Lesson 4: Constant-Coefficient SOLDEs
Preview
How do car manufacturers dampen the effects of driving over bumpy roads? The answer boils down to making use of the mathematical and physical properties of springs. We will explore these properties in this and the next lesson---as well as in Capstone 2---and discover that the simplest models of spring dynamics yield homogeneous SOLDEs whose coefficients are numbers. We will spend the rest of this lesson---and the next---studying how to solve these constant-coefficient SOLDEs, leveraging the theoretical foundation we built in the previous lesson to help us find the general solution of such systems.
In Module I we'll derive the SOLDE modeling the one-dimensional motion of a spring connected to a mass, obtaining a constant-coefficient homogeneous SOLDE.
In Module II we'll begin studying how to solve the SOLDE obtained in Module I. We will discover that doing so involves the consideration of three possible cases. We'll solve the first (simplest) case in this Module.
In Module III we'll solve the next simplest case, leaving the last case---which requires a discussion of complex numbers---for the next lesson.
Review
Learn
Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson 4.pdfClass Notes A
Class Notes B
Class Notes C
Class Notes D
Class Notes E
Class Notes F
Class Notes G
Reflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 4 - Practice Problems.pdf