Moment of Inertia Calculator: Rotational Dynamics

What is Moment of Inertia?

Moment of inertia (I) is the rotational equivalent of mass — it measures resistance to angular acceleration. A larger I means more torque is needed to achieve the same angular acceleration. It depends on both mass and the distribution of that mass relative to the rotation axis.

🔄 Moment of Inertia Calculator

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📐 Formula

τ = I × α

Newton's second law for rotation. τ = torque (N·m), I = moment of inertia (kg·m²), α = angular acceleration (rad/s²). For a solid disk: I = ½mr².

📝 Worked Example

Solid disk: m=2kg, r=0.3m → I = ½×2×0.09 = 0.09 kg·m²
Hollow cylinder (same m, r): I = mr² = 0.18 kg·m²
Hollow is harder to spin (twice the I for same m and r)

📝 How to Use

Select ShapeSolid disk, hollow cylinder, solid sphere, hollow sphere, rod, or point mass.

Enter DimensionsMass and relevant geometrical parameters (radius, length).

Choose AxisCenter, edge, or end axis — moment of inertia varies with axis choice.

Use Parallel Axis TheoremAdd md² to shift the axis from center to any parallel axis.

❓ FAQ

What is the parallel axis theorem?

I_new = I_cm + md². Allows calculating I about any axis parallel to the center-of-mass axis. Essential for compound objects.

Why do figure skaters spin faster when they pull in their arms?

Pulling arms in reduces I. Angular momentum L=Iω is conserved, so ω (spin rate) must increase when I decreases.

What is Moment of Inertia?