Dynamics of rotation about a fixed axis

The law of rotation

For a body turning about a fixed axis, the turning version of $F = ma$ is

Every straight-line idea has its turning twin:

A rope on a drum

A mass $m$ hangs from a rope wound round a drum of moment of inertia $I$ and radius $R$. Two laws, one for each body, joined by the rope:

• the mass: $mg - T = ma$;
• the drum: $TR = I\alpha$;
• the rope does not slip: $a = \alpha R$.

For a solid drum, $I = \tfrac12 MR^2$, and these give $a = \dfrac{mg}{m + \tfrac12 M}$. Explore it.

You should find that a heavy drum slows the fall a lot, and the tension then nearly equals the weight; a very light drum lets the mass fall almost freely, with almost no tension.

Watch out. The tension is not $mg$ when the mass accelerates. Find it from the mass's own equation.

A worked example

A grindstone with $I = 0.50\ \text{kg m}^2$ is driven by a motor torque of $4.0\ \text{N m}$ against a friction torque of $1.0\ \text{N m}$.

1. Net torque. $4.0\ \text{N m} - 1.0\ \text{N m} = 3.0\ \text{N m}$.
2. Angular acceleration. $\alpha = \dfrac{3.0\ \text{N m}}{0.50\ \text{kg m}^2} = 6.0\ \text{rad/s}^2$.
3. After $5.0\ \text{s}$ from rest. $\omega = 30\ \text{rad/s}$, and $K = \tfrac12 \times 0.50\ \text{kg m}^2 \times (30\ \text{rad/s})^2 = 225\ \text{J}$.

Think first. Draw each body on its own, write $\tau = I\alpha$ for what turns and $F = ma$ for what moves, then link them.

Practise with new numbers

Try it in Python

The program follows the grindstone as it spins up.

Remember these

• $\tau_{\text{net}} = I\alpha$.
• $K = \tfrac12 I\omega^2$; work $\tau\theta$; power $\tau\omega$.
• Rope on a drum: $mg - T = ma$, $TR = I\alpha$, $a = \alpha R$.

Check yourself

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

1. A torque of $12\ \text{N m}$ acts on a wheel with $I = 3.0\ \text{kg m}^2$. What is its angular acceleration?

2. How much work does a torque of $5.0\ \text{N m}$ do in turning a shaft through $10$ turns?

3. A motor turns at $100\ \text{rad/s}$ and delivers $1.5\ \text{kW}$. What torque does it give?

4. A $0.50\ \text{kg}$ mass hangs from a rope round a solid drum of mass $1.0\ \text{kg}$. What is its acceleration? ($g = 9.8\ \text{m/s}^2$.)