Moment of Inertia Calculator — Rotational Inertia Calculator
Calculate moment of inertia for common geometric shapes including cylinders, spheres, rods, and plates. Essential for rotational dynamics, mechanical engineering, and physics calculations. See also our Angular Acceleration Calculator and Torque Calculator.
How to Calculate Moment of Inertia
- Select the shape from the dropdown menu.
- Enter the mass and required dimensions.
- Click Calculate to find the moment of inertia.
- The result shows the rotational inertia about the specified axis.
Moment of Inertia Formulas
Common Moment of Inertia Formulas:
Solid Cylinder (about axis): I = ½mr²
Hollow Cylinder (about axis): I = ½m(r₁² + r₂²)
Solid Sphere (about diameter): I = ⅖mr²
Hollow Sphere (about diameter): I = ⅔mr²
Rod (about center): I = 1/12 ml²
Rod (about end): I = ⅓ml²
Rectangular Plate (about center): I = 1/12 m(w² + h²)
Disk (about center axis): I = ½mr²
Where:
I = Moment of inertia (kg⋅m²)
m = Mass (kg)
r = Radius (m)
l = Length (m)
w, h = Width, height (m)Example Calculation
Problem: Find the moment of inertia of a solid cylinder with mass 10 kg and radius 0.5 m about its axis.
Given: m = 10 kg, r = 0.5 m
Solution: I = ½mr² = ½ × 10 × (0.5)² = 1.25 kg⋅m²
Answer: The moment of inertia is 1.25 kg⋅m².
Moment of Inertia Reference Table
| Shape | Axis of Rotation | Formula |
|---|---|---|
| Point mass | Distance r from axis | I = mr² |
| Solid cylinder | Central axis | I = ½mr² |
| Hollow cylinder | Central axis | I = ½m(r₁² + r₂²) |
| Solid sphere | Any diameter | I = ⅖mr² |
| Hollow sphere | Any diameter | I = ⅔mr² |
| Thin rod | Through center ⊥ rod | I = 1/12 ml² |
| Thin rod | Through end ⊥ rod | I = ⅓ml² |
| Rectangular plate | Through center ⊥ plate | I = 1/12 m(a² + b²) |
Technical Details
Moment of inertia quantifies an object's resistance to rotational acceleration about a specific axis. It depends on both mass distribution and the axis of rotation. Objects with mass farther from the rotation axis have higher moment of inertia. The parallel axis theorem allows calculation for any axis: I = I_cm + md², where I_cm is the moment about the center of mass and d is the distance between axes. Moment of inertia is crucial in rotational dynamics, appearing in Newton's second law for rotation: τ = Iα, where τ is torque and α is angular acceleration.
Frequently Asked Questions
What is moment of inertia?
Moment of inertia is a measure of an object's resistance to rotational acceleration. It's the rotational analog of mass in linear motion and depends on mass distribution relative to the rotation axis.
How does axis location affect moment of inertia?
Moment of inertia increases as the rotation axis moves farther from the object's center of mass. The parallel axis theorem quantifies this: I = I_cm + md².
Why do hollow objects have different moments of inertia?
Hollow objects have mass distributed farther from the center, resulting in higher moment of inertia than solid objects of the same mass and outer dimensions.
How is moment of inertia used in engineering?
Engineers use moment of inertia to design rotating machinery, calculate torque requirements, analyze vibrations, and determine structural beam properties for bending resistance.
Can moment of inertia be negative?
No, moment of inertia is always positive because it involves squared distances. However, products of inertia (for asymmetric objects) can be negative.