Rotational Momentum in AP Physics 1: Understanding Angular Momentum

Angular momentum

The rotational analog of linear momentum; measures how hard it is to stop or redirect rotational motion, and is always defined about a specific axis or point.

Linear momentum

Momentum for straight-line motion, given by (\vec{p}=m\vec{v}); the linear analog to angular momentum.

Origin/axis (for angular momentum)

The chosen point or axis about which angular momentum is calculated; the same object can have different (\vec{L}) values depending on this choice.

Angular momentum of a particle (definition)

For a particle, (\vec{L}=\vec{r}\times \vec{p}), where (\vec{r}) is position from the origin and (\vec{p}) is linear momentum.

Angular momentum magnitude formula

(L=r p\sin(\theta)), where (\theta) is the angle between (\vec{r}) and (\vec{p}).

Sideways (tangential) momentum contribution

Only the component of momentum perpendicular to (\vec{r}) contributes to angular momentum; if motion is directly toward/away from the origin, (L=0).

Circular-motion angular momentum (particle)

For motion in a circle of radius (r) with speed (v) ((\vec{v}\perp\vec{r})), (L=mvr).

Right-hand rule for angular momentum

For (\vec{L}=\vec{r}\times\vec{p}), curl fingers from (\vec{r}) toward (\vec{p}); thumb points in the direction of (\vec{L}).

Sign convention (clockwise vs. counterclockwise)

Whether rotation is “positive” or “negative” depends on the chosen axis direction (e.g., out of the page); don’t assume clockwise is always negative.

Moment of inertia ((I))

A measure of how mass is distributed relative to a rotation axis; appears in (L=I\omega) for rigid rotation about a fixed axis.

Angular momentum of a rigid object (fixed axis)

For a rigid body rotating about a fixed axis, (L=I\omega).

Angular speed ((\omega))

Rate of rotation about an axis (direction by right-hand rule); units of rad/s.

Torque ((\vec{\tau}))

A measure of the twisting effect of forces about an axis; it causes changes in angular momentum rather than being angular momentum itself.

Rotational Newton’s second law

The relationship (\sum \vec{\tau}_{\text{ext}}=\dfrac{d\vec{L}}{dt}), connecting net external torque to the rate of change of angular momentum.

External torque

Torque on a system due to forces from outside the chosen system; determines whether the system’s total angular momentum can change.

Angular impulse

The torque-time effect that changes angular momentum: (\int \sum \vec{\tau}{\text{ext}}dt=\Delta\vec{L}); if torque is constant, (\sum\tau{\text{ext}}\Delta t=\Delta L).

Lever arm (perpendicular distance, (b))

The perpendicular distance from the axis/origin to the line of action of a force; used to compute torque magnitude (e.g., (\tau=bF)).

Off-center hit result (puck example)

If a force (F) acts for time (\Delta t) at perpendicular distance (b) from the center, the change in angular momentum is (\Delta L=bF\Delta t).

Conservation of angular momentum

If net external torque about the axis is zero/negligible, total angular momentum stays constant: (\vec{L}i=\vec{L}f) (or (Li=Lf) in fixed-axis scalar form).

Internal torques cancel

Torques from internal forces within a system cancel in pairs (Newton’s third law), so internal forces cannot change the system’s total angular momentum.

Choosing the system and axis

A required step for using angular momentum conservation: define what objects are included (system) and which axis you’re considering, then check external torque about that axis.

Changing moment of inertia (skater/stool idea)

If external torque is negligible and (I) changes, angular momentum conservation gives (Ii\omegai=If\omegaf); decreasing (I) increases (\omega).

Rotational inelastic collision (clay sticks)

When objects stick and rotate together, angular momentum can be conserved (if external torque is negligible) even though mechanical energy/rotational kinetic energy typically decreases.

Rotational kinetic energy

Energy of rotation: (K=\tfrac{1}{2}I\omega^2); often not conserved in sticking (inelastic) rotational collisions even when angular momentum is conserved.

Negligible external torque criteria (AP Physics style)

Angular momentum about an axis is often conserved if major external forces act through the axis (no lever arm), the event is very brief (collision), or bearings/pivots are low-friction; not conserved when forces like tension, brakes, or friction apply a clear torque about the axis.