Moment of Inertia Calculator

Calculate the moment of inertia for various common shapes. Select a shape, enter the required dimensions, and get the result instantly.

This tool helps you determine an object's resistance to rotational motion about a specific axis.

Practical Examples

Explore these common scenarios to see how the Moment of Inertia Calculator works.

Solid Cylinder (Axis)

Solid Cylinder (Axis)

A solid cylinder rotating around its central axis.

Mass: 10 kg

Radius: 0.5 m

Solid Sphere (Center)

Solid Sphere (Center)

A solid sphere rotating around an axis passing through its center.

Mass: 15 kg

Radius: 0.2 m

Thin Rod (End)

Thin Rod (End)

A thin rod rotating around an axis at its end.

Mass: 2 kg

Length: 1 m

Hollow Cylinder (Axis)

Hollow Cylinder (Axis)

A hollow cylinder rotating around its central axis.

Mass: 8 kg

Inner Radius: 0.4 m

Outer Radius: 0.5 m

Other Titles
Understanding Moment of Inertia: A Comprehensive Guide
An in-depth look at rotational inertia, its calculation, and its significance in physics and engineering.

What is Moment of Inertia?

  • Defining Rotational Inertia
  • The Role of Mass Distribution
  • Units of Measurement
Moment of Inertia, often denoted by the symbol 'I', is the rotational analog of mass for linear motion. It quantifies an object's resistance to angular acceleration. While mass measures resistance to change in linear velocity, moment of inertia measures resistance to change in angular velocity.
Key Factors
Two primary factors determine an object's moment of inertia: its total mass and the distribution of that mass relative to the axis of rotation. The farther the mass is from the axis, the greater the moment of inertia. This is why a figure skater can spin faster by pulling their arms in, effectively reducing their moment of inertia.
Units
The standard unit for moment of inertia is kilogram meter squared (kg·m²). This unit derives directly from its fundamental formula, which involves mass (kg) and the square of a distance (m²).

Conceptual Example

  • Imagine two wheels of the same mass. Wheel A has its mass concentrated at the rim, while Wheel B has its mass concentrated near the center. Wheel A will have a higher moment of inertia and will be harder to start or stop spinning.

Step-by-Step Guide to Using the Calculator

  • Selecting the Shape
  • Entering Dimensions
  • Interpreting the Results
Our calculator simplifies the process of finding the moment of inertia for various objects. Follow these steps for an accurate calculation.
1. Choose the Object's Shape
Start by selecting the shape of your object from the dropdown menu. We support common shapes like solid/hollow cylinders, solid/hollow spheres, rectangular plates, and thin rods.
2. Pick the Axis of Rotation
Next, select the axis around which the object is rotating. The moment of inertia is highly dependent on the chosen axis.
3. Input Mass and Dimensions
Enter the required parameters such as mass, radius, length, width, or height. Ensure you are using consistent units. The labels will guide you on which dimensions are needed for the selected shape.
4. Calculate and Analyze
Click the 'Calculate' button to see the result. The calculator will display the moment of inertia and the specific formula used for the calculation, helping you understand how the result was derived.

Real-World Applications of Moment of Inertia

  • Engineering and Design
  • Astronomy and Physics
  • Sports and Biomechanics
The concept of moment of inertia is not just theoretical; it has profound implications across various fields.
Flywheel Energy Storage
In mechanical engineering, flywheels are designed with a large moment of inertia to store rotational energy efficiently. They resist changes in speed, making them excellent for smoothing out power delivery from a fluctuating source like an engine.
Vehicle Dynamics
The moment of inertia of a car's wheels affects its acceleration and handling. Lighter wheels with lower rotational inertia allow for quicker acceleration.
Planetary Motion
In astronomy, the moment of inertia of planets helps scientists understand their internal composition and rotational behavior. It's a key parameter in modeling celestial mechanics.

Common Formulas and Derivations

  • Point Mass and Rigid Bodies
  • Parallel Axis Theorem
  • Perpendicular Axis Theorem
The calculation of moment of inertia is based on a set of fundamental formulas and theorems.
Basic Formula for a Point Mass
For a single point mass 'm' at a distance 'r' from the axis of rotation, the moment of inertia is I = mr². For a system of multiple point masses, it is the sum: I = Σ(mᵢrᵢ²).
Formulas for Common Shapes
Solid Sphere (about center): I = (2/5)mr²
Hollow Sphere (about center): I = (2/3)mr²
Solid Cylinder (about axis): I = (1/2)mr²
Thin Rod (about center): I = (1/12)mL²
Thin Rod (about end): I = (1/3)mL²
Parallel Axis Theorem
This theorem is crucial for finding the moment of inertia about an axis parallel to an axis passing through the center of mass. The formula is I = Icm + md², where Icm is the moment of inertia about the center of mass, 'm' is the mass, and 'd' is the perpendicular distance between the two axes.

Common Misconceptions and Key Insights

  • Inertia vs. Moment of Inertia
  • Dependence on Axis
  • Mass vs. Mass Distribution
Clarifying common points of confusion can deepen your understanding of this topic.
It's Not Just About Mass
A frequent mistake is to think that moment of inertia depends only on mass. However, the distribution of that mass is equally, if not more, important. An object with its mass spread far from the axis of rotation has a much higher moment of inertia than an object of the same mass with a more compact distribution.
An Object Can Have Multiple Moments of Inertia
Unlike mass, which is a scalar quantity, an object does not have a single moment of inertia. Its value is always defined relative to a specific axis of rotation. The same object will have different moments of inertia when rotated about different axes.