Angular Impulse Calculator

Angular impulse

This is the accumulated effect of net torque over time.

It equals the resulting change in angular momentum.

Positive angular impulse

Positive impulse changes angular momentum in the chosen positive axial direction.

It can increase, decrease, stop, or reverse rotational motion depending on the initial angular momentum.

Negative angular impulse

Negative impulse changes angular momentum in the opposite direction.

It does not automatically mean the object ends with negative angular velocity.

Torque-time result

The result is valid directly for constant torque or when the entered torque represents the true average over the interval.

Variable torque generally requires integration.

IΔω result

This form assumes moment of inertia remains constant during the rotational change.

If I changes, use the full angular-momentum difference rather than one fixed I multiplied by Δω.

Scenario 2: Braking torque

A constant −8 N·m net torque acts for 4 seconds.

Angular impulse = −32 N·m·s, so angular momentum changes by −32 kg·m²/s.

Scenario 3: Find torque from impulse

A rotor receives 50 N·m·s of angular impulse over 5 seconds.

Average net torque = 10 N·m.

Scenario 4: Find time

A constant 6 N·m torque must create 24 N·m·s of angular impulse.

Δt = 24/6 = 4 s.

Scenario 5: Inertia and spin-speed change

A rigid rotor has I = 5 kg·m² and Δω = 8 rad/s.

ΔL = IΔω = 40 kg·m²/s.

Scenario 6: Find angular velocity change

A rigid body with I = 10 kg·m² receives 50 kg·m²/s of angular-momentum change.

Δω = 5 rad/s.

Angular impulse is the rotational analog of linear impulse

Linear impulse equals the time integral of force and changes linear momentum.

Angular impulse equals the time integral of torque and changes angular momentum.

The fundamental relationship is dL/dt = τ_net

OpenStax derives net external torque as the rate of change of angular momentum.

Integrating through time gives ΔL = ∫τ_net dt.

Constant torque simplifies the integral

If τ_net is constant, it can be taken outside the integral.

The result is ΔL = τ_net Δt.

Scenario 7: Constant torque

τ = 4 N·m for 6 s.

Angular impulse = 24 N·m·s.

Average torque can also reproduce total impulse

By definition, τ_avg Δt equals the torque-time integral over the same interval.

The average must be a time-average, not simply an arbitrary midpoint torque.

Variable torque is the area under a torque-time graph

Positive graph area adds positive angular momentum.

Negative graph area subtracts angular momentum.

Scenario 8: Two torque intervals

+10 N·m acts for 2 s, then −4 N·m acts for 3 s.

Total angular impulse = 20 − 12 = 8 N·m·s.

Opposing torque can partially cancel previous impulse

Angular impulse is signed.

The net change depends on algebraic or vector accumulation through time.

Scenario 9: Equal positive and negative impulse

+5 N·m for 4 s gives +20 N·m·s.

−10 N·m for 2 s gives −20 N·m·s.

Net angular impulse is zero.

Zero net impulse means no net change in angular momentum

It does not mean torque was zero at every instant.

Positive and negative torque-time areas may cancel.

Torque is not angular impulse

Torque has units N·m.

Angular impulse adds a time dimension and has units N·m·s.

Angular impulse and angular momentum share units

1 N·m·s = 1 kg·m²/s.

This follows because 1 N = 1 kg·m/s².

Scenario 10: Unit reduction

N·m·s = (kg·m/s²)·m·s.

The result reduces to kg·m²/s.

Angular impulse equals change, not necessarily final angular momentum

ΔL = L_f − L_i.

Therefore L_f = L_i + Jθ.

Scenario 11: Nonzero initial angular momentum

L_i = 30 kg·m²/s and Jθ = 20 kg·m²/s.

L_f = 50 kg·m²/s.

Negative impulse can reduce positive angular momentum

If L_i is positive and impulse is negative, the final magnitude can decrease.

A sufficiently large negative impulse can stop or reverse the rotation.

Scenario 12: Rotor brought to rest

L_i = +40 kg·m²/s.

Jθ = −40 kg·m²/s.

L_f = 0.

Impulse larger than initial opposite momentum reverses direction

Signs are essential.

Angular momentum can cross zero and change axial direction.

Scenario 13: Rotation reversal

L_i = +20.

Jθ = −35.

L_f = −15 kg·m²/s.

The Angular Impulse Momentum Calculator should emphasize this state transition

Angular Impulse Calculator focuses on the impulse quantity.

Impulse-Momentum mode can make L_i + Jθ = L_f the center of the workflow.

The second calculator mode follows from L = Iω

If I is constant, ΔL = Iω_f − Iω_i.

Factor out I to obtain ΔL = I(ω_f − ω_i) = IΔω.

Scenario 14: Constant-I rotor

I = 3 kg·m².

ω changes from 5 to 15 rad/s.

ΔL = 3 × 10 = 30 kg·m²/s.

The sign of Δω matters

Δω = ω_f − ω_i.

A decrease in signed angular velocity produces negative Δω under the selected convention.

Scenario 15: Spin slowing

ω_i = +20 rad/s and ω_f = +5 rad/s.

Δω = −15 rad/s.

For positive I, ΔL is negative.

Spin reversal requires signed angular velocities

A change from +10 to −10 rad/s is Δω = −20 rad/s, not zero.

Using only speed magnitudes would miss the reversal.

Scenario 16: Reversal with fixed I

I = 2 kg·m², ω_i = +10, ω_f = −10 rad/s.

ΔL = 2 × (−20) = −40 kg·m²/s.

IΔω fails when I changes

The general change is ΔL = I_fω_f − I_iω_i.

One common I can be factored only when moment of inertia remains constant.

Scenario 17: Changing inertia

I_i = 4, ω_i = 2 gives L_i = 8.

I_f = 2, ω_f = 4 gives L_f = 8.

Δω is +2, but ΔL = 0, so using one fixed IΔω would be wrong.

A figure skater illustrates why changing I matters

Pulling mass inward changes rotational inertia.

Angular velocity can increase without an external angular impulse when total angular momentum is conserved.

Angular impulse is caused by external torque

Internal forces can redistribute angular momentum inside a system.

The total system change is governed by net external torque.

Choice of axis matters

Both torque and angular momentum depend on the chosen origin or axis.

The impulse relation must use the same reference throughout.

Scenario 18: Different torque origins

A force can produce zero torque about one point and nonzero torque about another.

Angular impulse therefore cannot be interpreted without specifying the reference axis.

Net torque means all external torque contributions combined

Motor torque, braking torque, friction torque, and load torque can act simultaneously.

Their signed sum determines the angular impulse rate.

Scenario 19: Motor and brake

Motor torque = +30 N·m.

Brake torque = −12 N·m.

Net torque = +18 N·m.

Torque direction follows the right-hand rule

In vector mechanics, torque is an axial vector.

A scalar calculator encodes opposite axial directions with positive and negative signs.

Counterclockwise-positive is common but not mandatory

The key requirement is consistency.

The final angular impulse sign inherits the torque sign under a positive time interval.

A zero time interval produces zero impulse for finite torque

Jθ = τΔt.

If Δt = 0, no finite time accumulates torque impulse.

Solving torque requires nonzero time

τ_avg = Jθ/Δt.

A nonzero impulse cannot be produced over zero duration by a finite average torque in this model.

Solving time requires nonzero torque

Δt = Jθ/τ_avg.

If torque is zero while impulse is nonzero, the entered assumptions are inconsistent.

Moment of inertia must be positive for ordinary rotating bodies

I describes mass distribution relative to the axis.

The IΔω rearrangements should reject zero I when solving Δω from nonzero impulse.

Angular velocity should normally use rad/s

If data are provided in RPM, convert using ω = RPM × 2π/60.

Then form Δω from consistently converted endpoint values.

Scenario 20: RPM change

Rotor speeds from 600 to 1,200 RPM.

Convert to 20π and 40π rad/s.

Δω = 20π rad/s.

Do not subtract RPM from rad/s

Angular-velocity differences require the same unit.

Convert both endpoints before subtraction.

Torque can be derived from force and lever arm when geometry is known

For one force, τ = rF sinφ.

The angular impulse then becomes rF sinφ Δt when these quantities remain constant.

Scenario 21: Perpendicular push on a wheel

r = 0.4 m, F = 25 N, φ = 90°.

Torque = 10 N·m.

If applied for 2 s, impulse = 20 N·m·s.

Radial force produces no torque about the center

If φ = 0, sinφ = 0.

That force contributes no angular impulse about that axis despite acting for a finite time.

Torque-time graphs can contain curved segments

The general integral handles arbitrary time dependence.

Numerical integration may be required when no simple analytic function is available.

Scenario 22: Linearly increasing torque

Torque rises from 0 to 20 N·m over 4 seconds.

The torque-time graph is a triangle with area 1/2 × 4 × 20 = 40 N·m·s.

Average torque for that ramp is 10 N·m

10 × 4 = 40 N·m·s.

This confirms that a correct time-average torque reproduces the integral.

Impulse depends on area, not peak torque alone

A large torque applied very briefly can produce the same impulse as a smaller torque applied for longer.

Peak torque by itself does not determine angular-momentum change.

Scenario 23: Equal impulses

100 N·m for 0.1 s gives 10 N·m·s.

10 N·m for 1 s also gives 10 N·m·s.

This does not mean the two load histories are mechanically equivalent

Peak stress, vibration, deformation, and transient response can differ.

Angular impulse describes only the total angular-momentum change.

Rotational impulse is useful in impacts

Short-duration contact torques can produce substantial changes in angular momentum.

The impulse framework avoids requiring the detailed instantaneous acceleration history when total torque-time integral is known.

Scenario 24: Brief impact

An average net impact torque of 500 N·m acts for 0.02 s.

Angular impulse = 10 N·m·s.

Angular impulse and angular acceleration are related through I when I is fixed

τ = Iα and Jθ = τΔt.

For constant I and α, Jθ = IαΔt = IΔω.

This links the Angular Acceleration Calculator

Angular Acceleration calculates α from torque and inertia or velocity change and time.

Angular Impulse accumulates the effect over time into ΔL.

Scenario 25: Same result through acceleration

I = 4 kg·m² and α = 3 rad/s² for 2 s.

Δω = 6 rad/s.

IΔω = 24 kg·m²/s.

Torque = 12 N·m, and τΔt also gives 24 N·m·s.

The equality of the two modes is a valuable internal check

When I is constant and the same physical interval is used, τΔt and IΔω should agree.

A mismatch signals inconsistent inputs, units, or assumptions.

Angular momentum itself can be calculated separately

For a fixed-axis rigid body, L = Iω.

The Angular Momentum Calculator handles the state value rather than only its change.

Scenario 26: Recover final state

L_i = 50 and Jθ = 20.

L_f = 70 kg·m²/s.

If I = 5 kg·m², then ω_f = 14 rad/s.

Angular impulse does not by itself determine energy change

Rotational kinetic energy depends on I and ω².

Two systems receiving the same angular impulse can experience different energy changes.

Scenario 27: Same impulse, different I

Both rotors receive ΔL = 20.

Their resulting Δω values differ if their moments of inertia differ.

Conservation of angular momentum is the zero-external-impulse case

If the net external torque integral is zero, ΔL = 0.

Total angular momentum remains constant.

A nonzero torque at some times can still yield zero net impulse

Only the integrated torque matters for total ΔL.

Positive and negative torque intervals can cancel.

The calculator should retain signed output

Do not convert angular impulse to an absolute magnitude automatically.

Direction can be essential to the physical interpretation.

Result cards should distinguish impulse and equivalent ΔL

Angular impulse: 36 N·m·s.

Equivalent angular-momentum change: 36 kg·m²/s.

The strongest interface should label torque as average net torque

This avoids implying that any instantaneous torque value can be multiplied by total duration.

The current page already describes the torque field this way conceptually.

The strongest future enhancement is a torque-time profile mode

Users could enter piecewise torque intervals or a time series.

The calculator could then integrate the signed area automatically.

Piecewise integration would broaden the tool without changing the core physics

Jθ = Στ_iΔt_i for piecewise-constant intervals.

Continuous functions can use analytic or numerical integration.

Do not merge this page conceptually with angular momentum

Angular momentum is a rotational state quantity.

Angular impulse is the change in that quantity produced over a time interval.

The strongest educational analogy is force impulse versus torque impulse

Linear: J = ∫F dt = Δp.

Rotational: Jθ = ∫τ dt = ΔL.