Introduction - Second-Year Dynamics

Welcome!

This resource is an open-source textbook for second-year classical mechanics and flipped classroom instruction that is currently under development for Queen’s University by Prof Sarah Sadavoy with support from Queen’s students Cora Sleegers and Lance Schonberg. This Textbook is licensed under a Creative Commons Attribution 4.0 International License (CC-BY-SA-NC). Materials can be used, shared, and adapted with credit, but materials should not be used for commercial purposes.

This textbook covers the main topics of the second-year dynamics course for Queen’s University, PHYS 206. These topics include:

Newtonian Mechanics
Simple Harmonic Motion
Non-inertial and Rotating Reference Frames
Linear and Angular Momentum
Torque and Rotation
Work and Energy Conservation
Central Forces
Motion in Space
Orbits and Kepler’s Laws
The Euler-Lagrange Method

How to Best Use this Book

This book follows the design and format from Introductory Physics: Building Models to Describe Our World, which is an open-source textbook for first-year physics developed by Prof Ryan Martin and students Emma Neary, Joshua Rinaldo, and Olivia Woodman among others.

To help broaden the physics instruction, this book contains a number of different elements to help with the learning process, including check point questions, worked examples, student commentary, and real-world applications of the physics. There are also videos and extra problems using python available via our online repository. Each textbook feature is represented by a different tag, as summarized below:

Physics Mindset and Strategies

Physics teaches problem solving. The main goals of this textbook are to develop your physics toolkit and to teach critical thinking so that you can break down big problems into smaller, easier to implement pieces. There are often multiple ways to solve a physics problem, and by building your foundation, you will be able to select which of those ways is most efficient or most ideal. When solving problems, think about the different methods you could use to solve physics problems and under which situations you would favour one method over another.

Thinking like a physicist: there are lots of ways to approach physics problems. As you practice solving problems, try to recognize and develop the strategies that make learning best for you. Everyone is different, but here are some common tips and strategies to consider:

I) Make sure you understand the system set up: Before attempting any problem, be sure you know what the system looks like given the description and all the moving parts. If it helps to visualize it, draw a diagram (it doesn’t need to be perfect). Taking a few moments to think about the system as a whole can save you a lot of time and trouble when solving the problem later on.

II) Plan how you will approach the problem: There are a number of ways you can solve most physics questions (e.g., Newton’s laws, conservation laws, Euler-Lagrange method). Before you decide on an approach, consider what information is given to you, what the question is asking, and what assumptions you can or cannot make. The more you practice, the better you will intuit how to approach most questions.

III) Stick with variables/symbols as long as possible: Symbols are less messy than numbers and you don’t have to worry about units. It is also easier to check your answer at the end if you only have to plug numbers into your calculator once.

IV) Look for tricks to simplify the problem: Often, physicists can use math tricks or approximations to simplify the calculations or set up of a physics problem. Examples include math substitutions, scaling relations, or Taylor series simplifications. The textbook will highlight these approaches and give insights on when they are appropriate. Use these to your advantage.

V) Check your answer: After working through a complex problem, consider what you got at the end. Do your units make sense? Does the order of magnitude of a numerical value make sense? For example, if you get the motion of a simple pendulum to be faster than the speed of light, that should be an automatic flag that something went wrong. Think about your answer critically before moving on.

Dimensional Analysis Tips and Tricks

Dimensional analysis can be an effective way to check your answer. Unit conversions can be tricky and in some cases, you may be given specialized units over standard units (e.g., distances in astronomy are often in parsec or light years). Dimensional analysis lets you verify that the units broadly match your expectations. For example, the dimensional analysis of kinetic energy, $K = \frac{1}{2} mv^{2}$ is:

[\mathrm{J}] = [\mathrm{kg}][\mathrm{m} \mathrm{s}^{-1}]^{2}

(1)

[\mathrm{J}] = [\mathrm{kg} \mathrm{m}^{2}\mathrm{s}^{-2}]

(2)

[\mathrm{J}] = [\mathrm{J}]

(3)

For more information, see lists of Standard International (SI) units and some simple conversions from Wolfram, National Physical Laboratory, and Wikipedia.

VI) Physics is best learned with others: Whether you are hearing the thoughts and perspective of a classmate or vocalizing your own understanding, you learn more and retain that knowledge better by sharing with others. Share ideas with your classmates. Work on problems with each other. And have fun!