Angular Momentum Calculator
Last Updated: October 29, 2025
Calculate angular momentum from moment of inertia, angular velocity, mass, velocity, and radius using rotational dynamics equations. Essential tool for physics rotational motion, conservation laws, and mechanical engineering analysis.
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Object's resistance to angular acceleration
Rate of rotation about an axis
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Table of Contents
What is Angular Momentum Calculator?
Understanding Rotational Motion and Conservation Laws
The Angular Momentum Calculator is a specialized physics tool that calculates angular momentum (L) from various parameters including moment of inertia, angular velocity, mass, velocity, and radius. Angular momentum is the rotational equivalent of linear momentum and is fundamental to understanding rotational dynamics, conservation laws, and mechanical systems.
This calculator is essential for analyzing rotational systems in physics, engineering, and mechanical applications. It provides accurate calculations for angular momentum using multiple methods, including moment of inertia and angular velocity relationships, point particle calculations, and linear momentum conversions.
Key Concepts
Angular Momentum (L): Rotational equivalent of linear momentum, measured in kilogram-square meters per second (kg⋅m²/s). A vector quantity representing the "rotational inertia" of motion, conserved in isolated systems.
Moment of Inertia (I): Resistance to angular acceleration, measured in kilogram-square meters (kg⋅m²). Depends on mass distribution relative to the axis of rotation - like "rotational mass."
Angular Velocity (ω): Rate of rotation about an axis, measured in radians per second (rad/s). Determines how fast an object rotates.
Conservation of Angular Momentum: Angular momentum is conserved in isolated systems with no external torques. This explains phenomena like spinning ice skaters speeding up when pulling in their arms, and why planets maintain their orbital motion.
Vector Nature: Angular momentum has both magnitude and direction (right-hand rule). For rotation about a fixed axis, it points along the axis.
Physical Interpretation
Angular momentum describes the "rotational motion" of a system, analogous to how linear momentum describes "translational motion." Just as linear momentum p = mv quantifies how difficult it is to stop an object's linear motion, angular momentum L = Iω quantifies how difficult it is to stop an object's rotational motion.
The conservation of angular momentum is one of the most fundamental principles in physics. When a figure skater pulls in their arms, they reduce their moment of inertia. Since angular momentum L = Iω must remain constant (no external torques), their angular velocity increases - they spin faster. This same principle governs planetary motion, the formation of galaxies, and the behavior of spinning tops and gyroscopes.
Relationship to Torque
Angular momentum relates to torque through Newton's second law for rotation: τ = dL/dt, where τ is torque. This means torque causes changes in angular momentum, just as force causes changes in linear momentum. When no external torques act on a system, angular momentum is conserved. This relationship is fundamental to understanding rotational dynamics and is essential in designing rotating machinery, controlling spacecraft attitude, and analyzing mechanical systems.
Historical Development
The concept of angular momentum was developed as part of rotational dynamics in the 18th and 19th centuries, building upon Newton's laws of motion and Euler's work on rigid body dynamics. The conservation of angular momentum is one of the fundamental conservation laws in physics, alongside conservation of energy and linear momentum. It explains phenomena from spinning ice skaters and gymnasts to planetary motion, the formation of black holes, and the behavior of gyroscopes. The conservation principle was crucial in the development of quantum mechanics, where angular momentum quantization explains atomic structure and spectral lines.
Units and Dimensions
SI Unit: kg⋅m²/s (kilogram-square meters per second)
Dimensions: [M][L]²[T]⁻¹ (mass × length² × time⁻¹)
Vector Quantity: Has both magnitude and direction
Relationship: L = Iω = mvr = pr (for point particles)
Formulas and Equations
Angular Momentum Calculation Methods
1. From Moment of Inertia and Angular Velocity
L = Iω
Where:
- • L = Angular momentum (kg⋅m²/s)
- • I = Moment of inertia (kg⋅m²)
- • ω = Angular velocity (rad/s)
Use case: Calculate angular momentum for extended objects rotating about an axis. This is the most general formula for angular momentum, analogous to p = mv for linear motion. Essential for analyzing rotating rigid bodies, flywheels, rotating machinery, and any object with distributed mass rotating about a fixed axis. Used extensively in mechanical engineering, physics, and rotational dynamics analysis.
2. From Mass, Velocity, and Radius (Point Particle)
L = mvr
Where:
- • L = Angular momentum (kg⋅m²/s)
- • m = Mass (kg)
- • v = Linear velocity (m/s)
- • r = Radius (m)
Use case: Calculate angular momentum for point particles moving in circular paths, common in orbital mechanics and rotating systems. This simplified formula is particularly useful for analyzing planets, satellites, electrons in atomic orbits, and any particle following a circular trajectory. The product mvr represents the angular momentum about the center of rotation.
3. From Linear Momentum and Radius
L = pr
Where:
- • L = Angular momentum (kg⋅m²/s)
- • p = Linear momentum (kg⋅m/s)
- • r = Radius (m)
Use case: Convert linear momentum to angular momentum when the motion is perpendicular to the radius vector, useful in collision and orbital problems.
Applications of Angular Momentum
Real-World Uses Across Industries
| Industry | Applications | Importance |
|---|---|---|
| Mechanical Engineering | Gyroscopes, flywheels, rotating machinery, stabilizers, spinning wheels | Critical for stability and control systems |
| Physics Research | Conservation laws, quantum mechanics, atomic structure, particle physics | Fundamental conservation principle |
| Aerospace | Satellite stabilization, reaction wheels, attitude control, navigation systems | Essential for spacecraft orientation |
| Astronomy | Planetary motion, stellar rotation, galaxy dynamics, orbital mechanics | Vital for understanding celestial motion |
| Sports Science | Figure skating, diving, gymnastics, baseball pitching, discus throwing | Key for performance optimization |
Examples of Angular Momentum Calculation
Real-World Applications and Use Cases
Example 1: Spinning Wheel (I and ω)
Given:
- • Moment of Inertia: I = 2.5 kg⋅m²
- • Angular Velocity: ω = 120 rad/s
Step-by-step calculation:
Step 1: Apply angular momentum formula
L = Iω
L = 2.5 × 120
L = 300 kg⋅m²/s
Final Answer
300 kg⋅m²/s
This represents the angular momentum of the spinning wheel
Example 2: Point Particle in Circular Motion
Given:
- • Mass: m = 0.5 kg
- • Velocity: v = 10 m/s
- • Radius: r = 2 m
Step-by-step calculation:
Step 1: Apply point particle angular momentum formula
L = mvr
L = 0.5 × 10 × 2
L = 10 kg⋅m²/s
Final Answer
10 kg⋅m²/s
This represents the angular momentum of the point particle in circular motion