moment of inertia calculator - Aaron Graves, PhDude Replica

Interactive Calculator

Use SI units: mass in kilograms (kg), distances in meters (m), and result in kg·m².

Select shape / case

Formula:

I = mr²

Mass, m (kg)

Radius, r (m)

Length, L (m)

Width, a (m)

Height, b (m)

Known center-of-mass inertia, I_cm (kg·m²)

Offset distance, d (m)

What is moment of inertia?

Moment of inertia is a measure of how strongly an object resists rotational acceleration about a specific axis. You can think of it as the rotational equivalent of mass in linear motion. In straight-line mechanics, larger mass means more force is needed for the same acceleration. In rotational mechanics, larger moment of inertia means more torque is required for the same angular acceleration.

The key detail is that moment of inertia depends on both mass and how that mass is distributed relative to the axis. Two objects with the same mass can have very different rotational behavior if one has more mass far away from the axis.

Why this calculator is useful

This moment of inertia calculator helps you quickly evaluate common engineering and physics cases without memorizing every formula. It is useful for:

Physics homework and exam preparation
Mechanical design (wheels, rotors, shafts, flywheels)
Robotics and control systems
Estimating angular acceleration using Newton's second law for rotation: τ = Iα
Using the parallel axis theorem when the rotation axis is shifted

Formulas included in this calculator

Standard body formulas

Point mass: I = mr²
Thin rod (center, perpendicular axis): I = (1/12)mL²
Thin rod (end, perpendicular axis): I = (1/3)mL²
Solid disc / cylinder (central axis): I = (1/2)mr²
Thin ring / hoop (central axis): I = mr²
Solid sphere (diameter axis): I = (2/5)mr²
Thin hollow sphere (diameter axis): I = (2/3)mr²
Rectangular plate (center, perpendicular axis): I = (1/12)m(a² + b²)

Axis-shift formula

When you know inertia at the center of mass and need inertia about another parallel axis:

Parallel axis theorem: I = I_cm + md²

How to use the calculator

Select the shape or theorem from the dropdown list.
Enter the required values only (fields update automatically).
Click Calculate to get the final moment of inertia.
Review the formula and substitution shown in the result area.

Tip: keep all values in SI units to avoid conversion mistakes. If your dimensions are in centimeters, convert to meters first.

Worked examples

Example 1: Solid disc flywheel

Suppose a disc has mass m = 8 kg and radius r = 0.30 m. For a solid disc, I = (1/2)mr².

So, I = 0.5 × 8 × (0.30)² = 0.36 kg·m². This value tells you how much torque is required to spin it up at a desired angular acceleration.

Example 2: Thin rod rotating about one end

A rod has mass 2 kg and length 1.5 m. For an end axis perpendicular to the rod, I = (1/3)mL².

I = (1/3) × 2 × (1.5)² = 1.5 kg·m².

Example 3: Shifted axis using parallel axis theorem

If I_cm = 0.20 kg·m², m = 3 kg, and axis shift d = 0.40 m:

I = 0.20 + 3 × (0.40)² = 0.20 + 0.48 = 0.68 kg·m².

Common mistakes to avoid

Wrong axis: The same object has different inertia for different axes.
Unit mismatch: Mixing cm and m leads to large errors because length is squared.
Using the wrong shape model: Thin ring and solid disc are not interchangeable.
Confusing area moment of inertia with mass moment of inertia: They are different physical quantities.

Mass moment vs. area moment of inertia

This page calculates mass moment of inertia (units: kg·m²), used in rotational dynamics. In structural analysis, you may encounter second moment of area (units: m⁴), which is used in beam bending. They are related conceptually but not the same quantity, so formulas and units are different.

FAQ

What are the output units?

All results are shown in kg·m² when you input kg and meters.

Can I use this for non-uniform objects?

The built-in formulas assume ideal, uniform bodies. For non-uniform distributions, you generally need integration, CAD software, or experimental methods.

Is a larger moment of inertia always better?

Not always. High inertia can smooth rotation (good for flywheels) but makes rapid acceleration and deceleration harder (often undesirable in responsive systems like robots).

Final notes

A reliable moment of inertia estimate is foundational in mechanics. Whether you are solving class problems, sizing a motor, or designing a rotating system, this calculator gives fast and consistent results for common geometries. Use it as a practical starting point, then refine with detailed modeling when your application demands higher precision.