Calculator
Pick what to solve, enter initial and final values, then calculate.
Example data table
These sample values show how changing inertia changes angular speed.
| Scenario | Iᵢ (kg·m²) | ωᵢ (rad/s) | I𝒻 (kg·m²) | ω𝒻 (rad/s) | Notes |
|---|---|---|---|---|---|
| Skater pulls arms in | 3.20 | 4.00 | 1.60 | 8.00 | Inertia halves, speed doubles. |
| Disk slows by friction | 2.00 | 10.00 | 2.00 | 9.50 | Nonzero torque reduces L slightly. |
| Mass moves outward | 1.10 | 12.00 | 2.20 | 6.00 | Inertia doubles, speed halves. |
Formula used
Angular momentum for rotation about a fixed axis is: L = I · ω
- L is angular momentum (kg·m²/s).
- I is moment of inertia (kg·m²).
- ω is angular speed (rad/s).
If the net external torque is negligible, angular momentum is conserved: Lᵢ ≈ L𝒻 which gives Iᵢωᵢ ≈ I𝒻ω𝒻.
When torque is present, the change is: ΔL = τ · Δt.
This calculator applies L𝒻 = Lᵢ + τΔt before solving.
How to use this calculator
- Select what you want to solve for: ω𝒻, ωᵢ, I𝒻, or Iᵢ.
- Choose an angular speed unit for both inputs and outputs.
- Enter moments of inertia directly, or compute them from a shape.
- Provide the known angular speeds, then press Calculate.
- Set τ and Δt to zero for ideal conservation problems.
- Use the download buttons to export your last result.
Conservation of Angular Momentum: Practical notes
1) What stays constant in an ideal case
When the net external torque is essentially zero, angular momentum stays constant: L = I·ω. That is why a skater spins faster after pulling arms in. In practice, small friction torques cause slow drift, so L is nearly constant for short intervals.
2) Typical input ranges you may encounter
For classroom and lab setups, I often falls between 0.01 and 10 kg·m². Small rotors and toys can be 0.001–0.05 kg·m², while flywheels may exceed 20 kg·m². Angular speed commonly ranges from 0.5 to 200 rad/s, or 5–1900 rpm.
A quick sanity test is to compute the percent difference: 100·|L𝒻−Lᵢ|/Lᵢ. In many demos, values within 1–3% indicate good isolation. If your ω is given in rpm, remember 600 rpm equals about 62.83 rad/s. For slow motion studies, 30 deg/s corresponds to roughly 0.524 rad/s in practice.
3) How inertia options change results
The shape selector helps estimate I from measurable geometry. A solid disk uses ½mr², a hoop uses mr², and a solid sphere uses ⅖mr². Because I scales with r², doubling radius quadruples inertia, strongly reducing ω for the same angular momentum.
4) Unit awareness for angular speed
This calculator converts rpm and deg/s to rad/s internally. Use 1 rpm = 2π/60 rad/s and 1 deg/s = π/180 rad/s. If you are validating against a textbook, keep units consistent: inertia in kg·m² and torque in N·m.
5) Including torque for real systems
When torque is not negligible, the change is ΔL = τ·Δt. For example, τ = 0.20 N·m applied for 3.0 s adds 0.60 kg·m²/s of angular momentum. If your predicted ω differs from measurements, the missing term is often torque or time.
6) Worked data-style check
Suppose Iᵢ = 3.2 kg·m² and ωᵢ = 4 rad/s, so Lᵢ = 12.8 kg·m²/s. If I𝒻 becomes 1.6 kg·m² with τ≈0, then ω𝒻 = 12.8/1.6 = 8 rad/s. This mirrors the example table above.
7) Interpreting outputs and exports
Compare Lᵢ and L𝒻 to judge conservation quality. A small mismatch usually reflects rounding or nonzero torque. Export CSV to store scenarios, and export PDF to share a clean report of inputs, units, and computed values for documentation.
FAQs
1) When is angular momentum truly conserved?
It is conserved when the net external torque about the axis is zero or negligible. Internal forces can redistribute mass and change inertia, but they cannot change the total angular momentum without external torque.
2) Why does pulling mass inward increase angular speed?
Pulling mass inward reduces the moment of inertia. With L = I·ω approximately constant, a smaller I requires a larger ω. The energy can increase because work is done while moving inward.
3) Do I need radians per second to use the calculator?
No. You can enter angular speed in rad/s, rpm, or deg/s. The calculator converts everything to rad/s internally, then converts the final angular speed back to your selected unit.
4) What does the torque and time section do?
It applies ΔL = τ·Δt to model non-ideal systems. Positive torque increases angular momentum, negative torque decreases it. Set both values to zero to compute the ideal conservation case.
5) Which inertia shape should I pick?
Choose the closest approximation: disk for solid cylinders, hoop for thin rings, sphere options for balls, and rod options for slender beams. If you already know I from a datasheet, use direct entry.
6) Why are my L values slightly different?
Small differences can come from rounding, unit conversion, or a torque you did not include. Try increasing decimal precision in your inputs and verify that τ and Δt represent the same interval as your measurements.