Moment of Inertia Calculator

About the Moment of Inertia Calculator

The Moment of Inertia Calculator computes the moment of inertia (\( I \)) for common shapes based on their mass and geometric properties. The moment of inertia measures an object’s resistance to angular acceleration and is used in rotational dynamics.

Supported shapes and formulas:

• Solid Cylinder (about central axis): \( I = \frac{1}{2} m r^2 \)
• Thin Disk (about central axis): \( I = \frac{1}{2} m r^2 \)
• Solid Sphere (about diameter): \( I = \frac{2}{5} m r^2 \)
• Thin Rod (about center, perpendicular to length): \( I = \frac{1}{12} m L^2 \)
• Rectangular Plate (about center, perpendicular to plane): \( I = \frac{1}{12} m (w^2 + h^2) \)

This tool is ideal for physics students, engineers, and educators analyzing rotational motion in systems like machinery, vehicles, or structures.

• Features:
  • Calculates moment of inertia for five common shapes.
  • Dynamic input fields based on selected shape.
  • Supports unit conversions: mass (kg, g), length (m, cm, mm).
  • Validates inputs: positive mass and geometric parameters; shape selection required.
  • Keypad includes digits, decimal point, scientific notation (E), and negative sign (-).
  • Clear and backspace functionality, with a "Copy" button for results.
• Practical Applications: Useful in mechanical engineering (e.g., designing rotating components), physics education, robotics, and aerospace for analyzing rotational dynamics.
• How to Use:
  • Select a shape from the dropdown (e.g., Solid Cylinder).
  • Enter the mass (e.g., 1 kg) and select the unit (kg, g).
  • Enter the required geometric parameters (e.g., radius for cylinder, width and height for plate) and select units (m, cm, mm).
  • Use the keypad to insert digits, decimal point, scientific notation (E), or negative sign (-).
  • Click "Calculate" to compute the moment of inertia, then use "Copy" to copy the result.
  • Use "Clear" to reset, or "⌫" to delete the last character.
  • Share or embed the calculator using the action buttons.
• Helpful Tips:
  • Ensure mass and geometric parameters are positive.
  • Use scientific notation (E) for small values (e.g., 1E-3 for 0.001).
  • Moment of inertia is in \( \text{kg·m}^2 \) (or \( \text{g·cm}^2 \) if < 0.0001 \( \text{kg·m}^2 \)).
  • Formulas assume uniform mass distribution and specified axes.
  • For complex shapes, use the parallel axis theorem separately.
• Examples:
  • Example 1: Solid Cylinder:
    • Input: Shape = Solid Cylinder, Mass = 2 kg, Radius = 0.1 m
    • Steps:
      • Formula: \( I = \frac{1}{2} m r^2 \)
      • \( I = \frac{1}{2} \cdot 2 \cdot (0.1)^2 = 0.5 \cdot 2 \cdot 0.01 = 0.01 \, \text{kg·m}^2 \)
    • Result: Moment of Inertia = 0.01 \( \text{kg·m}^2 \)
  • Example 2: Thin Rod:
    • Input: Shape = Thin Rod, Mass = 500 g, Length = 50 cm
    • Steps:
      • Convert: 500 g = 0.5 kg, 50 cm = 0.5 m
      • Formula: \( I = \frac{1}{12} m L^2 \)
      • \( I = \frac{1}{12} \cdot 0.5 \cdot (0.5)^2 = \frac{1}{12} \cdot 0.5 \cdot 0.25 \approx 0.0104167 \, \text{kg·m}^2 \)
    • Result: Moment of Inertia = 0.0104 \( \text{kg·m}^2 \)
  • Example 3: Rectangular Plate:
    • Input: Shape = Rectangular Plate, Mass = 1 kg, Width = 20 cm, Height = 30 cm
    • Steps:
      • Convert: 20 cm = 0.2 m, 30 cm = 0.3 m
      • Formula: \( I = \frac{1}{12} m (w^2 + h^2) \)
      • \( I = \frac{1}{12} \cdot 1 \cdot ((0.2)^2 + (0.3)^2) = \frac{1}{12} \cdot 1 \cdot (0.04 + 0.09) = \frac{1}{12} \cdot 0.13 \approx 0.0108333 \, \text{kg·m}^2 \)
    • Result: Moment of Inertia = 0.0108 \( \text{kg·m}^2 \)

Calculate moment of inertia with detailed steps using this calculator. Share or embed it on your site!