# Angular Momentum Calculator

Calculate angular momentum (L = Iω) and rotational kinetic energy (½Iω²) from moment of inertia and angular velocity.

## What this calculates

Angular momentum is the rotational equivalent of linear momentum. Defined as L = Iω, it depends on the moment of inertia (how mass is distributed relative to the rotation axis) and the angular velocity. This calculator computes angular momentum and rotational kinetic energy, with the option to calculate moment of inertia for a point mass.

## Inputs

- **Moment of Inertia Input** — options: Enter I directly (kg·m²), Point mass (I = mr²) — Choose how to provide the moment of inertia.
- **Moment of Inertia (I)** (kg·m²) — min 0 — Moment of inertia of the rotating body (used in direct mode).
- **Mass** (kg) — min 0 — Mass of the point mass (used in point mass mode).
- **Radius** (m) — min 0 — Distance from the axis of rotation (used in point mass mode).
- **Angular Velocity (ω)** (rad/s) — Angular velocity of the rotating body in radians per second.

## Outputs

- **Moment of Inertia** (kg·m²) — I used in the calculation
- **Angular Momentum** (kg·m²/s) — L = Iω
- **Rotational Kinetic Energy** (J) — KE = ½Iω²

## Details

Angular momentum L = Iω is a conserved quantity in the absence of external torques. This conservation law explains why a spinning ice skater speeds up when pulling in their arms (reducing I increases ω to keep L constant) and why gyroscopes resist tilting.

The moment of inertia I depends on how mass is distributed relative to the rotation axis. For a point mass, I = mr². For common shapes: solid sphere I = (2/5)mr², hollow sphere I = (2/3)mr², solid cylinder I = (1/2)mr², thin rod (center) I = (1/12)mL². Larger I means more resistance to changes in rotational motion.

The rotational kinetic energy KE = ½Iω² is the rotational analog of ½mv². A flywheel stores energy in its rotation, used in energy storage systems, vehicles, and power grids. Understanding angular momentum is essential in astrophysics (planetary rotation), engineering (gyroscopes, turbines), and atomic physics (electron orbital angular momentum).

## Frequently Asked Questions

**Q: What is angular momentum?**

A: Angular momentum (L) is the rotational equivalent of linear momentum. It measures the quantity of rotation of an object and is calculated as L = Iω, where I is the moment of inertia and ω is the angular velocity. Like linear momentum, it is conserved when no external torque acts on the system.

**Q: Why does an ice skater spin faster when pulling in their arms?**

A: Conservation of angular momentum: L = Iω is constant. When the skater pulls in their arms, the moment of inertia I decreases (mass is closer to the axis). To keep L constant, ω must increase, so the skater spins faster. Extending the arms increases I and slows the spin.

**Q: What is moment of inertia?**

A: Moment of inertia (I) is the rotational equivalent of mass. It measures how difficult it is to change an object's rotational motion. It depends on both the mass and how that mass is distributed relative to the rotation axis. Mass farther from the axis contributes more to I (I = Σmr²).

**Q: How is rotational KE different from linear KE?**

A: Rotational KE (½Iω²) is the kinetic energy due to spinning, while linear KE (½mv²) is due to straight-line motion. A rolling ball has both: it translates and rotates simultaneously. The total KE is the sum of both components.

**Q: What are the units of angular momentum?**

A: Angular momentum is measured in kg·m²/s (kilogram-meters-squared per second) in SI units. This is equivalent to J·s (Joule-seconds) or N·m·s (Newton-meter-seconds). In quantum mechanics, angular momentum is quantized in units of ħ (reduced Planck constant ≈ 1.055 × 10⁻³⁴ J·s).

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Source: https://vastcalc.com/calculators/physics/angular-momentum
Category: Physics
Last updated: 2026-04-21