6.2.3Energy and Momentum of Rotating Systems

Impulse–Momentum Theorem in Rotational Form — AP PHYSICS 1 Revision Notes & Topical Questions

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6.2.3 Impulse–Momentum Theorem in Rotational Form — AP PHYSICS 1 notes: State and use the rotational impulse–momentum theorem to relate net torque, time and change in angular momentum, including cases where I changes.ΣτextΔt = ΔL = Lf − Li — the angular impulse of the net externa…

Rigid body about a fixed axis
Constant I
Changing I
Conservation
System choice
Average torque

Section 1 of 9The theorem

1ΣτextΔt

ΔL = Lf − Li

the angular impulse of the net external torque equals the change in angular momentum.

2Rigid body about a fixed axis

ΣτΔt = I_fω_f − I_iω_i.

3Origin

Origin a direct result of Στ = ΔL/Δt, the rotational analogue of FΔt = Δp.

4Constant I

Constant I reduces to Στ = Iα.

Section 2 of 9Why it is useful

Why it is useful

Changing I

Changing I works when the moment of inertia changes, unlike Στ = Iα.
- (e.g. changing shape)

Conservation

Conservation if Στ_ext = 0 then ΔL = 0.

System choice

System choice include all objects whose angular momentum changes;
- only torques from outside count, internal torques cancel in pairs.

Section 3 of 9Variable torque

Angular impulse

Angular impulse the sum of τΔt over intervals, or the area under the τ–t graph.

Average torque

τ_avgΔt = ΔL gives the average torque in a collision or brief push.

Trade-off

Trade-off a longer time lets a smaller torque produce the same ΔL; a shorter torque must be larger.

Section 4 of 9Collisions about a pivot

Collisions about a pivot

Short collision

Short collision external torques have small angular impulse compared with the internal collision torques.
- (gravity, pivot forces)

Angular momentum about the pivot

Angular momentum about the pivot is the same just before and after.

Linear momentum

Linear momentum is not conserved because the pivot supplies an external force.

Section 5 of 9Collision about a pivot: pattern

Choose the pivot as the axis so the unknown pivot force has no torque.

Write the incoming object's angular momentum about the pivot just before, mvr_⊥, with sign.

Write the combined system's angular momentum just after using the total moment of inertia.

State the external angular impulse during the brief collision is negligible and equate the two.

For the swing afterward use energy conservation: kinetic energy converts to gravitational potential energy.

Energy during impact

Energy during impact kinetic energy is not conserved in the collision, since it is inelastic.

Swing

Swing mechanical energy is conserved if pivot friction is negligible.

Section 6 of 9Checks

Direction

Direction the change in L must match the direction of the torque.

Axis

Axis use the same axis for torque and angular momentum.

Section 7 of 9More from the specification

Velocity

Rate of change of displacement.

Distance

Total length of path travelled, a scalar.

Newton

Unit of force, kg m s^-2.

Section 8 of 9Key terms

Term	Meaning
Impulse–momentum theorem (rotation)	ΣτΔt = ΔL.
Average torque	Constant torque giving the same angular impulse in the same time.
External torque	Torque from outside the chosen system.
Internal torque	Torque between parts of the system; cancels in pairs.
Pivot	Fixed axis supplying force but no torque about itself.

Section 9 of 9Exam technique

Including internal torques in the total impulse.

Assuming I is constant when the object changes shape.

Using different axes for torque and angular momentum.

Ignoring the sign of a second torque acting in turn.

Using linear momentum conservation in a pivoted collision where the pivot supplies an external force.

Last updated: 14 October 2026