Mass Moment of Inertia Calculator
Calculate moment of inertia for various geometric shapes and objects with precise physics formulas
Calculate Moment of Inertia
Solid sphere rotating about center axis
Moment of Inertia Results
Shape: Solid Sphere
Formula: I = (2/5)mr²
Description: Solid sphere rotating about center axis
Physical Interpretation
Example Calculation
Spinning Solid Sphere
Shape: Solid sphere
Mass: 5.0 kg
Radius: 0.3 m
Axis: Through center
Calculation Steps
Formula: I = (2/5)mr²
I = (2/5) × 5.0 kg × (0.3 m)²
I = (2/5) × 5.0 × 0.09
I = 0.18 kg⋅m²
Common Shape Formulas
Point Mass
I = mr²
Solid Sphere
I = (2/5)mr²
Solid Cylinder
I = (1/2)mr²
Rod (center)
I = (1/12)mL²
Thin Ring
I = mr²
Units & Conversions
1 kg⋅m² = 0.737562 lb⋅ft⋅s²
1 kg⋅m² = 23.7304 lb⋅ft²
Physics Tips
Moment of inertia depends on mass distribution and axis choice
Units are always mass × length²
Used in rotational dynamics: τ = Iα
Parallel axis theorem: I = I_cm + md²
Understanding Moment of Inertia
What is Moment of Inertia?
Moment of inertia is a measure of an object's resistance to changes in rotational motion. It plays the same role in rotational motion that mass plays in linear motion. The moment of inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation.
Basic Formula
I = Σ miri²
Sum of mass × distance² for all particles
Key Properties
- •Always positive (mass and distance² are positive)
- •Depends strongly on axis location and orientation
- •Units are always mass × length²
- •Additive for composite objects
Applications in Physics
Rotational Dynamics
Newton's Second Law for rotation: τ = Iα
Torque = Moment of Inertia × Angular Acceleration
Rotational Kinetic Energy
KE = ½Iω²
Similar to linear KE = ½mv²
Angular Momentum
L = Iω
Angular momentum = I × angular velocity
Parallel Axis Theorem: I = Icm + md²
Relates moment of inertia about any axis to that about the center of mass