Calculator
Plotly Graph
The graph scales the active body dimension while keeping mass fixed.
Example Data Table
| Shape | Mass | Main Dimension | Formula | Inertia |
|---|---|---|---|---|
| Solid Disc | 8 kg | 0.4 m radius | I = (1/2)mr² | 0.640000 kg·m² |
| Thin Rod About Center | 6 kg | 1.8 m length | I = (1/12)mL² | 1.620000 kg·m² |
| Solid Sphere | 12 kg | 0.3 m radius | I = (2/5)mr² | 0.432000 kg·m² |
| Rectangular Plate | 5 kg | 1.2 m by 0.8 m | I = (1/12)m(w²+h²) | 0.646667 kg·m² |
Formula Used
Moment of inertia measures resistance to angular acceleration about an axis. It depends on mass amount, mass placement, and selected rotation axis.
Common formulas included in this calculator:
- Point mass: I = m r²
- Thin rod about center: I = (1/12) m L²
- Thin rod about end: I = (1/3) m L²
- Solid disc: I = (1/2) m r²
- Thin ring: I = m r²
- Solid sphere: I = (2/5) m r²
- Hollow sphere: I = (2/3) m r²
- Rectangular plate: I = (1/12) m (w² + h²)
- Solid cylinder: I = (1/2) m r²
- Cylindrical shell: I = m r²
When the rotation axis is offset from the centroidal axis, the calculator also applies the parallel axis theorem:
Ishifted = Icentroid + m d²
How to Use This Calculator
- Select the body shape that matches your part.
- Enter the body mass and choose its unit.
- Fill the needed dimensions for that shape.
- Enter axis offset distance if the axis is shifted.
- Choose the desired output unit.
- Press calculate to show the result above the form.
- Review the graph, result table, and formula text.
- Use the export buttons to save CSV or PDF.
About This Calculator
Why inertia matters
Rotational moment of inertia is a core quantity in mechanics. It describes how strongly a body resists changes in angular motion. Designers, students, and analysts use it in flywheels, shafts, rotors, robotic joints, and machine components.
Why shape selection matters
Different shapes place mass at different distances from the rotation axis. That distance changes inertia more than many users expect. A ring and a disc can share the same mass and radius, yet the ring produces a larger inertia.
Why the parallel axis option helps
Real projects often rotate around supports, hinges, or shaft centers that do not pass through the centroid. The offset distance option helps estimate that case quickly. It gives a more practical value for torque planning and angular acceleration studies.
FAQs
1. What is rotational moment of inertia?
It is a measure of how much a body resists angular acceleration about a chosen axis. Higher values mean harder rotational speed changes under the same torque.
2. Why does radius affect inertia so strongly?
Most inertia formulas include squared distance from the axis. Small increases in radius can create much larger inertia values because the distance term is multiplied by itself.
3. When should I use the axis offset field?
Use it when the actual rotation axis is not through the centroidal axis used by the base formula. The calculator then applies the parallel axis theorem automatically.
4. Can I compare several shapes quickly here?
Yes. Change the shape dropdown, keep mass and dimensions as needed, and recalculate. This helps compare how geometry changes resistance to rotational motion.
5. Which unit is best for engineering work?
kg·m² is the standard SI output and is usually best for engineering calculations. Other units can help when matching lab notes, legacy documents, or classroom exercises.
6. Does this calculator work for composite bodies?
Not directly as one entry. For composite bodies, calculate each part separately about a common axis, then add the inertia values together.
7. What is the difference between disc and ring results?
A disc spreads mass across the radius, while a thin ring keeps mass at the outer radius. The ring therefore gives a larger inertia for equal mass and radius.
8. Why is my shifted inertia always larger?
Because the parallel axis addition uses m d², which is never negative. Any nonzero offset increases total inertia relative to the centroidal value.