Moment of Inertia Calculator

What is Moment of Inertia?

The moment of inertia, also known as the mass moment of inertia, rotational inertia, or angular mass, is a measure of an object's resistance to rotational acceleration. It is the rotational analog of mass for linear motion. The moment of inertia plays a crucial role in the analysis of rotational dynamics, just as mass does in linear dynamics.

Formula

The formula for the moment of inertia depends on the shape of the object and its axis of rotation. Here are some common formulas:

• Solid Sphere: \( I = \frac{2}{5}mr^2 \)
• Hollow Sphere: \( I = \frac{2}{3}mr^2 \)
• Solid Cylinder: \( I = \frac{1}{2}mr^2 \)
• Hollow Cylinder: \( I = mr^2 \)
• Thin Rod (about end): \( I = \frac{1}{3}ml^2 \)

Where:

• \( I \) is the moment of inertia (in kg·m²)
• \( m \) is the mass of the object (in kg)
• \( r \) is the radius (in m)
• \( l \) is the length of the rod (in m)

Calculation Steps

Let's calculate the moment of inertia for a solid sphere:

1. Given:
   • Mass (\( m \)) = 2 kg
   • Radius (\( r \)) = 0.1 m
2. Apply the solid sphere formula: \[ I = \frac{2}{5}mr^2 \]
3. Substitute the known values: \[ I = \frac{2}{5} \times 2 \text{ kg} \times (0.1 \text{ m})^2 \]
4. Perform the calculation: \[ I = \frac{2}{5} \times 2 \text{ kg} \times 0.01 \text{ m}^2 = 0.004 \text{ kg·m}^2 \]

Example and Visual Representation

Let's visualize the moment of inertia for different shapes:

This diagram illustrates:

• A solid sphere (blue)
• A solid cylinder (green)
• A thin rod (red)
• The angular velocity (\( \omega \)) representing rotation (yellow arc)