Moment of inertia measures how strongly a rigid body resists angular acceleration about a chosen rotation axis.
It is the rotational analogue of mass, but it also depends on how the mass is distributed relative to the axis.
1. General definition
For point masses, moment of inertia is
\[
I=\sum_i m_i r_i^2,
\]
where \(r_i\) is the perpendicular distance from each mass to the rotation axis. For a continuous homogeneous body,
the sum becomes an integral:
\[
I=\int r^2\,dm.
\]
This is why two objects with the same mass can have different moments of inertia. More mass farther from the axis
gives a larger \(I\).
2. Common formulas
For standard homogeneous shapes, the integral has already been evaluated. The correct formula depends on both the
shape and the axis.
| Object |
Axis |
Moment of inertia |
| Thin hoop / thin ring |
Central symmetry axis |
\(I=MR^2\) |
| Thin hoop / thin ring |
Diameter axis |
\(I=\frac{1}{2}MR^2\) |
| Solid disk / solid cylinder |
Central symmetry axis |
\(I=\frac{1}{2}MR^2\) |
| Solid cylinder |
Diameter axis through center |
\(I=\frac{1}{12}M(3R^2+h^2)\) |
| Thick hollow cylinder |
Central symmetry axis |
\(I=\frac{1}{2}M(R_1^2+R_2^2)\) |
| Thin rod |
Perpendicular axis through center |
\(I=\frac{1}{12}ML^2\) |
| Thin rod |
Perpendicular axis through end |
\(I=\frac{1}{3}ML^2\) |
| Rectangular plate |
Perpendicular axis through center |
\(I=\frac{1}{12}M(a^2+b^2)\) |
| Solid sphere |
Any diameter |
\(I=\frac{2}{5}MR^2\) |
| Thin spherical shell |
Any diameter |
\(I=\frac{2}{3}MR^2\) |
3. Why the axis matters
Moment of inertia is not just a property of the object; it is a property of the object about a particular axis.
A rod rotating about its center has
\[
I_{\mathrm{center}}=\frac{1}{12}ML^2.
\]
The same rod rotating about its end has
\[
I_{\mathrm{end}}=\frac{1}{3}ML^2.
\]
The end-axis value is larger because the mass is, on average, farther from the rotation axis.
4. Parallel-axis theorem
If the moment of inertia about a center-of-mass axis is known, the moment of inertia about a parallel axis shifted
by distance \(d\) is
\[
I=I_{\mathrm{cm}}+Md^2.
\]
This theorem explains formulas such as the rod about an end. Starting from the center-axis formula,
\[
I_{\mathrm{cm}}=\frac{1}{12}ML^2,
\]
and shifting the axis by \(d=L/2\),
\[
I_{\mathrm{end}}
=
\frac{1}{12}ML^2
+
M\left(\frac{L}{2}\right)^2.
\]
\[
I_{\mathrm{end}}
=
\frac{1}{12}ML^2+\frac{1}{4}ML^2
=
\frac{1}{3}ML^2.
\]
5. Units
The SI unit of moment of inertia is
\[
\mathrm{kg\,m^2}.
\]
This comes from the definition \(I=\int r^2\,dm\), where distance is squared and multiplied by mass.
6. Worked example
A solid cylinder has mass
\[
M=8\ \mathrm{kg},
\]
and radius
\[
R=0.15\ \mathrm{m}.
\]
About its central symmetry axis, the formula is
\[
I=\frac{1}{2}MR^2.
\]
Substitute the values:
\[
I=\frac{1}{2}(8)(0.15)^2.
\]
\[
I=4(0.0225)=0.09\ \mathrm{kg\,m^2}.
\]
Therefore,
\[
\boxed{I=0.09\ \mathrm{kg\,m^2}}.
\]
Key idea: mass close to the axis is easier to spin than mass far from the axis. This is why a hoop has a larger
moment of inertia than a solid disk of the same mass and radius.
