Interactive module Rigid-body dynamics
Mass Moments of Inertia
How hard is it to spin a flywheel, balance a crankshaft or point a satellite? The answer is in how a body's mass is spread around its axes. This module, written for second-year mechanical and aerospace engineering students, takes you from the moment of inertia of a single particle to the inertia tensor, angular momentum and principal axes, in eight lessons with 3D figures and questions that check your work.
- 8 lessons
- about 4½ h of lessons
- Works offline
Learning outcomes
By the end of the module you can:
- Explain what the mass moment of inertia measures, compute it for particles, and use the radius of gyration. (Lesson 1)
- Derive moments of inertia of standard bodies by integration, and use a table of standard results. (Lesson 2)
- Apply the parallel-axis theorem, and find moments of inertia of composite bodies, holes included. (Lessons 3–4)
- Compute products of inertia, use symmetry to show that they vanish, and transfer them to parallel axes. (Lesson 5)
- Assemble the inertia tensor and use it to find angular momentum \(\Hvec = \Imat\wvec\) and kinetic energy \(T = \tfrac12\wvec^\mathsf{T}\Imat\wvec\). (Lesson 6)
- Find the moment of inertia about any axis, and transform the tensor to rotated axes. (Lesson 7)
- Find principal axes and principal moments with the plane formulas, Mohr's circle or an eigen-solver, and relate them to balancing and spin stability. (Lesson 8)
How to use this module
Before you start: you should be comfortable with 3D vectors (the dot and cross products), single-variable integrals and simple double integrals, centers of mass, and plane rigid-body kinetics (\(\sum M = I\alpha\)). Matrix multiplication is used from Lesson 6 on, and eigenvalues in Lesson 8.
Work through the lessons in order. The times allow for working the examples on paper and doing the checks. A suggested plan in four sittings:
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Lessons 1–260 min
What inertia measures, particles and the radius of gyration, and integration for standard bodies.
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Lessons 3–465 min
The parallel-axis theorem and composite bodies.
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Lessons 5–670 min
Products of inertia, the inertia tensor, angular momentum and kinetic energy.
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Lessons 7–8, Practice Lab and Self-Check Quiz135 min
Any axis and rotated axes, principal axes, about 30 min of practice, then the 35-minute quiz.
About 5½ h in total:265 min of lessons, 30 min of practice, 35 min for the quiz
Get the most out of it
- Keep pencil and paper handy
Try each worked example yourself before you open its solution, and set out composite bodies in a table.
- Do the checks
Every lesson has “Check your understanding” questions with hints and full solutions. You can type answers like
2/5*3*0.1^2orsqrt(0.29). - Practice until it is routine
The Practice Lab generates unlimited problems, each with a worked solution.
- Test yourself at the end
The Self-Check Quiz gives a score, solutions and a results page you can print or save as a PDF.
- Track your progress
Tick “Mark this lesson complete” at the end of each lesson. A ✓ appears here and in Contents; it is saved in this browser only.
- Keep the formulas close
Print the Formula Sheet, with its table of standard bodies, and look up terms in the Glossary.
Lessons
Each lesson has interactive figures, worked examples and questions with instant feedback.
Practice, tools and reference
Use these alongside the lessons, or on their own when you revise.
Requirements
Any modern browser. Works offline.
- A current version of Chrome, Edge, Firefox or Safari, with JavaScript on. A laptop or desktop screen works best; the pages also fit tablets and phones.
- The 3D figures need WebGL, which every current browser has. If it is turned off, the figures show a short message and the rest of the page still works.
- No internet connection, installation or account. Everything is inside this folder.
- Your progress is saved in this browser on this device. Private or incognito windows do not keep it.
Something not working? Read README.txt in the module folder for how to open the module and fix common problems.