1. What is Moment of Inertia?

The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation and is often called "rotational inertia".

2. How Does the Calculator Work?

The calculator uses the fundamental formula for moment of inertia:

Where:

• \( I \) — Moment of inertia (kg·m²)
• \( r \) — Distance from axis of rotation (m)
• \( dm \) — Mass element (kg)

Explanation: For common shapes, we use simplified formulas derived from integrating the mass distribution.

3. Importance of Moment of Inertia

Details: Moment of inertia is crucial in rotational dynamics, affecting how quickly an object can spin up or slow down under applied torque. It's essential in engineering applications like flywheels, gears, and rotating machinery.

4. Using the Calculator

Tips: Enter mass in kilograms, radius in meters, and select the object shape. All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: Why does shape affect moment of inertia?
A: Different shapes distribute mass differently relative to the axis of rotation, changing the integral of r²dm.

Q2: What's the difference between point mass and ring?
A: Both have I=mr², but point mass assumes all mass at distance r, while ring has mass distributed at constant radius r.

Q3: How does moment of inertia relate to kinetic energy?
A: Rotational kinetic energy is ½Iω², where ω is angular velocity, showing I's role in energy storage.

Q4: What about objects rotating about different axes?
A: Moment of inertia changes with axis position (parallel axis theorem) and orientation (perpendicular axis theorem).

Q5: Can I calculate for complex shapes?
A: Complex shapes can be broken into simpler components whose moments are summed (additive property).