Calculator Input
Example Data Table
| Case | Body | Mass (kg) | Main Dimensions (m) | Formula |
|---|---|---|---|---|
| 1 | Solid Disk | 10 | r = 0.5 | J = (1/2)mr² |
| 2 | Thin Ring | 8 | r = 0.4 | J = mr² |
| 3 | Annular Disk | 12 | rᵢ = 0.2, rₒ = 0.5 | J = (1/2)m(rₒ²+rᵢ²) |
| 4 | Rectangular Plate | 6 | L = 1.0, W = 0.6 | J = (m/12)(L²+W²) |
| 5 | Point System | Varies | (2,0.2), (3,0.4), (5,0.6) | J = Σmᵢrᵢ² |
Formula Used
The polar mass moment of inertia measures resistance to angular acceleration about a perpendicular axis. It is mass weighted by squared radial distance.
General equation: J = Σmᵢrᵢ²
Solid disk or solid cylinder: J = (1/2)mr²
Thin ring: J = mr²
Annular disk: J = (1/2)m(rₒ² + rᵢ²)
Rectangular plate about centroidal z-axis: J = (m/12)(L² + W²)
Use consistent mass and length units before applying any formula. This calculator converts the entered units into kilograms and meters internally.
How to Use This Calculator
Choose either the shape formula method or the custom point masses method.
Select your preferred mass and length units.
For shape mode, pick a body type and enter the required dimensions.
For point mode, enter one mass and radius pair per line.
Click Calculate to show the result below the header and above the form.
Review the plotted graph and output table.
Use the CSV or PDF buttons to export the result for documentation.
FAQs
1. What does polar mass moment of inertia represent?
It shows how strongly a body's mass resists angular acceleration about a perpendicular axis. Larger mass farther from the axis produces a larger value.
2. Is this the same as area moment of inertia?
No. Mass moment of inertia uses mass and affects dynamics. Area moment of inertia uses geometry and affects bending and structural stiffness.
3. When should I use the point masses method?
Use it when the body can be approximated by several masses located at known radii, or when measured discrete data is available.
4. Why are squared radii used in the formula?
Distance from the axis has a strong effect on rotational resistance. Squaring the radius captures that increasing leverage effect.
5. Can I use centimeters and grams?
Yes. The calculator accepts several unit choices and converts them internally to kilograms and meters before computing the final value.
6. Why does a ring have more inertia than a disk?
For the same mass and radius, more ring mass lies farther from the axis. That raises the polar mass moment of inertia.
7. What axis does this calculator assume?
It assumes a central axis perpendicular to the body plane, or the centroidal z-axis for rectangular bodies.
8. Can I export the results for reports?
Yes. The tool includes CSV and PDF export options for quick sharing, record keeping, and engineering documentation.