Moment of Inertia Calculator
Find the rotational inertia of cylinders, spheres, and rods.
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Formulas
Solid cylinder: I = ½mr²Solid sphere: I = 2/5 mr². Rod (center): I = 1/12 mL².
Moment of Inertia
Rotational equivalent of mass. Higher I = harder to spin up or slow down. Depends on mass distribution: mass far from axis = higher I. Used in flywheel design, robotics, vehicle dynamics.
What Moment of Inertia Means
The mass moment of inertia measures an object's resistance to rotational acceleration — the rotational equivalent of mass. It depends on both the mass and how that mass is distributed relative to the axis:
Solid cylinder: I = ½mr²I is the moment of inertia (kg·m²), m is mass, and r is radius. The formula varies with the object's shape.
Mass located farther from the axis contributes much more (with the square of distance), which is why a hollow ring resists rotation more than a solid disc of the same mass.
Worked Example
A 10 kg solid cylinder of radius 0.2 m:
I = 0.5 × 10 × 0.2² = 0.2 kg·m²
Moment of Inertia by Shape
| Shape (about central axis) | Formula |
|---|---|
| Solid cylinder / disc | ½mr² |
| Hollow cylinder (thin ring) | mr² |
| Solid sphere | ⅕mr² |
| Rod (about centre) | (1/12)mL² |
Frequently Asked Questions
How does moment of inertia relate to torque?
Through the rotational form of Newton's second law: τ = Iα, where α is angular acceleration. More inertia means more torque is needed to spin up.
Why does mass distribution matter?
Because each mass element contributes mr², mass far from the axis counts far more. This is why figure skaters spin faster when they pull their arms in.
Is this the same as the second moment of area?
No. This is the mass moment of inertia (rotational dynamics). The second moment of area is a geometric property used in beam bending.