Moment of Inertia Dimensional Formula [ML²] with Derivation

The moment of inertia dimensional formula is [M¹ L² T⁰], which most books shorten to [ML²]. Rotational inertia depends on mass and on the square of distance from the axis, and time plays no part.

Moment of inertia is to turning what mass is to straight line motion: it measures how hard a body is to spin up or slow down. The formula I = Σmr² shows where the bracket comes from. A flywheel, a cricket bat and a merry go round all have very different values, yet the same dimensions. Below: derivation, unit, a conversion, the radius of gyration link, checks and slips. Standard shapes and their values are on the moment of inertia formula page.

Quick facts

Dimensional formula
[M¹ L² T⁰], short form [ML²]
Defining formula
$I = \sum m r^{2}$
SI unit
kg m²
CGS unit
g cm²; 1 kg m² = 10⁷ g cm²
Linear partner
Mass
Chapter
System of particles and rotational motion, Class 11

Working out the moment of inertia dimensional formula

For a set of point masses about an axis: $I = \sum m_i r_i^{2}$ .
Each mass gives $[M]$ ; each perpendicular distance gives $[L]$ .
Squaring the distance: $[r^{2}] = [L^{2}]$ .
So every term, and therefore the sum, is $[M][L^{2}] = [M L^{2}]$ , with T⁰ if time is shown.

Adding terms never changes dimensions, so it does not matter how many particles there are. The same holds for the integral form $I = \int r^{2}\, dm$ used for continuous bodies: dm is a mass and r² is a length squared.

kg m² and a worked disc

Reading [ML²] as base units gives kg m², the SI unit of moment of inertia. It has no special name. CGS work uses g cm² instead.

A uniform disc of mass 2.0 kg and radius 0.50 m spins about its central axis. Find I in SI and CGS units.

For a disc, $I = \frac{1}{2} M R^{2} = \frac{1}{2}(2.0)(0.50)^{2} = 0.25\ \text{kg m}^2$ .

For CGS, use the powers M¹ and L²: one kg m² equals 10³ g × 10⁴ cm², or 10⁷ g cm².

So I = 0.25 × 10⁷ = 2.5 × 10⁶ g cm².

The ½ in the disc formula is a shape factor. Shape factors change the value of I but never its dimensions.

Every shape, one bracket

Every body has its own expression for I, but all of them are a number times a mass times a length squared:

Moment of inertia of common bodies, all [ML²]
Body and axis	Moment of inertia
Thin ring about its center	$M R^{2}$
Solid disc about its center	$\frac{1}{2} M R^{2}$
Thin rod about its middle	$\frac{1}{12} M L^{2}$
Solid sphere about a diameter	$\frac{2}{5} M R^{2}$

The radius of gyration k is defined by $I = M k^{2}$ . Since I is [ML²] and M is [M], k must be a pure length, [L]. See the radius of gyration page.

One false friend: engineers use a second moment of area for beams, also called an area moment of inertia. It is area times distance squared, [L⁴], measured in m⁴. It is not the mass moment of inertia of rotational dynamics.

Rotational formulas checked with [ML²]

Check: rotational kinetic energy K = ½Iω²

$[I][\omega^{2}] = [M L^{2}][T^{-2}] = [M L^{2} T^{-2}]$ .

This matches energy, so the formula holds up.

Check: angular momentum L = Iω

$[M L^{2}][T^{-1}] = [M L^{2} T^{-1}]$ , the known bracket for angular momentum.

Torque follows too: τ = Iα gives [ML²][T⁻²] = [ML²T⁻²].

These two checks link this page to the angular momentum and torque brackets.

Slips with rotational inertia

Writing [ML]. The distance is squared, so L must carry the power 2.
Writing [ML²T⁻²]. That is energy or torque. Moment of inertia has no time in it.
Thinking a different axis changes the dimensions. It changes the value only.
Mixing up the mass and area moments. [ML²] for rotation, [L⁴] for beams.

The linear partner of I is plain mass, and the full table of rotational brackets is on the dimensional formulas page.

Frequently asked questions

Does moment of inertia depend on time?

No. Its bracket is [M¹ L² T⁰], so the time power is zero. It depends only on mass and how that mass is spread around the axis.

What is the dimension formula of moment of inertia in short form?

The short form is [ML²]. In full it reads [M¹ L² T⁰].

In which SI unit is moment of inertia measured?

kg m², kilogram metre squared. In CGS it is g cm², and 1 kg m² = 10⁷ g cm².

Is moment of inertia a scalar or a vector?

At school level it is treated as a scalar about a chosen axis. In advanced work it is a tensor, but its dimensions are still [ML²].

Why does radius of gyration have the dimensions of length?

Because I = Mk². Dividing [ML²] by [M] leaves [L²], and the square root is [L].