Moment of Inertia

The Meaning of Moment of Inertia

When you push a stationary object, it starts moving in a straight line. The resistance it offers to that change in its state of motion is called mass — a measure of its inertia in linear motion. But what about rotation? Try spinning a bicycle wheel by applying a force at its rim. The wheel resists being set into rotation. That resistance is not simply its mass; it depends on how far the mass is from the axis of rotation. A heavy wheel with all its mass concentrated near the axle is easier to spin than a light wheel with the same mass spread out at the rim.

This rotational analogue of mass is called the moment of inertia, denoted by $I$. For a single particle of mass $m$ at a perpendicular distance $r$ from a fixed axis, the moment of inertia is defined as:

The word "moment" here hints at the geometric factor $r^2$ — the farther the mass is from the axis, the greater its rotational inertia. For a system of many particles, the total moment of inertia about the same axis is simply the sum of the individual moments:

where $r_i$ is the perpendicular distance of the $i$-th particle from the axis. This is a scalar quantity (unlike torque or angular momentum, which are vectors). Its SI unit is $\text{kg m}^2$, and its dimensional formula is $[M L^2 T^0]$.

Why $r^2$ and Not $r$?

The $r^2$ factor emerges naturally from the rotational analogue of Newton's second law. For a particle moving in a circle of radius $r$, the tangential acceleration $a_t = r \alpha$, where $\alpha$ is the angular acceleration. The tangential force required is $F = m a_t = m r \alpha$. The torque about the axis is $\tau = F r = (m r^2) \alpha$. So $\tau = I \alpha$, with $I = m r^2$. The square appears because torque involves force times distance, and the force itself already contains one factor of $r$ from the acceleration relation.

Moment of Inertia of a Rigid Body

For a continuous rigid body, we replace the sum by an integral. Imagine the body divided into infinitesimal mass elements $dm$, each at a perpendicular distance $r$ from the axis. Then:

The integration is carried out over the entire volume of the body. The value of $r$ depends on the shape of the body and the chosen axis. This integral is often easier to evaluate using the body's density $\rho$. If $\rho$ is uniform, $dm = \rho \, dV$, and $I = \rho \int r^2 \, dV$.

Properties of Moment of Inertia

The textbook lists three key properties that follow directly from the definition $I = \sum m_i r_i^2$. Each one is proved below.

›Proof

Property (I): Moment of inertia is additive.

If a body consists of several parts (say, a disc with a hole, or a composite of two shapes), the total moment of inertia about a given axis is the sum of the moments of inertia of its individual parts about the same axis.

Proof: Let the body be divided into $n$ parts, each with mass $m_k$ and moment of inertia $I_k = \sum_{i \in \text{part }k} m_i r_i^2$. Then the total moment of inertia is:

$I = \sum_{\text{all }i} m_i r_i^2 = \sum_{k=1}^n \left( \sum_{i \in \text{part }k} m_i r_i^2 \right) = \sum_{k=1}^n I_k$

This holds because the sum over all particles can be grouped into disjoint subsets. The additivity is a direct consequence of the linearity of the summation.

›Proof

Property (II): The moment of inertia depends on the mass and the distribution of mass about the axis.

This is not a theorem to prove but a qualitative observation. Two bodies of the same mass can have very different moments of inertia if their mass distributions differ. For example, a thin ring of mass $M$ and radius $R$ has $I = MR^2$ about its central axis, while a solid disc of the same mass and radius has $I = \frac{1}{2} MR^2$ — half as much, because the disc's mass is spread closer to the axis on average.

›Proof

Property (III): For a given body and a given axis, the moment of inertia is a constant.

Proof: The definition $I = \sum m_i r_i^2$ involves only the fixed masses $m_i$ and the fixed perpendicular distances $r_i$ from the axis. As long as the body is rigid (distances between particles do not change) and the axis is fixed in space, each $r_i$ is a constant. Hence $I$ is a constant for that body-axis combination. This constancy is what makes $I$ a useful parameter in rotational dynamics — it plays the same role as mass does in linear motion.

The Radius of Gyration

It is often convenient to express the moment of inertia of a body of mass $M$ in terms of a single length $k$, called the radius of gyration. It is defined as the distance from the axis at which the entire mass of the body could be concentrated to produce the same moment of inertia:

The radius of gyration tells you, in a single number, how "spread out" the mass is relative to the axis. For a thin ring about its centre, $k = R$; for a solid disc about its centre, $k = R/\sqrt{2}$; for a solid sphere about its diameter, $k = \sqrt{2/5}\,R$.

Table 6.1: Moments of Inertia of Some Regular-Shaped Bodies

This is Table 6.1 from the textbook itself — it lists the moment of inertia of these standard shapes about the specific axes shown, for bodies of uniform density. Their derivations are beyond the scope of this book (the textbook itself says you'll work them out in higher classes), but these are the values you must know and be able to apply for exams.