Moment of inertia and the two theorems on axes

Turning inertia

Mass measures how hard a body is to speed up in a straight line. For turning, what matters is not just the mass, but how far it lies from the axis. That is the moment of inertia:

in $\text{kg m}^2$. Mass far from the axis counts much more than mass near it.

The radius of gyration $k$ is the distance at which all the mass would give the same $I$: $I = Mk^2$.

Two theorems that move the axis

Parallel axes. If $I_{cm}$ is about an axis through the centre of mass, then about any parallel axis a distance $d$ away,

Perpendicular axes (for flat bodies only). With $x$ and $y$ in the plane and $z$ at right angles to it, all through one point:

So a disc, with $I_z = \tfrac12 MR^2$, has $\tfrac14 MR^2$ about any diameter.

You should find the moment of inertia smallest about the middle, and four times as big about an end.

Watch out. The parallel axis theorem must start from the axis through the centre of mass. Moving between two other axes needs two steps.

A worked example

Two $2.0\ \text{kg}$ balls are joined by a light rod $0.60\ \text{m}$ long.

1. About the middle. Each ball is $0.30\ \text{m}$ away: $I = 2 \times 2.0\ \text{kg} \times (0.30\ \text{m})^2 = 0.36\ \text{kg m}^2$.
2. About one ball. One ball is on the axis, the other $0.60\ \text{m}$ away: $I = 2.0\ \text{kg} \times (0.60\ \text{m})^2 = 0.72\ \text{kg m}^2$.
3. Check by the theorem. $I = I_{cm} + Md^2 = 0.36\ \text{kg\,m}^2 + 4.0\ \text{kg} \times (0.30\ \text{m})^2 = 0.72\ \text{kg m}^2$.

Think first. For point masses, add $mr^2$. For standard shapes, start from the table and move the axis with a theorem.

Practise with new numbers

Try it in Python

The program cuts a rod into a thousand small pieces, adds up $mr^2$, and compares it with the formula.

Remember these

• $I = \sum m_ir_i^2$, in $\text{kg m}^2$; $I = Mk^2$.
• Ring $MR^2$; disc $\tfrac12 MR^2$; rod $\tfrac{1}{12}ML^2$ (middle), $\tfrac13 ML^2$ (end); solid sphere $\tfrac25 MR^2$.
• $I = I_{cm} + Md^2$; for flat bodies $I_z = I_x + I_y$.

Check yourself

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

1. A $0.5\ \text{kg}$ ball whirls on a light string $0.8\ \text{m}$ long. What is its moment of inertia about the centre?

2. A ring and a disc have the same mass and radius. Which is harder to set spinning about its axis? Why?

3. What is the moment of inertia of a $4.0\ \text{kg}$ uniform rod $3.0\ \text{m}$ long about one end?

4. A flat square plate has $I = 0.8\ \text{kg m}^2$ about an axis through its centre at right angles to it. What is $I$ about an axis in its plane, through its centre, parallel to a side?