SCIENCE · INVESTIGATE AND EXPLAIN

Keep signed angular momentum while coupling converts energy

For fixed-axis rotation, angular momentum is Iω with the axis sign convention retained. A point contribution more generally uses the perpendicular component of momentum times its moment arm, and contributions sum about one common origin. In this coaxial packet, the first rotor has inertia 2 kg m² and angular velocity+3 rad/s.

Learning outcome: Compare same-direction, opposite-direction and exact-cancellation states. Retain each rotor’s angular impulse and the whole-pair total. Explain why the kinetic conversion cannot be found by subtracting angular-speed magnitudes alone.

Starting recap

Declare the axis, angular units, recipient and valid contact or material branch. Distinguish supplied shape factors from independently measured inertia.

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Try

For J=1,Ω=4, finalω=10/3 rad/s. Initial energy 17 J and final50/3 J leave1/3 J internal conversion. Rotor1 receives angular impulse 2/3 N m s and rotor2 receives−2/3 N m s.

Open the experiment

Predict

Two rotors have equal and opposite total angular momenta before sticking. Their common final rotation is:

Your prediction and reason

Experiment

Compare same-direction, opposite-direction and exact-cancellation states. Retain each rotor’s angular impulse and the whole-pair total. Explain why the kinetic conversion cannot be found by subtracting angular-speed magnitudes alone.

Retain angular momentum and converted energy in coupling

Worked teaching model · synthetic values · Two coaxial ideal rotors, first inertia 2 kg m² and initial omega +3 rad/s; second selected inertia and signed omega. They lock to common rotation with negligible external angular impulse. Internal energy receives lost kinetic energy; no physical coupling or spinning apparatus.

Second rotor inertia: 2 kg m² Second initial angular velocity: -2 rad/s

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Converted kinetic energy versus second initial rotation

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: coupling loss (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Converted kinetic energy versus second initial rotation

coupling loss

second omega (rad/s)	internal energy increase (J)
-4	24.5
-3.8	23.12
-3.6	21.78
-3.4	20.48
-3.2	19.22
-3	18
-2.8	16.82
-2.6	15.68
-2.4	14.58
-2.2	13.52
-2	12.5
-1.8	11.52
-1.6	10.58
-1.4	9.68
-1.2	8.82
-1	8
-0.8	7.22
-0.6	6.48
-0.4	5.78
-0.2	5.12
0	4.5
0.2	3.92
0.4	3.38
0.6	2.88
0.8	2.42
1	2
1.2	1.62
1.4	1.28
1.6	0.98
1.8	0.72
2	0.5
2.2	0.32
2.4	0.18
2.6	0.08
2.8	0.02
3	0
3.2	0.02
3.4	0.08
3.6	0.18
3.8	0.32
4	0.5

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Initial total angular momentum	2	kg m²/s
Common final angular velocity	0.5	rad/s
Final total angular momentum	2	kg m²/s
Initial rotational kinetic energy	13	J
Final rotational kinetic energy	0.5	J
Internal energy increase	12.5	J
Angular impulse on first rotor	-5	N m s
Angular impulse on second rotor	5	N m s

Internal angular impulses cancel for the pair while each rotor changes its own angular momentum.

Conserve signed L=I1 omega1+I2 omega2 about the same axis, then impose the common final rotation. Kinetic energy can become internal energy during coupling. Exact cancellation of total L gives zero final rotation without requiring zero initial energy. An external axle torque during the interaction would require its angular impulse in the budget.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Internal angular impulses cancel for the pair while each rotor changes its own angular momentum.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Choose the axis, reference or material model

Build the algebraic relationship

The second rotor has selected inertia J and signed initial angular velocityΩ. They lock to a common final rate while net external angular impulse is negligible. Total angular momentum 6+JΩ is conserved, so ωfinal=(6+JΩ)/(2+J). Internal torques can change each rotor’s angular momentum even though their equal-and-opposite impulses cancel for the pair.

Check the original numerical packet

Kinetic energy is not generally conserved by locking. Initial energy is 9+JΩ²/2 and final energy is(2+J)ωfinal²/2. For J=2,Ω=−2, total angular momentum is 2, final rate 0.5 rad/s, initial energy 13 J and final energy 0.5 J. The remaining12.5 J becomes internal energy under the isolated coupling account.

Test the boundary and independent evidence

For J=3,Ω=−2, total angular momentum cancels and final rotation is zero, despite initial energy 15 J. Zero total angular momentum does not require every initial component to be at rest. An external axle torque during coupling would add an angular impulse to the budget; merely calling an axle friction small does not measure it. No actual spinning apparatus or coupling experiment is requested.

Worked original case

For J=1,Ω=4, finalω=10/3 rad/s. Initial energy 17 J and final50/3 J leave1/3 J internal conversion. Rotor1 receives angular impulse 2/3 N m s and rotor2 receives−2/3 N m s.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

Rotors with I1=2,ω1=3 and I2=1,ω2=0 lock without external angular impulse. Find the final common angular velocity in rad/s.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated teaching evidence. No rotating equipment, applied loads, material tests or physical assembly is requested. An admissible virtual model state does not certify real equipment safety, supervised practical skill or independent mastery.

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What independent evidence would justify neglecting the external angular impulse during a real coupling?

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