Moment of Inertia Formula: I = Σmr², Shapes, Unit, Examples

The moment of inertia formula is $I = \sum m_i r_i^2$ : split the body into small masses, multiply each mass by the square of its perpendicular distance from the axis of rotation, and add the results. For a single small mass the sum shrinks to $I = mr^2$ . The answer is measured in kilogram metre squared (kg m²).

Moment of inertia plays the part in turning that mass plays in straight line motion. A large value means the body is hard to start spinning and hard to stop. Unlike mass, though, it depends on where the mass sits, so the same object has a different I about every axis. This page sits in the rotation and circular motion section, and the matching dimensional formula of moment of inertia page derives [M L²] in detail.

Quick facts

General formula
$I = \sum m_i r_i^2$ or $I = \int r^2\, dm$
Point mass
$I = mr^2$
SI unit
kilogram metre squared (kg m²)
Dimensional formula
[M¹ L² T⁰]
Type
Scalar for a fixed axis (a tensor in advanced work)
Depends on
Mass, its spread, and the position of the axis
Shift of axis
$I = I_{cm} + Md^2$

Reading the moment of inertia formula term by term

What each symbol stands for
Symbol	Meaning	SI unit
$I$	Moment of inertia about one stated axis	kg m²
$m_i$	Mass of one small piece of the body	kg
$r_i$	Perpendicular distance of that piece from the axis, not from the centre	m
$\sum$	Add over every piece; for a smooth body it becomes an integral	none

The distance is squared, so mass far from the axis counts much more than mass close to it. Move a 1 kg lump from 0.1 m to 0.3 m and its share of I grows nine times, from 0.01 kg m² to 0.09 kg m². That is why a ring is harder to spin than a disc of the same mass and radius: all of the ring's mass sits at the rim.

The total moment of inertia formula for a body made of separate parts is simply the sum of the parts about the same axis: $I_{total} = I_1 + I_2 + I_3 + \dots$ . You may add moments of inertia only when every one is taken about the same line.

Standard results for common shapes

Moment of inertia of uniform bodies of mass M
Body	Axis	I
Thin ring or hoop, radius R	Through centre, perpendicular to plane	$MR^2$
Solid disc or cylinder, radius R	Through centre, along the symmetry axis	$\frac{1}{2}MR^2$
Thin rod, length L	Through centre, perpendicular to rod	$\frac{1}{12}ML^2$
Thin rod, length L	Through one end, perpendicular to rod	$\frac{1}{3}ML^2$
Solid sphere, radius R	Any diameter	$\frac{2}{5}MR^2$
Thin spherical shell, radius R	Any diameter	$\frac{2}{3}MR^2$
Ring, radius R	Any diameter	$\frac{1}{2}MR^2$
Disc, radius R	Any diameter	$\frac{1}{4}MR^2$

Each line comes from $I = \int r^2\,dm$ . Students in Class 11 and in AP Physics C are expected to know the first five without looking them up. The number in front, such as $\frac{2}{5}$ , tells you how the mass is spread: the smaller it is, the closer the mass lies to the axis. The radius of gyration packs the same idea into one length, $K = \sqrt{I/M}$ .

Worked examples

Example 1: two masses on a light rod

A 2 kg mass sits 0.5 m from an axis and a 3 kg mass sits 0.2 m from the same axis on a rod of negligible mass. Find I.

$I = (2)(0.5)^2 + (3)(0.2)^2 = 0.50 + 0.12 = 0.62\ \text{kg m}^2$

The lighter mass contributes more because it is farther out.

Example 2: a ball whirled on a string

A 0.2 kg ball moves in a circle of radius 50 cm. Treat it as a point mass and find its moment of inertia about the centre.

Convert: r = 0.50 m. $I = mr^2 = 0.2 \times 0.25 = 0.05\ \text{kg m}^2$ . The calculator above gives the same value.

Example 3: rod about its end, exam style

A uniform metre scale style rod of mass 500 g and length 120 cm pivots at one end. Find I about the pivot using the parallel axis theorem.

Convert: M = 0.5 kg, L = 1.2 m. About the centre, $I_{cm} = \frac{1}{12}(0.5)(1.2)^2 = 0.06\ \text{kg m}^2$ .

Shift by d = 0.6 m: $I = 0.06 + 0.5(0.6)^2 = 0.06 + 0.18 = 0.24\ \text{kg m}^2$ , which matches $\frac{1}{3}ML^2$ .

Parallel and perpendicular axis theorems

Parallel axis theorem: $I = I_{cm} + Md^2$ . If you know I about an axis through the center of mass, often spelled centre in Indian books,, you get I about any parallel axis a distance d away by adding $Md^2$ . It works only when one of the two axes passes through the center of mass. See the center of mass formula for locating that point.

Perpendicular axis theorem: $I_z = I_x + I_y$ for a flat (laminar) body, where x and y lie in its plane and z is normal to it. It is how a disc about a diameter gets $\frac{1}{4}MR^2$ from $\frac{1}{2}MR^2$ about its centre. For a cross section of a beam, the same sum gives the polar moment of inertia.

Never apply the perpendicular axis theorem to a sphere or any thick 3D body. It holds only for flat plates.

Where inertia shows up in other rotation formulas

Linear quantity and its rotational partner
Straight line	Rotation	Link
Mass $m$	Moment of inertia $I$	$I = \sum mr^2$
$F = ma$	$\tau = I\alpha$	Torque
$p = mv$	$L = I\omega$	Angular momentum
$\frac{1}{2}mv^2$	$\frac{1}{2}I\omega^2$	Spin kinetic energy

Every rotational law above uses the same I, so a mistake in I spreads through the whole problem. In a lab you can measure I directly; one neat method is the bifilar suspension.

Common slips with inertia

Using distance from the centre instead of from the axis. r is always measured at right angles to the axis.
Forgetting to square r. Check that your answer has units of kg m², not kg m.
Applying $Md^2$ between two axes that both miss the center of mass. Go to the center of mass first, then out again.
Leaving grams and centimetres. 500 g and 120 cm become 0.5 kg and 1.2 m before you substitute.
Quoting I without its axis. "I of a rod" means nothing until you say which axis.

Frequently asked questions

What is the formula of moment of inertia?

For a set of particles it is I = Σ mr², the sum of each mass times the square of its perpendicular distance from the axis. For a single point mass this is I = mr², and for a smooth body it becomes the integral of r² dm.

What is the formula of inertia in physics?

In straight line motion, inertia is measured by mass alone, in kg. In rotation, the inertia formula physics students use is the moment of inertia I = Σ mr², which also depends on how far the mass is from the axis.

In which unit is moment of inertia measured?

Use kilogram metre squared, written kg m². In base dimensions that reads [M¹ L² T⁰], one power of mass and two of length.

Is moment of inertia a vector?

No. For rotation about a fixed axis it is a scalar with no direction of its own. In advanced mechanics it is written as a tensor, but school and AP problems treat it as a positive number.

What is the moment of inertia general formula for a continuous body?

Write I = ∫ r² dm, express dm using the density and a small length, area or volume, and integrate over the whole body. The standard results for rods, discs and spheres all come from this one integral.