Moment of Inertia (Rod, Center)

I = \frac{1}{12}ML^2

Moment of inertia of a thin uniform rod rotating about its center, perpendicular to its length.

Variables

Moment of inertia

Rotational inertia

Allowed values: 0 or more

kg·m²

Mass

Total mass of rod

Allowed values: 0 or more

Length

Length of rod

Allowed values: greater than 0

Derivation

I = ∫r²dm. For a rod: I = ∫₋L/2^L/2 x²(M/L)dx = (1/12)ML².

Your notes

0/5000

Check value

Round inputs run through this page's calculator. Use it to test your own working or a rearranged form. It is a test case, not a real-world example.

M = 2 kg (Mass)
L = 3 m (Length)

Result: I = 1.5 kg·m²

Other mechanics formulas with the same quantities

Newton's Second Law (Rotation) (Rotational Motion) shares Moment of inertia
Rotational Kinetic Energy (Rotational Motion) shares Moment of inertia
Angular Momentum (Rotational Motion) shares Moment of inertia
Rolling Kinetic Energy (Rotational Motion) shares Moment of inertia

Where this formula is listed

Mechanics formulas, under Rotational Motion, alongside Angular Velocity, Angular Acceleration, Linear-Angular Velocity Relation, Centripetal Acceleration and Centripetal Force.
In the mechanics list it comes after Moment of Inertia (Solid Sphere) and before Parallel Axis Theorem.
Mechanics formula sheet, one of 58 formulas printed with symbol keys and SI units.
The full formula index, grouped by category and topic.

Common questions

What is the Moment of Inertia (Rod, Center) formula?

Moment of Inertia (Rod, Center) is written I = (1/12)ML². Moment of inertia of a thin uniform rod rotating about its center, perpendicular to its length. It belongs to Mechanics, under Rotational Motion.

What are the units in I = (1/12)ML²?

In SI units, I is moment of inertia, measured in kg·m²; M is mass, measured in kg; L is length, measured in m.

How is the Moment of Inertia (Rod, Center) formula derived?

I = ∫r²dm. For a rod: I = ∫₋L/2^L/2 x²(M/L)dx = (1/12)ML². The result is I = (1/12)ML².