Angular momentum and its conservation

Momentum for turning

A particle of momentum $\vec{p}$ at $\vec{r}$ from a point O has angular momentum

For a body turning about a fixed axis, adding up all its particles gives

in $\text{kg m}^2/\text{s}$.

Torque changes angular momentum

Just as force is the rate of change of momentum, torque is the rate of change of angular momentum:

Conservation

With no external torque, angular momentum stays the same:

A skater who pulls her arms in makes $I$ smaller, so $\omega$ grows.

You should find the spin rate rising in proportion as $I$ falls, and the kinetic energy rising too.

Watch out. Kinetic energy is not conserved here. $K = \dfrac{L^2}{2I}$ grows as $I$ shrinks. The skater's muscles do work pulling her arms in against their tendency to fly out.

A worked example

A merry-go-round with $I = 200\ \text{kg m}^2$ turns at $1.5\ \text{rad/s}$. A $40\ \text{kg}$ child, running toward its centre, jumps on and stays at its rim, $2.0\ \text{m}$ from the axis.

1. Before. $L = 200\ \text{kg m}^2 \times 1.5\ \text{rad/s} = 300\ \text{kg m}^2/\text{s}$ (the child brings none, running toward the axis).
2. After. $I = 200\ \text{kg m}^2 + 40\ \text{kg} \times (2.0\ \text{m})^2 = 360\ \text{kg m}^2$.
3. New spin. $\omega = \dfrac{300\ \text{kg m}^2/\text{s}}{360\ \text{kg m}^2} = 0.833\ \text{rad/s}$.

Think first. Check that no external torque acts about the axis. Then write $L$ before and after, and set them equal.

Practise with new numbers

Try it in Python

The program follows a skater as she pulls her arms in, step by step.

Remember these

• Particle: $\vec{L} = \vec{r} \times \vec{p}$; body on a fixed axis: $L = I\omega$.
• $\vec{\tau}_{\text{ext}} = \dfrac{d\vec{L}}{dt}$.
• No external torque: $I_1\omega_1 = I_2\omega_2$, while $K = \dfrac{L^2}{2I}$ changes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A wheel with $I = 0.60\ \text{kg m}^2$ spins at $25\ \text{rad/s}$. What is its angular momentum?

2. A diver halves her moment of inertia in mid-air. What happens to her spin rate?

3. Why does a cat, falling upside down, manage to turn over without breaking the conservation of angular momentum?

4. A constant torque of $2.0\ \text{N m}$ acts on a wheel for $5.0\ \text{s}$. By how much does its angular momentum change?