7. Angular Momentum

Definitions

Angular Momentum (L)

• A measure of the amount of rotational motion an object has about a particular axis, measured in kg⋅m²/s

$$\vec{L} = \vec{r} \times \vec{p}$$

where $\vec{r}$ is the position vector and $\vec{p}$ is the linear momentum

Conservation of Angular Momentum

• When the net external torque acting on a system is zero, the total angular momentum of the system remains constant

$$\vec{L} = \text{constant when } \sum \vec{\tau} = 0$$

 

Derivations

starting with the definition of angular momentum

$$\vec{L} = \vec{r} \times \vec{p}$$

for a particle with momentum $\vec{p} = m\vec{v}$ and since $\vec{v} = \omega \vec{r}$

$$\vec{L} = \vec{r} \times (m \vec{v}) = m(\vec{r} \times \vec{v}) = m \vec{r}^2 \omega$$

for a rotating rigid body, the total angular momentum is given by

$$\vec{L} = I \omega$$

taking the time derivative of angular momentum

$$\dfrac{d\vec{L}}{dt} = \dfrac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \dfrac{d\vec{p}}{dt}$$

assuming $\vec{r}$ is constant (or $\dfrac{d\vec{r}}{dt}$ is parallel to $\vec{p}$), and knowing that $\dfrac{d\vec{p}}{dt} = \vec{F}$ from Newton's second law

$$\dfrac{d\vec{L}}{dt} = \vec{r} \times \vec{F} = \vec{\tau}$$

therefore, the rate of change of angular momentum equals the net torque

when the net external torque is zero

$$\sum \vec{\tau} = 0$$

then

$$\dfrac{d\vec{L}}{dt} = 0$$

and angular momentum is conserved

$$\vec{L} = \text{constant}$$

 

Useful Equations

$$\vec{L} = \vec{r} \times \vec{p}, \quad \vec{L} = I \omega$$ $$\dfrac{d\vec{L}}{dt} = \vec{\tau}$$

 

Example 1

A point mass of 2 kg moves in a circle of radius 3 m with angular velocity 4 rad/s. What is its angular momentum?

Answer

Using $L = I\omega$ where $I = mr^2$ for a point mass

$$L = mr^2\omega = 2(3)^2(4)$$ $$L = 2(9)(4) = 72 \text{ kg⋅m}^2\text{/s}$$

 

Example 2

A figure skater with moment of inertia 4 kg⋅m² spins at 2 rad/s. She pulls her arms in, reducing her moment of inertia to 2 kg⋅m². What is her new angular velocity?

Answer

Using conservation of angular momentum $L_i = L_f$

$$I_1\omega_1 = I_2\omega_2$$ $$4(2) = 2\omega_2$$ $$\omega_2 = \frac{8}{2} = 4 \text{ rad/s}$$