What Is a Moment of Inertia Calculator?
A moment of inertia calculator finds an object's resistance to rotational acceleration about an axis — the rotational equivalent of mass. The moment of inertia depends on both the mass and how that mass is distributed relative to the axis. This calculator computes it for common shapes such as rods, disks, spheres, and point masses. Enter the shape, mass, and dimensions, and the calculator returns the moment of inertia.
How to Use the Calculator
- Select the shape — point mass, rod, disk, sphere, and so on.
- Enter the mass and relevant dimension (radius or length).
- Calculate — see the moment of inertia.
Moment of Inertia Formulas
| Object (axis) | Formula |
|---|---|
| Point mass | I = m r² |
| Solid disk (center) | I = ½ m r² |
| Solid sphere (center) | I = (2/5) m r² |
| Thin rod (center) | I = (1/12) m L² |
| Thin rod (end) | I = (1/3) m L² |
Why Mass Distribution Matters
The farther mass is from the axis, the larger the moment of inertia, because each bit of mass contributes its mass times the square of its distance. This is why a figure skater spins faster when pulling their arms in — reducing their moment of inertia increases their rotation speed.
Where It Is Used
- Rotational dynamics: torque equals moment of inertia times angular acceleration (τ = Iα).
- Engineering: designing flywheels, gears, and rotating machinery.
- Sports and physics: understanding spins and rotations.
The Parallel Axis Theorem
To find the moment of inertia about an axis parallel to one through the center of mass, use I = I_cm + m d², where d is the distance between the axes. This lets you adapt the standard formulas to off-center axes.
Frequently Asked Questions
What is moment of inertia?
It is an object's resistance to changes in its rotation about an axis — the rotational analog of mass, depending on how mass is distributed relative to the axis.
What is the moment of inertia of a solid disk?
For a solid disk rotating about its center, the moment of inertia is ½ m r², where m is mass and r is radius.
Why does mass distribution affect it?
Each mass element contributes its mass times the square of its distance from the axis, so mass farther out increases the moment of inertia much more.
What is the parallel axis theorem?
It gives the moment of inertia about a parallel axis: I = I_cm + m d², where d is the distance from the center-of-mass axis to the new axis.
Is this moment of inertia calculator free?
Yes — it is completely free, requires no signup, and covers common shapes.