Angular Momentum, Equilibrium and Rolling Motion: Quick Revision

A brief note on Angular Momentum, Equilibrium and Rolling Motion. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Angular momentum: $L = I\omega$ (body on an axis), $\vec{L} = \vec{r} \times \vec{p}$ (particle). Torque is its rate of change.
• Conservation: no external torque, so $I_1\omega_1 = I_2\omega_2$; kinetic energy $\dfrac{L^2}{2I}$ can change.
• Equilibrium: net force zero and net torque zero about every point.
• Rolling without slipping: $v = \omega R$; the contact point is at rest, the top moves at $2v$.
• Rolling energy: $\tfrac12 Mv^2\left(1 + \dfrac{I}{MR^2}\right)$; down a slope $a = \dfrac{g\sin\theta}{1 + I/MR^2}$.

The facts at a glance

A worked example

A solid ball, a disc and a ring roll from rest down a slope, dropping $0.80\ \text{m}$.

1. Ball. $v = \sqrt{\dfrac{10 \times 9.8\ \text{m/s}^2 \times 0.80\ \text{m}}{7}} = 3.35\ \text{m/s}$.
2. Disc. $v = \sqrt{\dfrac{4 \times 9.8\ \text{m/s}^2 \times 0.80\ \text{m}}{3}} = 3.23\ \text{m/s}$.
3. Ring. $v = \sqrt{9.8\ \text{m/s}^2 \times 0.80\ \text{m}} = 2.80\ \text{m/s}$. The more of its energy a body spends turning, the slower it goes.

Practice with new numbers

A program to try

The program works out the three speeds and the turning shares.

Mistakes to avoid

• Conserving kinetic energy when a skater pulls in. Conserve angular momentum.
• Forgetting a body's own weight in an equilibrium problem; it acts at its centre of mass.
• Distances along a ladder as moment arms. Use the perpendicular (horizontal) distances.
• Friction doing work in rolling. Static friction at the contact point does none.
• Mass or size deciding the rolling race. Only $\dfrac{I}{MR^2}$ does.

Check yourself

1. A turntable with $I = 0.20\ \text{kg m}^2$ spins at $5.0\ \text{rad/s}$. A $0.10\ \text{kg}$ lump sticks to it $0.30\ \text{m}$ from the axis. What is the new spin?

2. A $40\ \text{kg}$ child sits $1.5\ \text{m}$ from a seesaw's pivot. Where must a $50\ \text{kg}$ child sit to balance?

3. A disc rolls at $4.0\ \text{m/s}$. How fast is the top of the disc moving?

4. What fraction of a rolling ring's kinetic energy is in its turning?

5. Why does a spinning top stay upright while it spins fast?