SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish conservation from independence of origin

Angular momentum of a point particle is r × p about a specified origin. Conservation means that this vector is constant in time under the applicable zero-torque condition. It does not mean that every origin assigns the same constant vector.

Learning outcome: Evaluate both cross products at two times and verify the origin-shift identity before interpreting signs.

Starting recap

Integrate mass moments about named axes, derive torque and angular-momentum constraints, and test orbital energy, angular and numerical consistency against independent evidence.

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At v = 3 m/s and d = 2 m, the original angular momentum is +6 and the shifted value is −6 kg m²/s. The sign change is an origin effect, not a reversal of the particle velocity.

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Predict

For this force-free particle, two fixed origins can assign:

Your prediction and reason

Experiment

Evaluate both cross products at two times and verify the origin-shift identity before interpreting signs.

Keep the angular-momentum origin explicit

Worked teaching model · synthetic values · Force-free2 kg particle has ground position(1,v t,0). Compare fixed inertial origins(0,0,0) and(d,0,0), not rotating or accelerating frames. Angular momentum is about each specified origin.

Constant north velocity: 2 m/s Shifted origin east coordinate: 1 m

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Straight trajectory and two fixed origins

Spatial axes use the same pixels per physical unit, preserving geometry. 1: trajectory over2 seconds (solid).2: origin separation (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Straight trajectory and two fixed origins

trajectory over2 seconds

east coordinate (m)	north coordinate (m)
1	0
1	4

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.

origin separation

east coordinate (m)	north coordinate (m)
0	0
1	0

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
Angular momentum about original origin	4	kg m²/s
Angular momentum about shifted origin	0	kg m²/s
Origin-shift angular momentum change	-4	kg m²/s
Torque about either fixed origin	0	N m

Different fixed origins can assign different conserved angular momenta to the same force-free motion.

For r=(1,vt,0), p=(0,2v,0), the z cross product is2v. Shifting the origin by(d,0,0) subtracts d×p, giving2(1−d)v. No external force implies zero torque about each fixed origin. This is not an argument that angular momentum is independent of origin.

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

Different fixed origins can assign different conserved angular momenta to the same force-free motion.

Observation to attach to a capture

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Understand

Distinguish conservation from independence of origin

A force-free 2 kg particle has position (1, vt, 0) m and velocity (0, v, 0). About the origin at (0, 0, 0), the z component is 2v kg m²/s. The term involving vt contributes nothing because it is parallel to momentum.

Shift the origin to the fixed inertial point (d, 0, 0). The position becomes (1 − d, vt, 0), so angular momentum is 2(1 − d)v in the z direction. The difference is minus the origin-displacement vector crossed with momentum. At v = 2 m/s and d = 1 m, the shifted-origin angular momentum is zero while the original value is 4 kg m²/s.

Both quantities are conserved because the force and torque are zero about both fixed origins. An accelerating or rotating reference would require additional care and is not represented by simply substituting a time-dependent d in this fixed-origin argument. The equal-scale diagram shows a straight trajectory, not a circular orbit.

A separate nonzero-torque check uses a 2 kg particle with r = (t, t², 0) in the supplied dimensioned polynomial convention. Its momentum is (2, 4t, 0), angular momentum is 2t² in the z direction, and net force is (0, 4, 0) N. The torque is 4t in the z direction, exactly dLz/dt. At t = 1 s, Lz = 2 kg m²/s and torque is 4 N m; the vector lever arm, not force magnitude alone, sets the torque.

Changed-case check

Predict before reading the comparison. At v = 3 m/s and d = 2 m, the original angular momentum is +6 and the shifted value is −6 kg m²/s. The sign change is an origin effect, not a reversal of the particle velocity.

Retained investigation

Evaluate both cross products at two times and verify the origin-shift identity before interpreting signs. Why is an origin label part of an angular-momentum result?

Source and boundary

Primary conceptual reference checked 30 October 2026: OpenStax University Physics Volume 1. All models, packets and teaching prose are original. These are bounded mathematical investigations, not measured apparatus, physical procedures, practical certification or institutional credit.

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Why is an origin label part of an angular-momentum result?

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