Chapter 1 - Introduction, Scalars and Vectors

Introduction

I would like to present this introduction to mechanics by talking both about how some of the fundamental aspects of physics arose, and about the people who came up with them. I aim to show that physics is not just made up of arbitrary rules, seemingly pulled out of the air, that the world must obey. Although it can often appear this way, the truth is that the laws of physics as we know them were put together over the course of hundreds of years, by people who simply looked at the world and attempted to describe it. It turns out that mathematics happens to be a very good language for this description, which is why equations are so prevalent in physics. However, I believe that it should be possible to provide an explanation of most, if not all of it, with words, resorting only occasionally to the maths that comes alongside.

Which Mechanics?

This course is called an An Introduction to Mechanics, but in fact there are two main strands of mechanics: Classical and Quantum. Therefore, to be clear, this is a course on classical mechanics. In fact, in almost all cases, when the word mechanics is used on its own it refers to classical rather than quantum. What's the difference? Well, classical mechanics is much older, and was developed over hundreds of years based largely upon the work of Sir Isaac Newton. It deals with the motion of objects as a result of the action of forces. Quantum mechanics, by comparison, has been around since the early twentieth century. It is arguably more complex and mathematical, has more of a focus on individual particles, and makes predictions based upon probabilities. For more information on quantum mechanics, see the course, for more information on classical mechanics, read on.

Vectors and scalars

In many areas of physics, special tools are used to to help model the world. I do not mean physical tools, like particle accelerators, although these have their place too. No, in fact you could call these conceptual tools, essentially new ways of thinking that make navigating the more complex areas of physics a lot simpler. Commonly these will go hand in hand with maths, and full understanding will require some mathematical ability. However, for the most part we will attempt to cover things from a conceptual perspective only, which should be enough to achieve a reasonable understanding of mechanics.

The first pair of tools that we will cover are called scalars and vectors. They are very important to understanding not only mechanics, but many other areas of physics as well. Now, from their names they may sound somewhat complicated, but in truth they are fairly simple. In fact, you already know what scalars are, even if you don't refer to them by that name, and may also have some inkling of how vectors work too.

Scalars

Scalars are a simple number used to describe a physical quantity. They are usually best described by example: temperature is a scalar, length is a scalar, mass is a scalar. For example, say you would like to know the temperature of a room. You measure it, and it turns out to be 20 degrees. That value is a scalar. Now, say you would like to know the height of a ladder. You use a tape measure and it turns out to be 10 metres. That value is a scalar. These are just numbers, nothing more complex than that. Hopefully this will make sense to you; most of us use scalars, like temperature and length, every day of our lives without realising or thinking about it.

Vectors

Now, on to vectors. These are only slightly more complicated. They are numbers, like scalars, only in addition to a value they also have a direction. Usually scalars and vectors are distinguished by saying that scalars have a size (also known as a magnitude) only, and vectors have both a size (a magnitude) and a direction. Vectors can probably be best understood by example and by comparison with scalars.

Consider a car travelling along a road. You may want to know some information about that car, for example, how fast it's travelling. That value is referred to as its speed, and is commonly measured in miles or kilometres per hour. That value is also a scalar, it's a simple number that tells you one thing, how fast the car is travelling. However, suppose you want to know more information about the car. Suppose the road is a straight line in the North/South direction and you want to know whether the car is travelling along it to the North or to the South. The speed, when combined with this information about the direction of travel, is known as the velocity, and is now no longer a scalar, it is a vector.

Hopefully you can see why vectors can be useful, they can tell you a lot about an object. They are also necessary for describing things like forces, which feature heavily in classical mechanics. But how do we represent vectors? Scalars are simple enough to write down, they are just numbers after all, but how do you write down a number that also contains a direction? Well, it can be done mathematically, or visually. We will just look at the visual method for now, which simply involves drawing arrows to represent the vector. The direction the arrow points can be used to give the direction of the vector, and the length of the arrow represents its size (magnitude).

SVGs of some arrows.

THERE WILL BE SOME PAGE NUMBERS HERE.