Unit 5 Rotation: Understanding and Using Angular Momentum

Angular momentum

Rotational analogue of linear momentum; measures how hard it is to stop or redirect rotational motion about a chosen origin or axis.

Origin dependence (of angular momentum)

Angular momentum is not an intrinsic property of an object alone; its value depends on the point (origin) about which it is calculated.

Angular momentum of a particle (definition)

For a particle at position vector r with momentum p, angular momentum about the origin is L⃗ = r⃗ × p⃗.

Magnitude of particle angular momentum

$L = r p \sin \theta$, where $\theta$ is the angle between $\tilde{r}$ and $\tilde{p}$.

Perpendicular-component rule (for L)

Only the component of momentum perpendicular to r⃗ contributes to angular momentum about that origin (via the sinθ factor).

Radial motion and angular momentum

If a particle moves directly toward or away from the origin (θ = 0 or π), then L = 0 about that origin.

Right-hand rule (for L⃗)

The direction of L⃗ = r⃗ × p⃗ is given by the right-hand rule for the cross product.

Units of angular momentum

kg·m²/s (equivalently N·m·s).

Angular momentum in uniform circular motion (particle)

If $\tilde{v} \bot \tilde{r}$, then $L = m v r$ about the center.

Angular momentum using angular speed

Using v = rω, a particle in circular motion has L = m r² ω.

Moment of inertia (definition)

For rotation about a fixed axis, I = Σ mᵢ rᵢ², where rᵢ is each mass element’s distance from the axis.

Rigid-body angular momentum (fixed-axis AP case)

For planar rotation about a principal axis, angular momentum is $\tilde{L} = I \tilde{\theta}$ (so magnitude $L = I\theta$).

Principal axis (context for L⃗ ∥ ω⃗)

An axis about which rotation makes $\tilde{L} \text{parallel} \tilde{\theta}$; in general 3D rotation, $\tilde{L}$ need not be parallel to $\tilde{\theta}$ unless rotating about a principal axis.

Total angular momentum of a system

Vector sum over particles: $\tilde{L}_{tot} = \textstyle\bigoplus ( \tilde{r}_i \times \tilde{p}_i )$ about the chosen origin.

Angular momentum decomposition about an origin

$\tilde{L}_{O} = \tilde{r}_{CM} \times M \tilde{v}_{CM} + \tilde{L}_{CM}$ (translation of CM plus “spin” about CM).

Torque (definition)

Torque about an origin is $\tilde{\tau} = \tilde{r} \times \tilde{F}$.

Rotational dynamics link (torque–angular momentum)

For a particle, $\frac{d\tilde{L}}{dt} = \tilde{\tau}$ (net torque changes angular momentum).

Angular impulse

Integral of torque over time: ΔL⃗ = ∫ τ⃗ dt; useful when forces act briefly and are hard to model directly.

Conservation of angular momentum (statement)

If net external torque about a chosen origin/axis is zero, then total angular momentum about that origin/axis is constant.

External torque condition for conservation

$\tilde{\tau}_{ext} = 0 \rightarrow \tilde{L}_{tot} = \text{constant}$ (must be about the same specified origin/axis).

Pivot choice strategy

Choosing the pivot as the origin often removes unknown pivot forces from torque because their lever arm is zero about the pivot.

Central force

A force always along (parallel or antiparallel to) r⃗ to a fixed center; it produces zero torque about that center (r⃗ × F⃗ = 0).

Conservation with changing moment of inertia

With negligible external torque and fixed-axis rotation, $I_i \theta_i = I_f \theta_f$ (e.g., skater pulls in arms).

Rotational kinetic energy (fixed-axis)

$K = \frac{1}{2} I \theta^2$; it generally is NOT conserved when $I$ changes even if angular momentum is conserved.

Inelastic rotational collision (sticking)

During a short collision with negligible external torque about the axis, angular momentum is conserved but kinetic energy is not (use L, not K, across sticking).