Angular Momentum Calculator – Calculate L = Iω & L = mvr with Formulas

  • Adam
  • November 27, 2025

Calculate angular momentum instantly using moment of inertia (L=Iω) or particle motion (L=mvr). Free calculator with multiple modes, common shapes, conservation principles, and step-by-step solutions for rigid bodies and particles.

Angular Momentum Calculator - Calculate L = Iω & L = mvr

An angular momentum calculator determines the rotational momentum of objects using either moment of inertia and angular velocity (L = Iω) or mass, velocity, and radius for particles (L = mvr sin θ). Angular momentum is a fundamental conserved quantity in physics that describes rotational motion in mechanics, astrophysics, and quantum systems.

Calculate Angular Momentum

Result:

What is Angular Momentum?

Angular momentum is a vector quantity that measures the rotational momentum of an object or system rotating about an axis. It is the rotational analog of linear momentum and represents the tendency of a rotating object to continue spinning. Angular momentum is conserved in isolated systems, making it one of the fundamental conservation laws in physics alongside energy and linear momentum conservation.

Key Characteristics of Angular Momentum:

  • Vector quantity: Has both magnitude and direction determined by the right-hand rule
  • Units: Kilogram meter squared per second (kg·m²/s) or Joule-seconds (J·s)
  • Symbol: L (for orbital/classical) or J (for quantum systems)
  • Conservation: Remains constant in isolated systems with no external torque
  • Dimension: ML²T⁻¹
  • Related to torque: Rate of change of angular momentum equals net torque (τ = dL/dt)

Angular Momentum Formula

Angular momentum can be calculated using different formulas depending on whether you're analyzing rigid body rotation or particle motion. The fundamental equations connect angular momentum to moment of inertia, angular velocity, mass, linear velocity, and position.

Angular Momentum for Rigid Bodies

For a rigid body rotating about a fixed axis with moment of inertia I and angular velocity ω:

L = Iω

Where:

  • L = angular momentum (kg·m²/s)
  • I = moment of inertia (kg·m²)
  • ω = angular velocity (rad/s)

Angular Momentum for a Point Particle

For a point particle with mass m moving at velocity v at distance r from the rotation axis:

L = mvr sin θ

or in vector form:

L = r × p

Where:

  • m = mass of particle (kg)
  • v = velocity of particle (m/s)
  • r = distance from rotation axis (m)
  • θ = angle between position vector r and momentum vector p
  • p = linear momentum = mv (kg·m/s)

Relationship Between Formulas

For a point particle in circular motion, both formulas give the same result:

Since I = mr² for a point mass and v = ωr:

L = Iω = (mr²)ω = mr(rω) = mrv

This confirms both approaches are consistent

Related Equations

Torque and Angular Momentum:

τ = dL/dt

Torque equals rate of change of angular momentum

Conservation of Angular Momentum:

Linitial = Lfinal

When net external torque is zero

Angular Momentum and Rotational Energy:

Erot = L²/(2I)

Moment of Inertia for Common Shapes

The moment of inertia depends on the object's mass distribution. Here are formulas for common geometric shapes:

Shape Axis of Rotation Moment of Inertia (I)
Point Mass Distance r from axis I = mr²
Solid Cylinder/Disk Through center, along axis I = ½mR²
Hollow Cylinder Through center, along axis I = mR²
Solid Sphere Through center I = (2/5)mR²
Hollow Sphere Through center I = (2/3)mR²
Thin Rod Through center, perpendicular I = (1/12)mL²
Thin Rod Through end, perpendicular I = (1/3)mL²

How to Calculate Angular Momentum

Calculating angular momentum requires identifying the system type and selecting the appropriate formula. Follow these systematic steps:

  1. Identify the system type: Determine whether you're analyzing a rigid body rotation or particle motion.
  2. Choose the appropriate formula: Use L = Iω for rigid bodies or L = mvr sin θ for particles.
  3. Find moment of inertia (if needed): For rigid bodies, calculate or look up the moment of inertia based on geometry and mass distribution.
  4. Determine angular velocity: Measure or calculate ω in rad/s (convert from RPM if necessary: ω = 2πn/60).
  5. Calculate magnitude: Substitute values into the formula and compute the angular momentum magnitude.
  6. Determine direction: Apply the right-hand rule: curl fingers in rotation direction, thumb points along angular momentum vector.
  7. Check units: Verify result is in kg·m²/s or equivalent units.

Calculation Example 1: Rotating Disk

Problem: A solid disk with mass 5 kg and radius 0.3 m rotates at 120 RPM. Calculate its angular momentum.

Solution:

Step 1: Calculate moment of inertia for solid disk

I = ½mR² = ½ × 5 × 0.3² = 0.5 × 5 × 0.09 = 0.225 kg·m²

Step 2: Convert RPM to rad/s

ω = 2πn/60 = 2π × 120/60 = 4π = 12.57 rad/s

Step 3: Calculate angular momentum

L = Iω = 0.225 × 12.57

L = 2.83 kg·m²/s

Calculation Example 2: Particle in Circular Motion

Problem: A 2 kg mass moves at 10 m/s in a circle of radius 3 m. Find the angular momentum about the center.

Solution:

Given: m = 2 kg, v = 10 m/s, r = 3 m, θ = 90° (perpendicular motion)

Using formula: L = mvr sin θ

L = 2 × 10 × 3 × sin 90°

L = 60 × 1

L = 60 kg·m²/s

Alternative method:

I = mr² = 2 × 3² = 18 kg·m²

ω = v/r = 10/3 = 3.33 rad/s

L = Iω = 18 × 3.33 = 60 kg·m²/s ✓

Calculation Example 3: Conservation Application

Problem: A figure skater spinning at 2 rev/s with arms extended (I = 4 kg·m²) pulls arms in, reducing moment of inertia to 2 kg·m². Find the new rotation rate.

Solution:

Using conservation of angular momentum: Linitial = Lfinal

I1ω1 = I2ω2

Convert: ω1 = 2 rev/s × 2π rad/rev = 4π rad/s

4 × 4π = 2 × ω2

ω2 = 16π/2 = 8π rad/s = 4 rev/s

The skater spins twice as fast (4 rev/s) after pulling arms in.

Conservation of Angular Momentum

Angular momentum is conserved in systems with no net external torque, leading to important phenomena in nature and technology:

Conservation Principle:

If net external torque τext = 0, then:

Linitial = Lfinal

I1ω1 = I2ω2

Key Implications:

  • When moment of inertia decreases, angular velocity increases proportionally
  • When moment of inertia increases, angular velocity decreases proportionally
  • Total angular momentum remains constant despite internal changes

Real-World Examples of Conservation

  • Figure skating: Pulling arms inward decreases I, increasing spin rate dramatically
  • Diving: Tucking into a ball reduces I for rapid spins, extending body slows rotation for entry
  • Planetary orbits: Planets move faster when closer to the sun (Kepler's second law)
  • Gyroscopes: Maintain orientation due to conservation of angular momentum
  • Neutron stars: Collapsing stars spin extremely fast due to conservation as radius decreases

Applications of Angular Momentum

Understanding and calculating angular momentum is essential across numerous scientific and engineering fields:

  • Astrophysics: Analyzing planetary motion, stellar rotation, black hole accretion disks, and galaxy formation
  • Mechanical Engineering: Designing flywheels, turbines, rotating machinery, and gyroscopic stabilization systems
  • Aerospace: Spacecraft attitude control, satellite stabilization, and orbital mechanics calculations
  • Quantum Mechanics: Describing electron orbital angular momentum, spin, and atomic energy levels
  • Sports Science: Analyzing diving, gymnastics, figure skating, and rotational techniques in various sports
  • Robotics: Controlling robotic arm movements and maintaining balance in humanoid robots
  • Automotive: Analyzing wheel dynamics, engine balancing, and vehicle stability control systems
  • Particle Physics: Understanding fundamental particle properties and conservation laws in collisions

Frequently Asked Questions

What is the formula for angular momentum?
The formula for angular momentum depends on the system. For rigid bodies rotating about a fixed axis, L = Iω where I is moment of inertia and ω is angular velocity. For a point particle, L = mvr sin θ where m is mass, v is velocity, r is distance from axis, and θ is the angle between position and velocity vectors. In vector form, L = r × p (cross product of position and momentum vectors). All formulas give results in units of kg·m²/s.
How do you calculate angular momentum from moment of inertia?
To calculate angular momentum from moment of inertia, multiply the moment of inertia (I) by the angular velocity (ω): L = Iω. First determine the moment of inertia based on the object's geometry (for example, I = ½mR² for a solid disk). Then find the angular velocity in rad/s (convert from RPM if needed using ω = 2πn/60). Finally, multiply: L = Iω. For example, if I = 3 kg·m² and ω = 10 rad/s, then L = 30 kg·m²/s.
What is the unit of angular momentum?
The SI unit of angular momentum is kilogram meter squared per second (kg·m²/s). This can also be expressed as Joule-seconds (J·s) since J = kg·m²/s². The dimensional formula is ML²T⁻¹ where M is mass, L is length, and T is time. In quantum mechanics, angular momentum is often expressed in units of reduced Planck's constant ℏ (h-bar), which equals 1.055 × 10⁻³⁴ J·s.
Is angular momentum conserved?
Yes, angular momentum is conserved in isolated systems with no net external torque. This is one of the fundamental conservation laws in physics. When τext = 0, then Linitial = Lfinal, meaning the total angular momentum remains constant even if individual components change. Conservation of angular momentum explains phenomena like ice skaters spinning faster when pulling arms inward, planetary orbital mechanics, and gyroscope stability. However, if external torques act on the system, angular momentum can change according to τ = dL/dt.
What is the difference between angular momentum and linear momentum?
Linear momentum (p = mv) describes motion in a straight line, while angular momentum (L = Iω or L = r × p) describes rotational motion about an axis. Linear momentum is a vector in the direction of motion with units kg·m/s. Angular momentum is a pseudovector perpendicular to the plane of rotation (determined by right-hand rule) with units kg·m²/s. Linear momentum is conserved when no external forces act; angular momentum is conserved when no external torques act. Both are fundamental conserved quantities but apply to different types of motion.
How does moment of inertia affect angular momentum?
Moment of inertia (I) directly affects angular momentum through the relationship L = Iω. Objects with larger moment of inertia have greater angular momentum for the same angular velocity. When angular momentum is conserved (no external torque), changes in moment of inertia cause inverse changes in angular velocity: if I decreases, ω increases proportionally (I₁ω₁ = I₂ω₂). This explains why figure skaters spin faster when pulling arms inward (reducing I) and slower when extending arms (increasing I).
Can angular momentum be negative?
Angular momentum's sign depends on the chosen coordinate system and rotation direction. By convention, counterclockwise rotation (viewed from above) gives positive angular momentum, while clockwise rotation gives negative angular momentum, following the right-hand rule. The sign indicates direction along the rotation axis. In calculations, negative angular momentum simply means rotation occurs in the opposite direction to the chosen positive orientation. The magnitude (absolute value) represents the strength of rotational motion regardless of direction.
What is the relationship between torque and angular momentum?
Torque is the rate of change of angular momentum: τ = dL/dt. This is Newton's second law for rotational motion, analogous to F = dp/dt for linear motion. A net torque causes angular momentum to change; when net torque is zero, angular momentum remains constant (conservation). For constant moment of inertia, τ = Iα where α is angular acceleration. Integrating over time: change in angular momentum equals impulse torque (ΔL = ∫τdt), similar to how impulse changes linear momentum.
How do you find angular momentum for a particle moving in a straight line?
Even for straight-line motion, a particle can have angular momentum about a point not on its path. Use L = r⊥p = r⊥mv, where r⊥ is the perpendicular distance from the reference point to the line of motion (moment arm). Alternatively, L = rp sin θ = rmv sin θ where θ is the angle between position vector r and momentum vector p. If the particle passes through the reference point, r⊥ = 0 and L = 0 at that instant. Angular momentum remains constant for straight-line motion with no forces (free particle).
What happens to angular momentum when an object stops rotating?
When an object stops rotating, its angular momentum becomes zero because ω = 0 in L = Iω. This requires a net external torque acting over time to change the angular momentum from its initial value to zero: ΔL = ∫τdt. The torque could come from friction, air resistance, or deliberate braking. In the absence of external torques, an object would continue rotating indefinitely (conservation of angular momentum). The energy of rotation converts to heat through friction during the stopping process.

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