In a cycle repair shop in Ludhiana, a mechanic mounts two things of the same mass on the same kind of axle. One is a bicycle wheel with a heavy steel rim and thin spokes. The other is a flat steel plate, cut round, of the same radius. He gives each the same flick. The plate starts spinning briskly. The wheel starts lazily, but once turning it keeps turning far longer.
What resists a change of spin
In translation, the mass of a body measures its inertia. A loaded lorry needs a larger force than a scooter to gain speed at the same rate, and a larger force to stop. Spinning bodies also resist a change, but mass is not the whole story. The wheel and the plate of the repair shop have the same mass, yet they respond to the same flick very differently.
The cleanest way to find the right measure is through energy. A spinning body has kinetic energy even when its centre stays still, because each of its particles moves. Recall from Work, Kinetic Energy and Power that a particle of mass \(m\) at speed \(v\) has kinetic energy \(\tfrac12 mv^2\). Now add these energies for every particle of a spinning body.
\[ K = \tfrac12 m_1 v_1^2 + \tfrac12 m_2 v_2^2 + \tfrac12 m_3 v_3^2 + \dots \]
Why. the kinetic energy of the body is the sum of the kinetic energies of its particles.
\[ v_i = \omega r_i \]
Why. every particle turns with the one angular velocity \(\omega\) of the rigid body, on a circle of radius \(r_i\) about the axis.
\[ K = \tfrac12 m_1 r_1^2\omega^2 + \tfrac12 m_2 r_2^2\omega^2 + \dots \]
Why. substitute \(v_i = \omega r_i\) in each term.
\[ K = \tfrac12\left(\sum m_i r_i^2\right)\omega^2 \]
Read it. \(\omega\) is the same for every term, so \(\tfrac12\omega^2\) comes out as a common factor.
Compare the result with \(K = \tfrac12 Mv^2\) for a body in translation. The angular velocity \(\omega\) takes the place of \(v\). The sum \(\sum m_i r_i^2\) takes the place of \(M\). This sum is the moment of inertia \(I\) of the body about the axis. It is the rotational twin of mass.
The moment of inertia of a system of particles about an axis is \(I = \sum m_i r_i^2\). Here \(r_i\) is the perpendicular distance of the particle of mass \(m_i\) from the axis. Its SI unit is kg m². It is a scalar.
A light rod turns about an axis through one end, perpendicular to the rod. It carries beads of \(0.5\,\text{kg}\) at \(0.2\,\text{m}\) and \(0.4\,\text{m}\) from the axis, and a bead of \(1\,\text{kg}\) at \(0.6\,\text{m}\). Find its moment of inertia, and its kinetic energy when it turns at \(10\,\text{rad/s}\).
\(I = 0.5 \times 0.2^2 + 0.5 \times 0.4^2 + 1 \times 0.6^2 = 0.02 + 0.08 + 0.36 = 0.46\,\text{kg m}^2\).
\(K = \tfrac12 I\omega^2 = \tfrac12 \times 0.46 \times 10^2 = 23\,\text{J}\).
Check by particles: the speeds are \(2\), \(4\) and \(6\,\text{m/s}\), so \(K = 1 + 4 + 18 = 23\,\text{J}\). The outer bead alone carries more than three quarters of the energy.
Which statement about the moment of inertia is correct?
What I depends on
Three facts follow straight from the sum. More mass gives a larger \(I\), because every term grows. Mass placed farther from the axis gives a larger \(I\), and much faster than in proportion. Each distance enters squared, so twice as far means four times the contribution. And the same body has a different \(I\) about every axis, because changing the axis changes every distance.
One thing \(I\) does not depend on is how fast the body turns. It is a property of the body and the axis, just as mass is a property of the body alone. A fan has the same moment of inertia at rest and at full speed.
A solid disc and a thin ring have the same mass, \(2\,\text{kg}\), and the same radius, \(0.25\,\text{m}\). Each turns freely on a fixed axle. A string wound round each rim is pulled with the same steady force. Which gains angular velocity faster?
The two bodies in the figure above differ only in where their mass lies. The ring keeps all \(2\,\text{kg}\) at \(0.25\,\text{m}\), so \(I_{\text{ring}} = 2 \times 0.25^2 = 0.125\,\text{kg m}^2\). The disc has half of that, as the next lesson shows. Under the same pull the disc gains angular velocity twice as fast: after \(3\,\text{s}\) it has turned through \(9\,\text{rad}\) and the ring through only \(4.5\,\text{rad}\). The chapter on Dynamics of Rotation explains exactly how the pull and \(I\) decide this.
Try the three chips of Fig 3.1. About the z axis, which passes through O perpendicular to the page, every mass counts: \(2 \times 1^2 + 2 \times 1^2 + 1 \times 2^2 + 1 \times 2^2 = 12\,\text{kg m}^2\). About the x axis the two \(2\,\text{kg}\) masses lie on the axis and add nothing, so \(I = 8\,\text{kg m}^2\). About the y axis the \(1\,\text{kg}\) masses add nothing, and \(I = 4\,\text{kg m}^2\). Nothing in the system changed; only the axis did.
The phrase "the moment of inertia of this rod" has no meaning until an axis is named. A book that states \(I = 2\,\text{kg m}^2\) for a body without an axis has left out half the information.
Two \(1\,\text{kg}\) balls are fixed on a light rod, each \(0.3\,\text{m}\) from the axis. Both are moved out to \(0.6\,\text{m}\). What happens to the moment of inertia?
The moment of inertia of a body depends on which of these?
Distances from the axis, not from the origin
For point masses the work has three steps. Name the axis. Find each particle's perpendicular distance from the axis. Then add \(m r^2\). The second step carries the common mistake. The \(r\) in the sum is never the distance from the origin; it is the distance from the axis, measured at right angles to it.
Take a particle at \((x, y, z)\). Its distance from the z axis is \(\sqrt{x^2 + y^2}\). The coordinate \(z\) does not enter, because moving along the axis does not change the distance from it. In the same way the distance from the x axis is \(\sqrt{y^2 + z^2}\), and from the y axis \(\sqrt{x^2 + z^2}\).
A \(2\,\text{kg}\) particle sits at the point \((3\,\text{m}, 4\,\text{m}, 12\,\text{m})\). What is its moment of inertia about the z axis, in kg m²?
Adding the parts
Because \(I\) is a sum over particles, the moment of inertia of a body made of parts is the sum of its parts' values. There is one condition: every part must be taken about the same axis. A dumbbell is a rod plus two balls; a bicycle wheel is a rim, a hub and many spokes. Each part is found separately, about the one axis, and the results are added.
Two small balls of \(0.5\,\text{kg}\) each are fixed at the ends of a light rod \(0.8\,\text{m}\) long. Find the moment of inertia about an axis through the middle of the rod, perpendicular to it, in kg m².
The same dumbbell of two small balls on a light rod is now spun about the rod itself, as a spit turns. What is its moment of inertia?
The Dutch scientist Christiaan Huygens built the first accurate pendulum clocks. In 1673 he published a book on them and faced a hard question. Where, in a swinging bar of real size, is the one point that swings like a simple pendulum? His answer needed the sum of each small mass times the square of its distance from the pivot. He used \(\sum mr^2\) nearly a century before Euler, in 1765, gave it the name moment of inertia.
A sugarcane juice machine at a roadside stall has a large iron wheel with a thick rim and thin spokes. That wheel is a flywheel. Its mass sits far from the axle, so its \(I\) is large for its weight. Between squeezes the wheel keeps turning steadily, and each stalk of cane barely slows it. A solid wheel of the same mass would have a smaller \(I\) and would jerk with every stalk.
Now return to the repair shop. The bicycle wheel keeps most of its mass in its rim, far from the axle, so its moment of inertia is large. The round plate spreads its mass evenly, much of it close to the axle, so its \(I\) is smaller. The same flick sets the plate spinning quickly. The wheel responds slowly, and once spinning it keeps its motion longer.
- Using the distance from the origin in \(\sum mr^2\).Use the perpendicular distance from the axis. For the z axis it is \(\sqrt{x^2 + y^2}\).
- Quoting a moment of inertia without an axis.Always name the axis. The same body has a different \(I\) about each axis.
- Doubling \(I\) when the distances double.Distances enter squared. Twice as far means four times the moment of inertia.
- Adding the \(I\) of parts taken about different axes.Parts add only when every one is taken about the same axis.