Physics overview
What is moment of inertia?
Moment of inertia is the rotational equivalent of mass. It describes how difficult it is to change an object's rotational motion around a selected axis.
Standard formulas
Moment of inertia formulas by shape
Point mass
I = mr²
Solid disk or cylinder
I = ½mr²
Thin hoop or ring
I = mr²
Solid sphere
I = ⅖mr²
Thin spherical shell
I = ⅔mr²
Rod through center
I = ¹⁄₁₂mL²
Rod through one end
I = ⅓mL²
Formula variables
What each symbol means
- I is moment of inertia in kg·m².
- m is mass in kilograms.
- r is radius in meters.
- L is rod length in meters.
Worked example
Solid disk with mass 4 kg and radius 3 m
- Use the solid disk formula: I = ½mr².
- Substitute the values: I = ½ × 4 × 3².
- Square the radius: 3² = 9.
- Calculate the result: I = 18 kg·m².
Rotation axis
Why the axis matters
The same object can have different moments of inertia depending on the selected axis. A rod rotating through its center has a lower moment of inertia than the same rod rotating through one end.
Mass distribution
Distance from the axis changes I
Mass farther from the axis contributes more strongly because radius or length is squared in the formula. This is why hoops usually have greater moment of inertia than solid disks of equal mass and radius.
Units
Standard SI units
Use kilograms for mass and meters for radius or length. The resulting moment of inertia is expressed in kilogram square meters, written as kg·m².
Real-world applications
Where moment of inertia is used
- Flywheels and rotating machinery.
- Wheels, gears, and pulleys.
- Sports equipment and body rotation.
- Turbines and electric motors.
- Satellites and spacecraft attitude control.
Calculation limits
Important assumptions
- The object is treated as an ideal geometric shape.
- Mass is assumed to be distributed uniformly.
- The specified axis matches the selected formula.
- Mass and dimensions must be greater than zero.
- Units are not converted automatically.
Related tools
Continue your rotational analysis
Use the Rotational Kinetic Energy Calculator to calculate energy using moment of inertia and angular velocity.
Use the Torque Calculator to calculate torque, force, or lever arm distance.
Questions and answers
Moment of inertia calculator FAQ
What is moment of inertia?
Moment of inertia measures how strongly an object resists changes in rotational motion about a selected axis.
Which shapes does this calculator support?
It supports point masses, solid disks or cylinders, thin hoops, solid spheres, thin spherical shells, rods through the center, and rods through one end.
What unit is used for moment of inertia?
The standard SI unit is kilogram square meter, written as kg·m².
Why does the axis of rotation matter?
Moment of inertia changes when the axis changes because the mass distribution relative to that axis also changes.