of Angular Momentum?

Property Value
SI Unit kg·m²/s (equivalently, J·s)
CGS Unit g·cm²/s
Dimensional Formula [M¹L²T⁻¹]
Type of Quantity Vector (axial vector, perpendicular to the plane of rotation)

How Is the Angular Momentum Equation Derived from Torque?

Just as force is the rate of change of linear momentum, torque is the rate of change of angular momentum:

τ = dL/dt

For a rigid body with constant moment of inertia I, since L = Iω:

τ = d(Iω)/dt = I(dω/dt) = Iα

This shows that the familiar rotational equation τ = Iα is a special case of the more general angular momentum formula τ = dL/dt — exactly parallel to how F = ma is a special case of F = dp/dt in linear motion. This relationship is also the basis of the law of conservation of angular momentum: when no external torque acts on a system, dL/dt = 0, so L remains constant.

How Does the Angular Momentum Formula Compare to the Linear Momentum Formula?

Feature Linear Momentum Angular Momentum
Formula p = mv L = Iω (or L = r × p)
Related quantity Mass (m) Moment of inertia (I)
Governing law F = dp/dt τ = dL/dt
SI Unit kg·m/s kg·m²/s
Conserved when No external force acts No external torque acts
Quantity type Vector (along direction of motion) Vector (perpendicular to plane of rotation)

This side-by-side comparison is one of the most exam-relevant ways to remember the angular momentum equation, since almost every rotational motion formula has a direct linear-motion counterpart.

What Is the Relationship Between Angular Momentum and Moment of Inertia?

Moment of inertia (I) measures how mass is distributed relative to the axis of rotation — the further the mass is from the axis, the larger the moment of inertia. Since L = Iω, for a fixed angular momentum, a smaller moment of inertia results in a larger angular velocity, and vice versa. This inverse relationship is the reason a figure skater spins faster when pulling their arms inward (reducing I) and slower when extending them outward (increasing I) — a classic real-world illustration of the angular momentum formula in action.

How Do You Solve Problems Using the Angular Momentum Formula? (Solved Examples)

Example 1: Basic angular momentum calculation using L = Iω A disc has a moment of inertia of 2 kg·m² and rotates with an angular velocity of 5 rad/s. Find its angular momentum. Solution: L = Iω = 2 × 5 = 10 kg·m²/s

Example 2: Angular momentum of a particle using L = r × p. A particle of mass 0.5 kg moves with a velocity of 4 m/s in a circle of radius 2 m. Find its angular momentum about the centre. Solution: L = rp sin θ = r(mv) sin 90° = 2 × (0.5 × 4) × 1 = 4 kg·m²/s

Example 3: Finding angular velocity from angular momentum. A rotating wheel has an angular momentum of 40 kg·m²/s and a moment of inertia of 8 kg·m². Find its angular velocity. Solution: ω = L/I = 40/8 = 5 rad/s

Example 4: Torque from rate of change of angular momentum. The angular momentum of a rotating body changes from 20 kg·m²/s to 32 kg·m²/s in 4 seconds. Find the average torque acting on it. Solution: τ = ΔL/Δt = (32 − 20)/4 = 12/4 = 3 N·m

Why Is the Angular Momentum Formula Important for JEE and NEET?

Angular momentum is one of the highest-weightage concepts in the System of Particles and Rotational Motion chapter, contributing 2–3 questions in JEE Main and 1–2 in NEET most years. It also connects directly to later topics — planetary motion in Gravitation (Kepler's second law is a direct consequence of angular momentum conservation), and quantization of angular momentum in Atomic Structure. A strong grasp of the angular momentum formula therefore pays off well beyond this single chapter.

How Should Students Practice Angular Momentum Formula Problems?

  1. Start with direct substitution problems (L = Iω) before attempting L = r × p particle-based questions.
  2. Always check whether the problem involves a rigid body (use L = Iω) or a single particle in circular/general motion (use L = r × p).
  3. Practice torque-angular momentum problems (τ = dL/dt) alongside conservation-of-angular-momentum problems, since exams frequently combine both.
  4. Use the linear-vs-angular momentum comparison table above to avoid mixing up formulas under exam pressure — this is one of the most common error sources in rotational motion numericals.

Which Other eSaral Physics Resources Should Class 11 Students Check Next?

Want to master angular momentum, torque, and rotational motion with solved JEE and NEET problems? eSaral's Class 11 Physics course covers System of Particles and Rotational Motion in depth with concept videos by IIT Bombay faculty and chapter-wise practice tests. Explore Class 11 Physics Courses on eSaral →

Frequently Asked Questions

Find answers to common questions.

Is angular momentum a scalar or a vector quantity?
Angular momentum is a vector quantity, specifically an axial vector, meaning its direction is always perpendicular to the plane of rotation, determined using the right-hand rule.
What is the SI unit of angular momentum?
The SI unit of angular momentum is kilogram-metre-squared per second (kg·m²/s), which is dimensionally equivalent to joule-second (J·s).
What is the formula for angular momentum?
The formula for angular momentum is L = Iω for a rigid body, where I is moment of inertia and ω is angular velocity. For a single particle, it is L = r × p, the cross product of the position vector and linear momentum.
What is the dimensional formula of angular momentum?
The dimensional formula of angular momentum is [M¹L²T⁻¹], derived from L = Iω, since moment of inertia has dimension [M¹L²] and angular velocity has dimension [T⁻¹].
What is the difference between linear momentum and angular momentum?
Linear momentum (p = mv) describes motion in a straight line and depends on mass, while angular momentum (L = Iω) describes rotational motion and depends on moment of inertia, which accounts for how mass is distributed relative to the axis of rotation.
How is the angular momentum formula related to torque?
Torque equals the rate of change of angular momentum: τ = dL/dt. For a rigid body with constant moment of inertia, this simplifies to τ = Iα, the rotational equivalent of F = ma.
Why is angular momentum important for JEE and NEET Physics?
Angular momentum is a high-weightage topic in Rotational Motion and also underlies concepts in later chapters like planetary motion in Gravitation and quantization of angular momentum in Atomic Structure, making it essential across multiple units of the syllabus.
How do you find angular momentum using L = r × p?
For a particle moving with linear momentum p at a position vector r from the axis of rotation, angular momentum is L = r × p, with magnitude L = rp sin θ, where θ is the angle between the position vector and the momentum vector.
What is the relationship between angular momentum and moment of inertia?
Since L = Iω, angular momentum and moment of inertia are directly related for a given angular velocity, but for a fixed angular momentum, a smaller moment of inertia results in a larger angular velocity, and vice versa.
When is angular momentum conserved?
Angular momentum is conserved when no external torque acts on a system, meaning dL/dt = 0. This is why a spinning ice skater rotates faster when pulling their arms in — their moment of inertia decreases, so angular velocity increases to keep L constant.

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