0.5
kg·m²
1
0.5
kg·m²
1
The Moment of Inertia Calculator determines the rotational inertia of common geometric shapes, a fundamental quantity in rotational mechanics. Moment of inertia, often denoted I, quantifies how difficult it is to change an object's rotational velocity about a given axis. Just as mass resists linear acceleration according to Newton's second law, moment of inertia resists angular acceleration according to the rotational analog: $$\tau = I\alpha$$
The value of I depends not only on the total mass of the object but critically on how that mass is distributed relative to the rotation axis. A hollow cylinder with all its mass concentrated at distance r has twice the moment of inertia of a solid cylinder of the same mass and radius. This principle underpins the design of flywheels, figure-skating spins, and planetary rotation. Our calculator covers the six most commonly encountered shapes in physics and engineering: solid sphere, hollow sphere, solid cylinder (disk), hollow cylinder, and thin rod pivoting at center or end.
Understanding moment of inertia is essential for solving problems in angular momentum conservation, rotational kinetic energy, torque analysis, and mechanical design. Engineers use these formulas to design crankshafts, turbine rotors, and robotic actuators.
Each shape has a well-known closed-form expression derived by integrating $$I = \int r^2 \, dm$$ over the object's geometry:
Solid Sphere (axis through center): $$I = \frac{2}{5}mr^2$$
Hollow Sphere (thin shell, axis through center): $$I = \frac{2}{3}mr^2$$
Solid Cylinder / Disk (axis along cylinder axis): $$I = \frac{1}{2}mr^2$$
Hollow Cylinder (thin-walled, axis along cylinder axis): $$I = mr^2$$
Rod — Center Pivot (axis through midpoint, perpendicular): $$I = \frac{1}{12}mL^2$$
Rod — End Pivot (axis through one end, perpendicular): $$I = \frac{1}{3}mL^2$$
You select the shape, enter mass and the relevant dimension (radius for spheres/cylinders, length for rods), and the calculator applies the corresponding formula.
A higher moment of inertia means the object resists angular acceleration more strongly. For the same applied torque, an object with larger I will spin up more slowly. When comparing shapes of equal mass and size, the hollow forms always have larger I because their mass is farther from the axis. This is why a hollow sphere rolls down an incline more slowly than a solid sphere of the same mass and radius — more energy goes into rotation.
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A 7.26 kg bowling ball with radius 10.9 cm has I = 0.0345 kg·m², consistent with its compact mass distribution.
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A 150 g meter stick pivoted at one end: I = (1/3)(0.15)(1²) = 0.05 kg·m². Pivoting at the center would give 0.0125 kg·m², four times less.
Moment of inertia is the rotational equivalent of mass. It measures how much an object resists being spun faster or slower. The farther the mass is from the rotation axis, the larger the moment of inertia.
In a hollow sphere, all the mass sits at the maximum distance from the center. Since I depends on r², mass farther out contributes disproportionately more. The hollow sphere's coefficient is 2/3 versus 2/5 for the solid sphere.
The SI unit is kg·m² (kilogram meter squared). In imperial units, it may appear as slug·ft² or lb·ft·s².
When objects roll down an incline, gravitational potential energy converts into both translational and rotational kinetic energy. Objects with larger I (relative to mr²) allocate more energy to rotation and thus roll more slowly. A hollow cylinder finishes last among common shapes.
Yes. If you need the moment of inertia about an axis offset by distance d from the center of mass, use $$I_{\text{offset}} = I_{\text{cm}} + md^2$$. Compute I_cm with this calculator, then add md².
For a composite object, compute I for each component about the same axis (using the parallel axis theorem if needed) and sum them: $$I_{\text{total}} = \sum I_i$$. This calculator handles individual standard shapes which you can then combine.
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