Accurately compute the mass moment of inertia for common homogeneous shapes: thin rods, rectangular plates, solid cylinders/disks. Visualize the geometry and rotational axis.
The mass moment of inertia (I) quantifies an object's resistance to angular acceleration about a given axis. It depends on both mass and its distribution relative to the axis. In rotational dynamics, it plays the same role as mass in linear motion: torque = I · α. Engineers and physicists rely on accurate I values for designing flywheels, robotic arms, pendulums, and rotating machinery.
General definition: \( I = \int r^2 \, dm \)
For discrete point masses: \( I = \sum m_i r_i^2 \)
1. Thin Rod (axis through center, ⟂ length): \( I = \frac{1}{12} m L^2 \). Derived from integrating \( \lambda x^2 dx \) from -L/2 to L/2.
2. Thin Rod (axis through end, ⟂ length): \( I = \frac{1}{3} m L^2 \) . Follows from parallel axis theorem or direct integration.
3. Rectangular Plate (axis through center, ⊥ plane): \( I = \frac{1}{12} m (a^2 + b^2) \) where a = width (x-direction), b = height (y-direction). This result assumes uniform thickness and perpendicular axis through centroid.
4. Solid Cylinder / Disk (central axis): \( I = \frac{1}{2} m R^2 \) . Derived by integrating concentric rings.
All formulas assume homogeneous density. The radius of gyration \( k = \sqrt{I/m} \) indicates how far from the axis the mass would be concentrated to produce the same inertia.
If you need the moment of inertia about an axis parallel to one through the center of mass, use: \( I = I_{cm} + m d^2 \), where d is the perpendicular distance between axes. Our calculator outputs \( I_{cm} \) for standard centroidal axes; you can manually apply PAT for custom offsets.
A solid steel flywheel (mass 250 kg, radius 0.4 m) has moment of inertia I = ½·250·0.4² = 20 kg·m². When spun at 3000 rpm, the stored kinetic energy \( E = \frac{1}{2} I \omega^2 \) equals approx 985 kJ, demonstrating the critical role of mass distribution in energy storage systems. Our calculator quickly verifies these values for preliminary design.
A robotic link modelled as a thin rod of length 0.6 m and mass 1.2 kg rotating about its joint (end axis) gives I = 0.144 kg·m². Accurate inertia values enable precise torque control and trajectory planning. The calculator supports such joint-level estimations.
| Shape | Axis Description | Moment of Inertia |
|---|---|---|
| Thin Rod | Center, ⟂ length | \( \frac{1}{12} m L^2 \) |
| Thin Rod | End, ⟂ length | \( \frac{1}{3} m L^2 \) |
| Rectangular plate | Center, ⊥ plane | \( \frac{1}{12} m (a^2 + b^2) \) |
| Solid cylinder/disk | Central axis | \( \frac{1}{2} m R^2 \) |
| Hollow cylinder | Central axis | \( \frac{1}{2} m (R_{outer}^2 + R_{inner}^2) \) |
| Solid sphere | Through center | \( \frac{2}{5} m R^2 \) |
Note: Additional shapes like hollow cylinders / spheres are not directly computed in the current version but can be added via extension. The formulas are provided for reference.
In three-dimensional motion, the inertia tensor generalizes moment of inertia. However, for planar rotation and many engineering problems, the scalar moment about a fixed axis suffices. Our tool provides the principal moments for symmetric bodies.