Case 1
- Mass (kg)
- 2kg
- Distance from the axis (m)
- 0.5m
0.50000kg·m²
Open with these valuesMoment of Inertia of a Point Mass
Change any value to update the result.
0.50000kg·m²
Moment of inertia: 0.50000 kg·m²Examples
How the result moves
Distance from the axis (m) (m) → Moment of inertia (kg·m²)Your current valid inputs, with one value varied across the range.
How the result moves
Held fixed: Mass (kg) 2.000 kg.
| Distance from the axis (m) (m) | Moment of inertia (kg·m²) |
|---|---|
| 0.000 | 0.00000 |
| 0.250 | 0.12500 |
| 0.500Your value | 0.50000 |
| 0.750 | 1.12500 |
| 1.000 | 2.00000 |
This is the point-mass case, I = m × r², with no shape factor in front: 2 kg at 0.5 m from the axis gives 0.5 kg·m². The same formula holds for a thin ring spinning about its own axis. A solid disc is half of this, a solid sphere two fifths — use those factors when the mass is spread out.
Worked examples
Case 2
- Mass (kg)
- 5kg
- Distance from the axis (m)
- 2m
20.00000kg·m²
Open with these valuesCase 3
- Mass (kg)
- 1000kg
- Distance from the axis (m)
- 0.25m
62.50000kg·m²
Open with these valuesHow it's calculated
I = m × r²
- StepEnter the mass in kilograms.Mass (kg)=2kg
- StepEnter how far that mass sits from the axis of rotation, in metres.Mass (kg)=2kg·Distance from the axis (m)=0.5m
- StepRead the moment of inertia in kg·m².
- ResultFor a disc, sphere or rod, multiply by that shape's factor.0.50000kg·m²
Reference table
| Mass, distance | Example | Moment of inertia |
|---|---|---|
| 0.145, 0.3 | A baseball held 30 cm out | 0.01305 |
| 2, 0.5 | A 2 kg mass on a 50 cm arm | 0.5 |
| 1, 1 | The unit case | 1 |
| 5, 2 | A 5 kg weight two metres out | 20 |
| 1000, 0.25 | A one-tonne flywheel rim | 62.5 |
Questions
How do I calculate the moment of inertia?
For a point mass, multiply the mass by the square of its distance from the axis: I = m × r². Kilograms and metres give kg·m². A 2 kg mass 0.5 m from the axis has 0.5 kg·m².
Does this work for discs, spheres and rods?
Not as it stands — those bodies carry a shape factor. A solid disc or cylinder is ½mr², a solid sphere ⅖mr², a rod about its centre 1/12 mL², and a thin ring mr². Take the number from this calculator and multiply it by the factor for your shape.
Why is the radius squared?
Because the distance from the axis enters the definition squared. Doubling the radius multiplies the moment of inertia by four, which is why mass far from the axis matters much more than the same mass close to it.
How is the moment of inertia different from mass?
Mass measures resistance to a change in straight-line motion; the moment of inertia measures resistance to a change in rotation. Mass depends only on the object, but the moment of inertia also depends on where the axis is.
What if the axis does not pass through the centre of mass?
Then add m × d² to the centre-of-mass value, where d is the distance between the two parallel axes. That is the parallel axis theorem, and it has its own calculator.