Interactive module Rigid-body dynamics
Angular Momentum
Why does a satellite spin faster as its booms fold in, a bullet set a heavy door swinging, and a gyroscope float sideways instead of falling? Angular momentum answers all three. This module, written for second-year mechanical and aerospace engineering students, starts with a single particle and builds up to the 3D motion of rigid bodies, gyroscopes and tops, in eight lessons with live simulations and questions that check your work.
- 8 lessons
- about 4¾ h of lessons
- Works offline
Learning outcomes
By the end of the module you can:
- Compute the angular momentum of a particle, \(\Hvec_O = \rvec \times m\vvec\), in 2D and 3D. (Lesson 1)
- Apply \(\sum\Mvec_O = \dot{\Hvec}_O\), angular impulse and conservation of angular momentum to particles and central-force motion. (Lesson 2)
- Use \(\Hvec_O = \rvec_G \times m\vvec_G + \Hvec_G\) and the moment equations for systems of particles. (Lesson 3)
- Find the angular momentum of a rigid body in plane motion about any point, and connect it to \(\sum M_G = I_G\alpha\). (Lesson 4)
- Solve impulse, impact and coupling problems for rigid bodies, including the center of percussion. (Lesson 5)
- Compute \(\Hvec = \Imat\wvec\) and the kinetic energy of a body in 3D, and apply Euler's equations. (Lessons 6–7)
- Analyze gyroscopic moments, steady precession of tops and gyroscopes, and torque-free motion. (Lesson 8)
How to use this module
Before you start: you should be comfortable with vector cross products, Newton's second law and impulse–momentum for particles, plane rigid-body kinematics, and moments of inertia (the Mass Moments of Inertia module covers the inertia tensor used in Lessons 6–8).
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 1–265 min
Angular momentum of a particle, the moment equation, angular impulse and conservation.
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Lessons 3–465 min
Systems of particles and rigid bodies in plane motion.
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Lessons 5–675 min
Impulse, impact and couplings of rigid bodies; angular momentum in 3D.
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Lessons 7–8, Practice Lab and Self-Check Quiz140 min
Euler's equations and gyroscopic motion, about 30 min of practice, then the 35-minute quiz.
About 5¾ h in total:280 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 draw momentum and impulse diagrams.
- Do the checks
Every lesson has “Check your understanding” questions with hints and full solutions. You can type answers like
0.02*400*0.9orsqrt(2)*3. - 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, 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.