Angular Momentum Calculator

Angular Momentum Formula

For a rigid body rotating about a fixed axis, the magnitude of angular momentum is:

L = Iω

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

This mirrors linear momentum p = mv: moment of inertia plays the role of mass, and angular velocity plays the role of linear velocity.

L = I\omega

Vector Form and the Point-Mass View

Angular momentum is a vector: it points along the axis of rotation (by the right‑hand rule). For a single particle, the most general definition is:

\mathbfL = \mathbfr × \mathbfp with \mathbfp = m\mathbfv.

For many particles (or a continuous body), angular momenta add up. When rotation is about a fixed axis and the body is rigid, this total reduces neatly to L = I\,\omega.

\mathbfL = \mathbfr × \mathbfp

Units and Physical Meaning

Angular momentum uses units of kg·m²/s. The factor m² reflects “distance‑squared from the axis,” capturing how mass distribution matters: mass farther from the axis contributes more strongly. A large L means an object is “rotationally stubborn” — it resists changes to its spin.

Conservation of Angular Momentum

If the net external torque is zero, total angular momentum stays constant:

L = constant

Everyday consequences: - Figure skaters spin faster by pulling in their arms (I ↓ → ω ↑ so that L stays the same) - Divers tuck during a flip to speed up, then open to slow down before entry - Planets sweep equal areas in equal times (orbital angular momentum conservation) - Gyroscopes resist tipping and stabilize motion

L = I\omega = \textconstant

Torque–Angular Momentum Connection

Torque is the rotational “push” that changes angular momentum:

\boldsymbol\tau = \dfracd\mathbfLdt.

A nonzero torque changes L over time: increasing or decreasing spin rate, or reorienting the axis.

\boldsymbol\tau = \dfracd\mathbfLdt

Common Moments of Inertia (about symmetry axes)

- Solid disk/cylinder (radius R, mass M): I = 1/2 MR² - Thin hoop/ring (radius R, mass M): I = MR² - Solid sphere (radius R, mass M): I = 2/5 MR² - Thin rod (length L, about center): I = 1/12 ML²

These formulas show how geometry sets I; larger I at the same ω means larger L.

I_\textdisk=\tfrac12MR^2,\; I_\texthoop=MR^2,\; I_\textsphere=\tfrac25MR^2,\; I_\textrod,center=\tfrac112ML^2

Practical Tips and Intuition

- To increase angular momentum, either spin faster (ω ↑) or increase moment of inertia (I ↑), e.g., extend mass outward. - If L is conserved and you reduce I (pull mass inward), ω must rise — you spin faster. - Devices like flywheels store energy in rotation; high I at high ω yields large L and stored energy.

Applications in Engineering

Angular momentum is important in: - Gyroscopes and navigation systems - Flywheels for energy storage - Satellite attitude control - Sports equipment (discus, hammer throw) - Vehicle stability and control - Turbine and rotor design

Worked Example

Inputs

• moment_of_inertia: 3.0
• angular_velocity: 8.0

Result: Angular Momentum: 24.00 kg·m²/s

Explanation

For an object with moment of inertia I = 3.0 kg·m² rotating at ω = 8.0 rad/s:

Calculate the angular momentum: L = Iω L = 3.0 × 8.0 L = 24.0 kg·m²/s

This object has 24.0 kg·m²/s of angular momentum due to its rotation.

Second Scenario

Inputs

• moment_of_inertia: 2.25
• angular_velocity: 8.0

Result: Angular Momentum: 24.00 kg·m²/s

Explanation

This scenario uses different inputs (moment_of_inertia = 2.25, angular_velocity = 8.0) to show how changing one variable affects the angular momentum result. Run the calculator above with these values to get the exact updated output with step-by-step work.