Torque, Moment of Inertia and Rotation About a Fixed Axis: Quick Revision

A brief note on Torque, Moment of Inertia and Rotation About a Fixed Axis. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Vector product: $|\vec{A} \times \vec{B}| = AB\sin\theta$, at right angles to both; $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$.
• Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, size $rF\sin\theta$ = force times moment arm.
• Moment of inertia: $I = \sum m_ir_i^2$; mass far from the axis counts most.
• Axis theorems: $I = I_{cm} + Md^2$; for flat bodies $I_z = I_x + I_y$.
• Law of rotation: $\tau = I\alpha$; energy $\tfrac12 I\omega^2$; work $\tau\theta$; power $\tau\omega$.

The facts at a glance

A worked example

A steady torque of $3.0\ \text{N m}$ acts on a solid disc of mass $8.0\ \text{kg}$ and radius $0.25\ \text{m}$, from rest.

1. Moment of inertia. $I = \tfrac12 \times 8.0\ \text{kg} \times (0.25\ \text{m})^2 = 0.25\ \text{kg m}^2$.
2. Angular acceleration. $\alpha = \dfrac{3.0\ \text{N\,m}}{0.25\ \text{kg\,m}^2} = 12\ \text{rad/s}^2$.
3. After $4.0\ \text{s}$. $\omega = 48\ \text{rad/s}$ and $K = \tfrac12 \times 0.25\ \text{kg\,m}^2 \times (48\ \text{rad/s})^2 = 288\ \text{J}$, which equals the torque's work, $\tau\theta = 3.0\ \text{N\,m} \times 96\ \text{rad}$.

Practice with new numbers

A program to try

The program spins up the disc of the worked example.

Mistakes to avoid

• $\cos$ instead of $\sin$ in $rF\sin\theta$. The part of the force along the arm gives no torque.
• $MR^2$ for a disc. That is a ring.
• The parallel axis theorem from a non-central axis. Start from the centre of mass.
• The tension equal to the weight when a load accelerates.
• Degrees or turns in $\tau\theta$. Use radians.

Check yourself

1. What torque does a $25\ \text{N}$ force give at $0.60\ \text{m}$ from an axis, at $30^\circ$ to the arm?

2. What is $\hat{i} \times \hat{k}$?

3. A $2.0\ \text{kg}$ rod $0.90\ \text{m}$ long turns about one end. What is its moment of inertia?

4. A flywheel with $I = 1.5\ \text{kg m}^2$ spins at $40\ \text{rad/s}$. How much work would a brake have to do to stop it?

5. A motor gives $25\ \text{N m}$ at $120\ \text{rad/s}$. What is its power?