Moment of inertia
Mass for rotation
Spin a bicycle wheel, then a solid disc of the same mass and size. The wheel is harder to get going, and harder to stop.
In straight-line motion, mass measures how hard it is to change velocity. For rotation, the matching quantity is the moment of inertia.
Think of the body as many tiny particles. Particle i, at distance rᵢ from the axis, moves with speed vᵢ = ωrᵢ.
Its kinetic energy is ½mᵢvᵢ² = ½mᵢrᵢ²ω². Add up all the particles: K = ½(Σmᵢrᵢ²)ω².
The sum in brackets is the moment of inertia, I = Σmᵢrᵢ². So rotational kinetic energy is K = ½Iω², just like ½mv².
Notice the r²: mass far from the axis counts much more than mass near it. Double the distance and that bit of mass counts four times as much.
So I depends not just on the mass, but on how the mass is spread out, and on which axis you choose. Its unit is kg m².
A bicycle wheel has most of its mass in the rim, far from the axle. That is why its moment of inertia is large.
Standard results and radius of gyration
For simple shapes, the sum Σmᵢrᵢ² has been worked out once and for all. Here are the standard results.
A thin ring of mass M and radius R, about the axis through its centre perpendicular to its plane: I = MR². All its mass is at distance R.
A uniform disc about the same axis: I = ½MR². Much of its mass is close to the centre, so it gets half the ring’s value.
A thin rod of length L, about an axis through its middle and perpendicular to it: I = ML²/12.
A solid sphere about a diameter: I = ⅖MR². A hollow cylinder about its axis: MR². A solid cylinder about its axis: ½MR².
We can always write I = Mk², where k is the radius of gyration: the distance at which all the mass could sit to give the same I.
For a ring, k = R. For a disc, ½MR² = Mk², so k = R/√2. For a rod about its middle, k = L/√12.
Engines carry a heavy wheel called a flywheel, with a large moment of inertia. It resists sudden changes in speed and keeps the ride smooth.
Try it: the inertia race
- Both bodies have the same mass, size and torque. Only the shape differs.
- Press Race and see which reaches 5 turns first.
- Compare the radius of gyration k with R for each body.
- Try a ring against two beads at ±R. Why do they tie?
Quick check
A ring and a disc have the same mass and radius. The same torque acts on each about its central axis. Which speeds up faster?
- The ring, because its mass is at the edge
- The disc, because its moment of inertia is half the ring’s
- Both speed up equally, because the masses are equal
- It depends on how thick the disc is
Answer
B. I(ring) = MR² and I(disc) = ½MR². The disc’s mass is, on average, closer to the axis, so its moment of inertia is half. With the same torque, the disc gets twice the angular acceleration.
Work it out
Two small balls of 2 kg each are fixed to the ends of a light rod 1 m long. What is the moment of inertia about an axis through the middle, perpendicular to the rod?
- 0.5 kg m²
- 1 kg m²
- 2 kg m²
- 4 kg m²
Answer
B. Each ball is 0.5 m from the axis. I = Σmr² = 2 × 0.5² + 2 × 0.5² = 1 kg m². The light rod adds nothing.
Summary
- Moment of inertia:
I = Σmᵢrᵢ² - Rotational kinetic energy:
K = ½Iω² - Radius of gyration:
I = Mk² - Ring / disc (central axis):
MR² / ½MR² - Rod (middle) / solid sphere (diameter):
ML²/12 / ⅖MR²
- Moment of inertia is the rotational version of mass: it measures how hard it is to change a body’s rotation.
- It depends on the mass, how the mass is spread out, and the axis. Mass far from the axis counts most.
- The radius of gyration k is where all the mass could be placed to give the same I.
- Flywheels use a large I to keep the speed of a machine steady.