Calculator
Example Data
Use these sample values to verify your setup.
| Case | Ri | Ro | L | Mass / Density | Axis | Result (kg·m²) |
|---|---|---|---|---|---|---|
| Known mass | 0.025 m | 0.040 m | 0.200 m | m = 3.200 kg | Along axis | 0.00328 |
| Density mode | 25 mm | 40 mm | 200 mm | ρ = 7850 kg/m³ | Transverse | ≈ 0.0152 |
Formulas Used
For a thick-walled (hollow) cylinder with inner radius Ri, outer radius Ro, length L, and mass m:
- Longitudinal axis (along cylinder centerline): I = 1/2 · m · (Ro² + Ri²)
- Transverse axis (through center, perpendicular): I = 1/12 · m · (3(Ro² + Ri²) + L²)
Density mode uses m = ρ · π(Ro² − Ri²) · L before applying the selected inertia formula.
How to Use This Calculator
- Select Known mass or Density + dimensions.
- Enter Ri and Ro, then choose your length unit.
- If using transverse axis or density mode, enter cylinder length.
- Choose the axis type and the output unit, then press Calculate.
- Use Download CSV or Download PDF to save results.
Tip: keep Ro > Ri and use consistent real-world units.
Hollow Cylinder Inertia Notes
1) Why moment of inertia matters
Rotational inertia controls how strongly a tube resists angular acceleration. For the same applied torque, a larger inertia produces a smaller angular acceleration, which affects motor sizing, spin‑up time, and vibration response. It also sets stored kinetic energy, E = 1/2·I·ω², useful for flywheels and braking checks.
2) Hollow versus solid shapes
A hollow cylinder can achieve high stiffness with less mass because material sits farther from the center. Since inertia scales with radius squared, moving mass outward often increases inertia more than simply adding mass near the axis.
3) Picking the correct axis
Use the longitudinal option for rotation about the cylinder’s centerline (shafts, rollers, flywheels). Use the transverse option when the tube swings like a pendulum or rotates about a diameter through its center; length becomes important there. Examples include a drum on trunnions or a tube mounted crosswise on a robotic arm.
4) Input data you should know
Measure inner radius Ri and outer radius Ro carefully. A small radius error can change inertia noticeably because Ro² and Ri² appear directly. For example, with Ri 25 mm and Ro 40 mm, increasing Ro by 1 mm raises Ro² by about 5%.
5) Density mode and common materials
If mass is unknown, the calculator can estimate it from density and volume, m = ρ·π(Ro² − Ri²)·L. Typical densities are about 7850 kg/m³ for steel, 2700 kg/m³ for aluminum, 8500 kg/m³ for brass, and 1400 kg/m³ for rigid PVC. For composites, use a tested average density because voids and fiber fraction can vary by batch.
6) Units and reporting
You can enter dimensions in m, cm, mm, inches, or feet and export results in kg·m², g·cm², lb·ft², or lb·in². Keeping one unit system through your design notes reduces conversion mistakes during reviews and simplifies supplier comparisons later.
7) Interpreting results for design
Compare candidate tubes by inertia‑to‑mass ratio when weight is limited. If the inertia is too low, increasing Ro usually helps more than decreasing Ri, but check stress and fit constraints. Always confirm Ro > Ri and use realistic length for transverse calculations. When possible, cross‑check with CAD properties and document assumptions about uniform density and straight geometry.
FAQs
1) What is the difference between longitudinal and transverse inertia?
Longitudinal inertia is for rotation about the cylinder’s centerline. Transverse inertia is for rotation about a diameter through the center. Transverse includes the length term L², so longer cylinders resist swinging rotation more.
2) Can I use diameter instead of radius?
Yes. Convert first: Ri = Di/2 and Ro = Do/2. The calculator expects radii, and the formulas use squared radii, so doubling mistakes become four‑times errors in inertia.
3) Why does outer radius affect the result so much?
Because inertia depends on Ro² and Ri². Material farther from the axis contributes more, so small changes to Ro can noticeably change I, especially for thick‑walled tubes.
4) When should I use density mode?
Use density mode when you know material density and geometry but not the mass. Enter Ri, Ro, and length, choose density units, and the tool computes mass from volume before calculating inertia.
5) What if my tube has holes or a keyway?
These formulas assume a uniform, continuous cylinder. Features like holes, slots, or keyways reduce mass and inertia. For accurate values, use CAD mass properties or apply correction factors based on removed volume.
6) Which output unit should I choose?
Pick the unit that matches your analysis tools and documentation. SI kg·m² is standard for engineering calculations. lb·ft² or lb·in² can be convenient for US-based torque and motor data.