Welcome and Introduction

Welcome to Physics 2210!

Overview

Let's start with the big picture: what are the main topics we'll be studying this semester? This is actually something of an unusual class, in that it's both a physics class AND a math class at the same time! The physics topics are all things you have seen before, at some level:

  1. Newton's laws
  2. Projectile motion
  3. Energy and momentum conservation, gravity
  4. Simple harmonic oscillators
  5. Driven oscillators and heat transfer

The main difference is that we'll be digging deeper into all of these problems than what you saw in first-year physics. In doing so, we'll discover that we need more sophisticated math tools in order to actually solve the problems that we find. Better yet, the math that we'll end up needing is central to a lot of the more advanced physics topics you'll see later! The math highlights will include:

  1. Curvilinear coordinates and vector algebra
  2. Ordinary differential equations
  3. Vector calculus
  4. Second-order differential equations
  5. Fourier series and partial differential equations

and we'll draw on some other topics that you may be more or less familiar with, including special functions, complex numbers, and power series.

This should be enough motivation, but since you've now seen modern physics, you might wonder: "why are we still studying classical mechanics, when we know that it's wrong?" Of course, the intuitive answer is that it's only wrong for really small things (quantum mechanics) or really fast things (special relativity). There is a more rigorous answer that involves one of the more deep and powerful concepts in physics, which is the idea of an effective theory. I'll talk a little bit about this later.

If we have extra time left at the end of the semester, we'll dig into a bit of special relativity. (You have seen the basics in modern physics, but at the point we'll be able to go further and talk about relativistic mechanics.)

Syllabus

We started class by briefly going over the syllabus. Remember: you're responsible for knowing all of the contents of the syllabus, and it is subject to update through the semester. (Any such updates will be announced to the class and indicated on the syllabus page.)

A few words on learning

One topic that doesn't come up enough in classes, at least in my opinion, is the topic of how to effectively learn. You've all made it as far as this classroom, which certainly means you each have your own effective ways of approaching this problem. But I think it's useful to go over a couple of scientific ideas related to learning, which will also help me explain a couple of reasons why the class is structured as it is.

I'm just going to give a couple of high points, but if you feel like you want to delve deeper into the subject, I recommend the Coursera course "Learning How To Learn" as a lightweight but insightful set of material.

I firmly believe that anyone can succeed in this class, as long as they put in the required time and effort.

Space and time

Mechanics is the study of motion, and motion is described using the spatial positions of a physical system as a function of time. So, the best place to start is by carefully defining how we will talk about space and time.

One warning Taylor gives that I'll amplify: if Taylor and my choices of notation are not what you're used to, that's a good thing! There are many conventions and notation choices out there; to read scientific books, papers, etc. more broadly, you need to get comfortable with changes to notation. (That doesn't mean I'll change arbitrarily mid-class, but I will comment on popular alternatives sometimes.)

Units and dimensions

The foundation of science is reproducibility, which means if you and I do the same experiment, we will find the same results. But defining what "the same results" means requires us to agree on some ground rules. One very important detail is choice of units (look up the sad story of the Mars Climate Orbiter!), but you should be familiar with that by now, so we'll just agree to work in SI and move on.

Actually, one piece of terminology first: we will work in SI units, but a slightly more general concept than units is the idea of dimensions. A dimension is a basic physical property of some system. Meters, micrometers, feet, and light-years are all different units, but they all measure the same dimension, which is length. The important physical dimensions for mechanics are:

and in fact, that's it: we can derive everything else. For example, energy (SI base unit: Joules, J) has dimensions of \( [M] \cdot [L]^2 / [T]^2 \), and correspondingly the Joules is equal to \( \textrm{kg} \cdot \textrm{m}^2 / \textrm{s}^2 \). As long as we only work in SI base units, we can use dimensions and units more or less interchangeably (and I will mix the terms together), but it's worth remembering that there is a difference. (Note: there are some other concepts that aren't just simply derived from this set of three dimensions, like charge and temperature. But length, time, mass is all we will need for mechanics!)

The existence of dimensions gives a really important distinction between physics problems and pure math problems. The simple requirement of matching dimensions can greatly restrict the space of allowed solutions, and sometimes lets you guess the form of the right answer without a full solution (this is called dimensional analysis.) Even if you solve a problem in full, checking dimensions is a nice filter to easily verify your results (this will save you hours of confusion and many mistakes over the course of your physics career!)


Clicker Question

(Note: in class I just did this as an example, since we didn't have clickers set up yet. But this is what it might have looked like as a clicker question...)

You remember that the formula for the period of a pendulum \( T \) involves the gravitational acceleration \( g \) and the length \( L \) of the pendulum, and a square root, but not how they fit together. Use dimensions to find the right combination below:

A. \( T \sim \sqrt{gL} \)

B. \( T \sim \sqrt{1/gL} \)

C. \( T \sim \sqrt{L/g} \)

D. \( T \sim \sqrt{g/L} \)

E. \( T \sim \sqrt{g/L^2} \)

Answer: C

Using dimensions, we know that period must have units of time \( [T] \), so whatever is under the square root should be \( [T]^2 \). The gravitational acceleration has dimensions \( [g] = [L] / [T]^2 \), so we must have \( T \sim \sqrt{L/g} \).


(A minor aside on notation: \( \sim \) is the similarity symbol. \( T \sim \sqrt{L/g} \) is read as "\( T \) goes like \( \sqrt{L/g} \)". It's a much weaker version of equality: the exact interpretation varies based on context, but it always indicates "there is some stuff missing here." It can denote an especially rough approximation, or that the two sides are only equal in some special limit, or that dependence on some other variables or constants is missing. Physicists love this symbol, because we are happy when we understand the gross details of a problem as a starting point. Here it denotes a missing constant: the correct formula is \( T = 2\pi \sqrt{L/g} \).)

Rectangular coordinates; vector products

Next, we have to agree on a coordinate system to describe where things are in our experiment. We live in three dimensions, so we need three coordinates - three numbers - to uniquely describe a given point in space. (You can take this as a definition of what "three dimensions" means, in fact.) We also need to agree on an origin, \( O \), from which our coordinates are measured.

All of the below should be review at least partly, but it will be a good opportunity to refresh your memory and let me set up some math notation, for which I'll generally try to follow Taylor.

We also have to agree on our time coordinates, but since there's only one dimension of time, that's easy: if my time axis is \( t \) and yours is \( t' \), the only possible difference is that we might disagree on the origin, i.e. my \( t=0 s \) might be your \( t'=2 s \). Unless explicitly said otherwise, I'll assume that we always have a single common time axis \( t \) with common origin, and just worry about the other coordinates.

Let's start with the most familiar coordinate system, called rectangular or sometimes rectilinear or Cartesian coordinates. These are the coordinates \( (x,y,z) \) that describe distances along a set of three perpendicular axes. Any choice of three axes will do, as long as they're all mutually perpendicular!