🌀 Moment of Inertia Calculator
Calculate moment of inertia
How to Use This Calculator
Select Shape
Choose the shape type: Solid Cylinder, Hollow Cylinder, Solid Sphere, Hollow Sphere, or Rod (about center or end).
Enter Values
Input the mass, radius, and length (if applicable) of your shape. Make sure all values use consistent units (e.g., kg for mass, m for radius/length).
Click Calculate
Press the "Calculate" button to compute the moment of inertia about the specified axis.
Formulas
Solid Cylinder (about central axis)
I = ½mr²
Hollow Cylinder (about central axis)
I = mr²
Solid Sphere (about center)
I = ⅖mr²
Hollow Sphere (about center)
I = ⅔mr²
Rod (about center)
I = (1/12)mL²
Rod (about end)
I = (1/3)mL²
Where:
- I = Moment of Inertia (kg·m²)
- m = Mass (kg)
- r = Radius (m)
- L = Length (m)
Example: Solid cylinder with mass = 10 kg, radius = 0.5 m
I = ½ × 10 × (0.5)² = ½ × 10 × 0.25 = 1.25 kg·m²
About Moment of Inertia Calculator
The Moment of Inertia Calculator computes the moment of inertia (rotational inertia) for various shapes. Moment of inertia measures an object's resistance to rotational motion about an axis, analogous to mass in linear motion.
When to Use This Calculator
- Physics: Calculate rotational motion and angular acceleration problems
- Engineering: Design rotating machinery, wheels, and mechanical systems
- Mechanical Design: Analyze flywheels, gears, and rotating components
- Education: Learn and practice moment of inertia calculations
- Sports Science: Analyze rotating objects in sports (e.g., discs, balls)
- Automotive: Design engine components and rotating assemblies
Why Use Our Calculator?
- ✅ Multiple Shapes: Supports cylinders, spheres, rods, and more
- ✅ Instant Results: Get moment of inertia immediately
- ✅ Different Axes: Calculate about center or end points
- ✅ Step-by-Step Display: See the calculation formula
- ✅ 100% Accurate: Precise physics calculations
- ✅ Completely Free: No registration required
Understanding Moment of Inertia
Moment of inertia (I) is the rotational equivalent of mass. Key concepts:
- Rotational Inertia: Measures resistance to angular acceleration
- Axis Dependent: Moment of inertia depends on the rotation axis
- Distribution Matters: Mass farther from axis = larger moment of inertia
- Units: kg·m² (kilogram meters squared)
- Parallel Axis Theorem: I = I_cm + md² (shift axis by distance d)
- Energy: Rotational kinetic energy = ½Iω² (where ω is angular velocity)
Real-World Applications
Flywheels: High moment of inertia flywheels store rotational energy. A solid cylinder has higher moment of inertia than a hollow cylinder of same mass and radius.
Gymnastics: Athletes tuck their bodies to reduce moment of inertia, allowing faster rotations. Extended positions increase moment of inertia, slowing rotation.
Wheels: Wheel design considers moment of inertia for acceleration and braking. Lower moment of inertia wheels respond faster to torque.
Frequently Asked Questions
What is moment of inertia?
Moment of inertia (I) is a measure of an object's resistance to rotational acceleration about an axis. It's the rotational equivalent of mass in linear motion. Higher moment of inertia means harder to start or stop rotation.
Why does the axis matter?
Moment of inertia depends on the distance of mass from the rotation axis. Same object rotating about different axes has different moments of inertia. Mass farther from axis contributes more to moment of inertia.
What are the units of moment of inertia?
Moment of inertia has units of kg·m² (kilogram meters squared). In physics, it represents mass times distance squared, reflecting how mass distribution affects rotational motion.
Why is hollow cylinder moment of inertia larger than solid?
For a hollow cylinder, more mass is farther from the axis (at radius r). Since I = mr², mass at larger radius contributes more. Both have mass m and radius r, but hollow has all mass at radius r, while solid has mass distributed from 0 to r.
What's the difference between rod about center vs. end?
Rod about center: I = (1/12)mL². Rod about end: I = (1/3)mL². The end case has 4× larger moment of inertia because mass is, on average, farther from the rotation axis.
How is moment of inertia used in real life?
It's used in designing flywheels (energy storage), wheels (acceleration), figure skating (spin control), and mechanical systems. Understanding moment of inertia helps optimize rotational performance.
Can moment of inertia be negative?
No, moment of inertia is always positive or zero. It's a physical quantity representing mass distribution, which cannot be negative. It's defined as the sum or integral of mass × distance², which is always non-negative.