Moment of Inertia Calculator: A Comprehensive Guide

When calculating the moment of inertia (MOI) of various geometric shapes, it can be overwhelming to navigate the complex formulas and variables involved. That's why this calculator is designed to simplify the process, providing accurate results for a wide range of objects with different axes of rotation.

Solid and Hollow Tetrahedrons

For solid and hollow tetrahedrons (four flat faces), the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center and one vertex: I = (2/5)mr²

* For a spherical shell with inner radius r₁, outer radius r₂, and mass m: I = (2/3)m(r₂³ - r₁³)

* For a hollow sphere with radius r and mass m: I = (2/3)mr³

Spherical Shells and Hollow Spheres

For spherical shells and hollow spheres, the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center: I = (2/3)m(r₂³ - r₁³)

* For a sphere with radius r and mass m: I = (2/5)mr²

Rods and Cones

For rods and cones, the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center or one end: I = ml²/12

* For a right circular cone with radius r, height h, and mass m: I = (1/2)mrh

Plane Polygons and Thin Rectangular Plates

For plane polygons and thin rectangular plates, the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center, perpendicular to the plane: I = mr²

* For a regular polygon with n vertices, radius R, and mass m: I = (1/2)mrR

Point Masses and Thin Circular Hoops

For point masses and thin circular hoops, the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center: I = mr²

* For a thin circular hoop with radius r and mass m: I = mr²

Solid Polyhedra and Ellipsoids

For solid polyhedra and ellipsoids, the moment of inertia can be calculated using the following formulas:

* About the axis passing through the center and one vertex: I = (2/5)ms²

* For a solid ellipsoid with semiaxes a, b, c, and mass m: I = (1/10)ma³ + (1/20)m(b³+c³)

By using this moment of inertia calculator, engineers and physicists can quickly and accurately determine the MOI for various geometric shapes, making it an essential tool in their work. Whether you're designing a new product or studying the motion of celestial bodies, understanding the moment of inertia is crucial for predicting behavior and optimizing performance.

Enter your shape's dimensions and mass to get started!

Moment of inertia calculator