Moment of Inertia Calculator Formula

Understand the math behind the moment of inertia calculator. Each variable explained with a worked example.

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Formulas Used

Moment of Inertia

inertia = (shape == 1) ? (2/5) * mass * pow(radius, 2) : ((shape == 2) ? (2/3) * mass * pow(radius, 2) : ((shape == 3) ? 0.5 * mass * pow(radius, 2) : (1/12) * mass * pow(2 * radius, 2)))

Variables

VariableDescriptionDefault
shapeShape1
massMass(kg)5
radiusRadius (or half-length for rod)(m)0.2

How It Works

Moment of Inertia

Moment of inertia (I) is the rotational analogue of mass. It depends on mass distribution relative to the rotation axis.

Formulas by Shape

  • Solid Sphere: I = (2/5) * m * r²
  • Hollow Sphere: I = (2/3) * m * r²
  • Solid Cylinder/Disk: I = (1/2) * m * r²
  • Thin Rod (center): I = (1/12) * m * L²

    • Worked Example

    A solid sphere of mass 5 kg and radius 0.2 m.

    shape = 1mass = 5radius = 0.2
    1. 01For a solid sphere: I = (2/5) * m * r²
    2. 02I = 0.4 * 5 * 0.04
    3. 03I = 0.4 * 0.2
    4. 04I = 0.08 kg·m²

    Frequently Asked Questions

    Why does shape affect moment of inertia?

    Moment of inertia depends on how mass is distributed relative to the rotation axis. Mass farther from the axis contributes more.

    What is the parallel axis theorem?

    I = I_cm + m*d², where I_cm is the moment of inertia about the center of mass and d is the distance to the new parallel axis.

    How is moment of inertia used?

    It appears in the rotational form of Newton's second law: tau = I * alpha, and in rotational kinetic energy: KE = 0.5 * I * omega².

    Learn More

    Guide

    How to Calculate Moment of Inertia

    Learn how to calculate the moment of inertia for common shapes, composite sections, and rotational systems. Includes the parallel axis theorem and worked examples.

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