Mass Moment of Inertia Calculator

General Physics

Select a shape and enter its dimensions to calculate the mass moment of inertia.

Practical Examples

Click on an example to load the data into the calculator.

Solid Cylinder

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A solid cylinder with a mass of 10 kg and a radius of 0.5 m.

Shape: Solid Cylinder (about central axis)

Mass: 10 kg

Radius 1: 0.5 m

Hollow Sphere

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A hollow sphere with a mass of 5 kg and a radius of 0.2 m.

Shape: Hollow Sphere (about any diameter)

Mass: 5 kg

Radius 1: 0.2 m

Thin Rod (about center)

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A thin rod with a mass of 2 kg and a length of 1.2 m, rotating about its center.

Shape: Thin Rod (about center)

Mass: 2 kg

Length: 1.2 m

Hollow Cylinder

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A hollow cylinder with 15kg mass, 0.3m inner and 0.6m outer radius.

Shape: Hollow Cylinder (about central axis)

Mass: 15 kg

Radius 1: 0.3 m

Radius 2: 0.6 m

Other Titles
Understanding Mass Moment of Inertia: A Comprehensive Guide
An in-depth look at the principles, calculation, and application of mass moment of inertia.

What is Mass Moment of Inertia?

  • Defining Rotational Inertia
  • The Role of Mass Distribution
  • Units and Dimensions
The mass moment of inertia, often denoted by 'I', is a measure of an object's resistance to angular acceleration. It is the rotational analog of mass in linear motion. While mass measures an object's resistance to being accelerated linearly, the moment of inertia measures its resistance to being spun or rotated.
Key Factors Influencing Moment of Inertia
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 further the mass is from the axis, the greater the moment of inertia.

Conceptual Examples

  • A solid disk and a ring of the same mass and radius are spun. The ring has a higher moment of inertia because its mass is concentrated farther from the center.
  • An ice skater pulling their arms in to spin faster is a classic demonstration of conservation of angular momentum, which involves changing the moment of inertia.

Step-by-Step Guide to Using the Calculator

  • Selecting the Object Shape
  • Entering Physical Parameters
  • Interpreting the Results
Our calculator simplifies the process of finding the mass moment of inertia for common shapes. First, select the shape of your object from the dropdown menu. Options include solid cylinders, hollow spheres, thin rods, and more.
Inputting Data Accurately
Next, input the required physical parameters such as mass, radius, and length in the specified units (kg and m). The required fields will change dynamically based on the selected shape. Ensure all values are positive.

Calculation Walkthrough

  • To find the inertia of a 2 kg solid sphere with a 0.1 m radius, select 'Solid Sphere', enter Mass=2 and Radius=0.1, and click 'Calculate'.
  • The result will be displayed in kilogram-meter squared (kg·m²), the standard unit for moment of inertia.

Real-World Applications of Mass Moment of Inertia

  • Engineering and Mechanical Design
  • Astrophysics and Celestial Mechanics
  • Sports Science and Biomechanics
The concept is crucial in mechanical engineering for designing rotating machinery like flywheels, turbines, and engine crankshafts. A flywheel, for instance, is designed with a large moment of inertia to store rotational energy efficiently.
Inertia in Space
In astrophysics, it helps describe the rotation of planets, stars, and galaxies. The Earth's moment of inertia, for example, is a factor in its rotational stability and the precession of its axis.

Application Scenarios

  • Designing a satellite's reaction wheels, which control its orientation by changing their angular velocity.
  • Analyzing the performance of a baseball bat, where the moment of inertia affects swing speed and power.

Common Misconceptions and Correct Methods

  • Inertia vs. Moment of Inertia
  • The Parallel Axis Theorem
  • Choosing the Correct Formula
A frequent point of confusion is the difference between linear inertia (mass) and rotational inertia (moment of inertia). While related, they are not interchangeable. Mass is an intrinsic property, whereas the moment of inertia depends on the axis of rotation.
The Parallel Axis Theorem
This theorem is essential when calculating the moment of inertia about an axis that is parallel to an axis passing through the center of mass. The formula is I = Icm + md², where Icm is the inertia about the center of mass, m is the mass, and d is the distance between the two axes.

Clarification Points

  • You cannot use the formula for a solid cylinder to find the inertia of a thin rod.
  • For a rod rotating about its end, the moment of inertia is higher than when it rotates about its center, as proven by the Parallel Axis Theorem.

Mathematical Derivation and Formulas

  • The Integral Definition
  • Formulas for Common Shapes
  • Composite Bodies
For a continuous mass distribution, the moment of inertia is calculated by integrating the square of the distance (r²) from the axis of rotation over the entire mass of the object: I = ∫ r² dm.
Standard Formulas
This integral is solved for various standard shapes to yield the formulas used in this calculator. For example, for a solid cylinder of mass M and radius R rotating about its central axis, the formula is I = ½ MR². For a thin rod of length L about its center, it's I = (1/12)ML².

Formula Quick Reference

  • Solid Sphere: I = (2/5)MR²
  • Hollow Cylinder: I = ½ M(R₁² + R₂²)