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Moment of inertia

Forces and motion · level 8, lesson 2 of 6

A question to start

Race a hoop, a solid disc and a solid ball down the same slope, all from rest. Which wins? Would the order change on Mars?

The idea

In a straight line, mass is the measure of inertia. In rotation, inertia depends on the mass and on how far it is from the axis. The moment of inertia is

\[I = \sum m_i r_i^2,\]

the sum over every bit of mass of the mass times the square of its distance from the axis. Mass far out counts much more than mass near the middle.

Shape, mass \(M\), radius \(R\), about its centre \(I\)
Thin hoop or ring \(MR^2\)
Solid disc or cylinder \(\tfrac12 MR^2\)
Solid ball \(\tfrac25 MR^2\)
Thin rod of length \(L\), about its middle \(\tfrac1{12} ML^2\)

A rolling object has two kinds of kinetic energy: moving along, \(\tfrac12 mv^2\), and spinning, \(\tfrac12 I\omega^2\). Rolling without slipping, \(v = \omega R\). Write \(I = kmR^2\). Energy from a drop \(h\): \(mgh = \tfrac12 mv^2(1 + k)\), so \(v^2 = 2gh/(1+k)\), and the acceleration along a slope of angle \(\theta\) is

\[a = \frac{g\sin\theta}{1 + k}.\]
ball: k = 2/5disc: 1/2hoop: 1

So the ball wins, then the disc, then the hoop. The mass and the radius cancel: any solid ball beats any hoop. And on Mars, every acceleration is 0.38 of Earth's, but the order is the same, because \(g\) scales them all together.

Rolling down 2 m of a 20° slope from rest Mars Earth
Ball, \(a = g\sin\theta / 1.4\) 0.91 m/s², 2.1 s 2.40 m/s², 1.3 s
Disc, \(/1.5\) 0.85 m/s², 2.2 s 2.24 m/s², 1.3 s
Hoop, \(/2\) 0.63 m/s², 2.5 s 1.68 m/s², 1.5 s

Worked examples

1. A disc on Mars. \(a = 3.71 \times 0.342 / 1.5 = 0.85\) m/s². Time for 2 m: \(\sqrt{4/0.85} = 2.2\) s.

2. A flywheel. A steel disc of 50 kg and radius 0.3 m: \(I = \tfrac12 \times 50 \times 0.09 = 2.25\) kg m².

3. Two masses on a rod. Two 1 kg masses at 0.5 m from the middle of a light rod: \(I = 2 \times 1 \times 0.25 = 0.5\) kg m². Move them in to 0.25 m: \(I = 0.125\) kg m², a quarter.

Try it

  1. Find \(I\) for a 2 kg hoop of radius 0.4 m.
  2. Find \(I\) for a 10 kg solid cylinder of radius 0.2 m.
  3. Find the acceleration of a solid ball rolling down a 30° slope on Mars.
  4. Why does a hoop roll down more slowly than a disc?
Answers
  1. \(2 \times 0.16 = 0.32\) kg m².
  2. \(\tfrac12 \times 10 \times 0.04 = 0.2\) kg m².
  3. \(3.71 \times 0.5 / 1.4 = 1.33\) m/s².
  4. All its mass is far from the axis, so more of its energy goes into spinning and less into moving along.

Common mistake

"A heavier ball rolls down faster." Mass and radius cancel: all solid balls roll down at the same rate (without air). Only the shape, the value of \(k\), matters.

Where it leads