How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Angular Momentum used?

Rotational Motion, engineering, research, and related scientific disciplines.

What formulas does this use?

All standard angular momentum formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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MECHANICSRotational MotionPhysics Calculator

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Angular Momentum - Rotational Conservation of Motion

Angular momentum is the rotational equivalent of linear momentum. This calculator helps you determine angular momentum using multiple methods: mass-velocity-radius, moment of inertia-angular velocity, or shape-based calculations.

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L = Iω is the rotational analog of p = mv Conservation applies when net external torque is zero Figure skater effect: I decreases → ω increases Vector direction follows right-hand rule

Key quantities

L = Iω

Angular Momentum

Key relation

I (kg·m²)

Moment of Inertia

Key relation

ω (rad/s)

Angular Velocity

Key relation

L₁ = L₂

Conservation

Key relation

Ready to run the numbers?

Why: Angular momentum conservation is fundamental to understanding rotating systems from figure skaters to galaxy formation. It explains why pulling in arms increases spin rate and why orbital mechanics work.

How: Angular momentum can be calculated from L = mvr for point masses, L = Iω for rotating bodies, or from the moment of inertia of common shapes.

L = Iω is the rotational analog of p = mvConservation applies when net external torque is zero

Sources:HyperPhysicsNIST

Run the calculator when you are ready.

Calculate Angular MomentumEnter your values and select the calculation method below

🌍 Earth Orbiting Sun

Earth's orbital angular momentum around the Sun

⛸️ Figure Skater Spin

Spinning figure skater with arms extended

🚴 Bicycle Wheel

Spinning bicycle wheel at cruising speed

🎯 Toy Gyroscope

Small gyroscope spinning rapidly

🎠 Merry-Go-Round

Playground merry-go-round with child

🚁 Helicopter Rotor

Main rotor blade during flight

🏀 Basketball Spin

Basketball spinning on finger

🪐 Earth Rotation

Earth spinning on its axis

⚛️ Electron Orbit

Electron in hydrogen atom (classical model)

💨 Wind Turbine

Large wind turbine at operating speed

Enter Your Values

Calculation Method

Calculation MethodSelect calculation method

Input Parameters

Mass (m)Mass of the object

Velocity (v)Linear velocity

Radius (r)Distance from rotation axis

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

⛸️

A figure skater can increase spin rate 5-10× by pulling in arms

— Physics LibreTexts

🌍

Earth's orbital angular momentum is 2.66×10⁴⁰ kg·m²/s

— NIST

⚛️

Electrons have quantized angular momentum L = nℏ

— HyperPhysics

🚁

Helicopter rotors store massive angular momentum at operating speed

— MIT OpenCourseWare

📋 Key Takeaways

• Angular momentum (L) is conserved when no external torque acts on a system — a fundamental conservation law
• For point masses, L = mvr where m is mass, v is velocity, and r is radius
• For rotating bodies, L = Iω where I is moment of inertia and ω is angular velocity
• The figure skater effect: pulling in arms decreases I, so ω increases to conserve L

💡 Did You Know?

⛸️A figure skater can increase spin rate by 5-10× by pulling in their arms, demonstrating conservation of angular momentumSource: Physics LibreTexts

🌍Earth's orbital angular momentum around the Sun is 2.66×10⁴⁰ kg⋅m²/s — one of the largest values in the solar systemSource: NIST

⚛️Electrons in atoms have quantized angular momentum: L = nℏ where n is a quantum number and ℏ is Planck's constantSource: HyperPhysics

🪐Black holes can have angular momentum up to L = GM²/c, the maximum allowed by general relativitySource: Wolfram MathWorld

🚁Helicopter rotors store massive angular momentum — a typical main rotor has L ≈ 10⁵ kg⋅m²/s at operating speedSource: MIT OpenCourseWare

🌌Galaxy rotation curves show conservation of angular momentum explains why outer stars orbit slower than expectedSource: Physics LibreTexts

📖 How Angular Momentum Works

Angular momentum is the rotational equivalent of linear momentum. It measures the "amount" of rotation an object has and is conserved when no external torque acts on the system.

Conservation Principle

When no external torque (τ = 0), angular momentum is conserved: L₁ = L₂. This means I₁ω₁ = I₂ω₂. If moment of inertia decreases, angular velocity must increase to keep L constant.

Two Types

Angular momentum comes in two forms:

• Spin: Rotation about object's own axis (Earth spinning)
• Orbital: Rotation about external point (Earth orbiting Sun)

🎯 Expert Tips

💡 Use Right Moment of Inertia

Moment of inertia depends on both mass distribution and rotation axis. A rod has different I when rotating about center vs end.

💡 Check Conservation Conditions

Angular momentum is only conserved when net external torque is zero. Friction, external forces, or collisions can change L.

💡 Vector Nature Matters

Angular momentum is a vector pointing along the rotation axis. Use right-hand rule: curl fingers in rotation direction, thumb points along L.

💡 Figure Skater Effect

Pulling mass closer to axis decreases I. To conserve L, angular velocity increases — this is why skaters spin faster with arms in.

⚖️ Angular Momentum vs Linear Momentum

Aspect	Angular Momentum (L)	Linear Momentum (p)
Unit	kg⋅m²/s	kg⋅m/s
Symbol	L	p
Formula (point)	L = mvr	p = mv
Formula (body)	L = Iω	N/A
Conservation	When τ = 0	When F = 0
Vector Direction	Along rotation axis	Along velocity
Example	Spinning top	Moving car

What is Angular Momentum?

Angular momentum (L) is the rotational equivalent of linear momentum. It measures the quantity of rotational motion an object has and is a fundamental conserved quantity in physics. Just as linear momentum describes how hard it is to stop an object moving in a straight line, angular momentum describes how hard it is to stop a rotating object.

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Basic Definition

Angular momentum is the product of an object's moment of inertia and its angular velocity, representing the "amount" of rotation.

Symbol: L

SI Unit: kg⋅m²/s

🌀

Types of Angular Momentum

There are two types: spin angular momentum (rotation about own axis) and orbital angular momentum (rotation about external point).

Examples:

Spin: Earth rotating on axis
Orbital: Earth orbiting Sun

⚖️

Conservation Law

Angular momentum is conserved when no external torque acts on a system, a fundamental principle in physics.

Conservation:

L₁ = L₂ (when τ = 0)

I₁ω₁ = I₂ω₂

How to Calculate Angular Momentum

📍 For Point Mass / Orbit

L = m × v × r

m = mass, v = linear velocity, r = radius from rotation axis

Use for: Planets orbiting stars, electrons in atoms, objects on circular paths

🔵 For Rotating Bodies

L = I × ω

I = moment of inertia, ω = angular velocity

Use for: Spinning wheels, rotating disks, gyroscopes, figure skaters

📐 Moment of Inertia for Common Shapes

Point Mass

I = ext{mr}^{2}

Solid Sphere

I = (2/5) ext{mr}^{2}

Hollow Sphere

I = (2/3) ext{mr}^{2}

Solid Cylinder/Disk

I = (1/2) ext{mr}^{2}

Hollow Cylinder/Hoop

I = ext{mr}^{2}

Thin Rod (center)

I = (1/12) ext{mL}^{2}

Thin Rod (end)

I = (1/3) ext{mL}^{2}

When to Use Angular Momentum Calculator

🌍

Astronomy & Astrophysics

Calculate orbital mechanics, planetary rotations, and stellar dynamics.

Planetary orbits
Black hole spin
Galaxy rotation

⚙️

Engineering

Design flywheels, gyroscopes, and rotating machinery.

Flywheel energy storage
Gyroscopic stabilization
Rotating equipment

⛸️

Sports Science

Analyze athletic movements and optimize performance.

Figure skating spins
Diving rotations
Gymnastics

Conservation of Angular Momentum

Classic Examples

⛸️ Figure Skater Spin

When a skater pulls their arms in, moment of inertia decreases, so angular velocity increases to conserve L.

I₁ω₁ = I₂ω₂ → ω₂ = ω₁(I₁/I₂)

🎯 Gyroscope Stability

A spinning gyroscope resists changes to its axis orientation due to conservation of angular momentum.

L = Iω (direction maintained)

Angular Momentum Formulas

Point Mass / Orbital

L = m × v × r = p × r

Rotating Body

L = I × ω

Vector Form

L = r × p (cross product)

Rotational Kinetic Energy

KE = ½Iω² = L²/(2I)

Frequently Asked Questions

What's the difference between angular momentum and linear momentum?

Linear momentum (p = mv) describes straight-line motion, while angular momentum (L = Iω or L = mvr) describes rotational motion. Angular momentum includes a distance component from the rotation axis.

Why does a figure skater spin faster when pulling in their arms?

Conservation of angular momentum: L = Iω must stay constant. When the skater pulls in their arms, moment of inertia (I) decreases. To keep L constant, angular velocity (ω) must increase, making them spin faster.

Is angular momentum always conserved?

Angular momentum is conserved only when there's no net external torque on the system. If external torque is present, angular momentum will change according to τ = dL/dt.

What is the direction of angular momentum?

Angular momentum is a vector pointing along the axis of rotation. Use the right-hand rule: curl your fingers in the direction of rotation, and your thumb points in the direction of L.

Angular Momentum Reference Values

Object	Angular Momentum (kg⋅m²/s)	Type
Electron in hydrogen	1.05 × 10⁻³⁴	Orbital
Spinning basketball	~0.03	Spin
Figure skater (slow)	~10	Spin
Bicycle wheel	~5	Spin
Earth (spin)	7.07 × 10³³	Spin
Earth (orbital)	2.66 × 10⁴⁰	Orbital
Sun	1.63 × 10⁴⁷	Spin
Milky Way Galaxy	~10⁶⁷	Spin

Tips for Angular Momentum Calculations

✅ Best Practices

• Use consistent units throughout calculations
• Check that moment of inertia matches the rotation axis
• Remember L is a vector - direction matters
• Verify conservation applies (no external torque)

❌ Common Mistakes

• Using wrong moment of inertia formula for shape
• Confusing orbital and spin angular momentum
• Forgetting to convert units (RPM to rad/s)
• Ignoring the vector nature of L

👈 START HERE

⬅️Jump in and explore the concept!

Angular Momentum - Rotational Conservation of Motion

🌍 Earth Orbiting Sun

⛸️ Figure Skater Spin

🚴 Bicycle Wheel

🎯 Toy Gyroscope

🎠 Merry-Go-Round

🚁 Helicopter Rotor

🏀 Basketball Spin

🪐 Earth Rotation

⚛️ Electron Orbit

💨 Wind Turbine

Enter Your Values

Calculation Method

Input Parameters

🔬 Physics Facts

📋 Key Takeaways

💡 Did You Know?

📖 How Angular Momentum Works

Conservation Principle

Two Types

🎯 Expert Tips

💡 Use Right Moment of Inertia

💡 Check Conservation Conditions

💡 Vector Nature Matters

💡 Figure Skater Effect

⚖️ Angular Momentum vs Linear Momentum

What is Angular Momentum?

Basic Definition

Types of Angular Momentum

Conservation Law

How to Calculate Angular Momentum

📍 For Point Mass / Orbit

🔵 For Rotating Bodies

📐 Moment of Inertia for Common Shapes

Point Mass

Solid Sphere

Hollow Sphere

Solid Cylinder/Disk

Hollow Cylinder/Hoop

Thin Rod (center)

Thin Rod (end)

When to Use Angular Momentum Calculator

Astronomy & Astrophysics

Engineering

Sports Science

Conservation of Angular Momentum

Classic Examples

⛸️ Figure Skater Spin

🎯 Gyroscope Stability

Angular Momentum Formulas

Point Mass / Orbital

Rotating Body

Vector Form

Rotational Kinetic Energy

Frequently Asked Questions

What's the difference between angular momentum and linear momentum?

Why does a figure skater spin faster when pulling in their arms?

Is angular momentum always conserved?

What is the direction of angular momentum?

Angular Momentum Reference Values

Tips for Angular Momentum Calculations

✅ Best Practices

❌ Common Mistakes

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