Recognize angular momentum in straight-line motion

1. Frame the question

Angular momentum is specified relative to a reference point or axis. A rigid body about a fixed principal rotation axis has planar angular momentum L=Iω. A moving point particle has magnitude mvr sinφ, where r runs from the reference point to the particle and φ is the angle between that radius and its velocity. The same expression is momentum times perpendicular distance to the line of motion.

2. Build the mechanism

A particle need not trace a circle to possess angular momentum. In this original case a 2 kg particle moves right along y=b at speed v. About the origin, its signed planar angular momentum is −2vb when counterclockwise tendency is positive. Above the origin the tendency is clockwise; below it the sign reverses. No three-dimensional direction construction is required.

As the particle moves along the line, its distance from the origin changes and so does φ, but r sinφ remains the perpendicular offset magnitude. Therefore L stays constant when no force acts. On a plot of L against horizontal position, the result is a horizontal line, not a representation of the particle’s spatial path.

Moving the reference point onto the line of motion makes this angular momentum zero without changing the particle’s linear momentum. This is not a contradiction: the reference is part of the quantity’s definition. When comparing before and after values or adding several members’ angular momenta, keep the same reference and inertial frame throughout.

3. Work an example

At v=3 m/s and y=+1 m, the 2 kg particle has linear momentum +6 kg m/s and angular momentum −6 kg m²/s about the origin. At x=0 its radius is perpendicular to motion; at x=√3 m the radius is 2 m and the angle is 30°, giving the same magnitude 2×3×2×½=6.

4. Predict and investigate

Predict first: If the particle is farther from the reference point later along the same straight path, must its angular momentum increase?

Compare offsets +1, 0 and −1 m at speed 3 m/s. Draw the actual horizontal paths separately from the angular-momentum plot. Explain the unchanged linear momentum, changing reference-dependent angular momentum and zero torque in each case. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

6. Reflect and retain evidence

Why would using mr times speed without the angle factor fail for a straight-line trajectory?

Retain your first response, system and axis choices, separate scalar energy and signed angular-momentum ledgers, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 6 topic mapping: 6.3. Checked 30 October 2026. Planar signed analysis without required three-dimensional angular-momentum direction. Slipping evolution is qualitative, in accordance with the framework boundary; no rolling-resistance model. Framework cross-reference, not AP authorization.