S5.7 Angular momentum

[A] Definition

Key Point 5.16

The angular momentum of a particle with respect to a point O is

\[ \vec{L} = \vec{r} \times \vec{p} = m \left( \vec{r} \times \vec{v} \right) \]

$\vec{r}$ is the position of the particle with respect to O.

Graphic - No title - angmom1
AVisualisation of Angular Momentum

[B] Angular momentum of a rigid body

Key Point 5.17

The magnitude of the angular momentum of a rigid body rotating about an axis with angular velocity ω is

\[ | \vec{L} | = I \omega \]
where I is the moment of inertia (Key Point 5.12) about the rotation axis. The direction of $\vec{L}$ is along the axis in the sense of the rotation vector.

Note: the moment of inertia depends on the rotation axis.

[C] Newton’s 2nd law for rotation

Key Point 5.18

Newton’s 2nd Law: angular form relates the total external torque, $\vec{\tau}_{ext}$, acting on a system of particles, to the total angular momentum $\vec{L}$ of the system about the same point.

\[ \vec{\tau}_{ext} = \frac{d \vec{L}}{d t} \]

$\vec{\tau}$ and $\vec{L}$ are expressed with respect to the same origin.

[D] Special case: Newton’s 2nd law for rigid bodies

For the special case of a rigid body rotating about an axis, Newton’s second law takes a particular form involving the moment of inertia.

Key Point 5.19

Newton’s 2nd law for rigid bodies: A torque, τ, applied to a rigid body rotating about a fixed axis produces an angular acceleration α about that axis

\[ \tau = I \alpha = I \frac{d\omega}{dt} = I \frac{d^2\theta}{dt^2} \]

where I is the moment of inertia about the rotation axis, ω the angular velocity and θ the angular displacement.

TAngular Momentum of a Particle Travelling in a Straight Line Past a Point
TThe Earth's Angular Momentum Direction
TPulling a yo-yo

Learning Resources

Textbook: HRW Chapter 10.9. 11.7-11.10
Course Questions:
Self-Test Questions: