Angular impulse and momentum formulas
Angular impulse is the accumulated effect of a net torque over time. The angular impulse-momentum theorem states that this impulse equals the change in angular momentum.
The relationship is useful for rotating machinery, sports mechanics, impacts, flywheels, and attitude-control systems where a torque changes rotational motion.
How to use the angular impulse-momentum calculator
- Choose the unknown: Select angular impulse, torque, time, or an angular-momentum value.
- Enter known values: Use values measured about the same axis and retain their signs.
- Calculate: Generate the result using J = τΔt = Lf − Li.
- Interpret direction: A negative result indicates impulse or momentum change opposite the chosen positive rotation.
Formula and variables
For constant net torque, angular impulse equals torque multiplied by time and equals final minus initial angular momentum.
J = τΔt = ΔL = Lf − Li- J — Angular impulse
- Torque accumulated over the interval (N·m·s)
- τ — Net torque
- External torque about the selected axis (N·m)
- Δt — Time interval
- Duration for which the torque acts (s)
- Li, Lf — Angular momentum
- Initial and final angular momentum (kg·m²/s)
Flywheel torque example
A constant 12 N·m torque acts on a flywheel for 4 seconds.
- Net torque
- 12 N·m
- Time
- 4 s
- J = τΔt
- J = 12 × 4 = 48 N·m·s
Result: The angular impulse and change in angular momentum are 48 kg·m²/s.
Angular momentum increases by 48 kg·m²/s in the torque direction.
Understanding your results
Sign and direction
Angular quantities depend on the selected axis. Use one sign convention throughout the calculation.
- Positive impulse increases angular momentum in the positive direction.
- Zero net external torque means angular momentum remains constant.
Assumptions
- Net torque is constant over the entered time interval.
- All quantities refer to the same rotation axis and inertial reference frame.
Limitations
- Variable torque requires the time integral of torque.
- The calculator does not determine moment of inertia or model deformation and friction automatically.
Common mistakes
- Using force instead of torque.
- Mixing milliseconds and seconds.
- Ignoring the direction of initial and final angular momentum.
Practical use cases
Machines and controls
Estimate momentum changes in flywheels, motors, and attitude-control systems.
Sports and impacts
Analyze rotational effects in swings, spins, and short-duration impacts.
Frequently asked questions
When is angular momentum conserved?
Angular momentum is conserved when the net external torque about the selected axis is zero.
Are N·m·s and kg·m²/s equivalent?
Yes. They are dimensionally equivalent SI units for angular impulse and angular momentum.
What if torque changes with time?
Integrate torque over time; a simple torque-times-time product is valid only for constant or average torque.
Sources and review
- OpenStax University Physics Volume 1: Angular Momentum — OpenStax. Accessed 2026-07-13.
- SI Units – Force — National Institute of Standards and Technology. Accessed 2026-07-13.
Reviewed 2026-07-13.