It’s time to put the entertainment industry that is SHMOP1 to one side and to take a peek at how we attempt to describe actual reality that doesn’t have anything to do with feelings.
I’m going to need this later on when I delve a bit further into quantum mechanics and so, for once, I’ve chosen a sensible title that I have half a chance of remembering.
Let’s start off with a question. You’ve got to move a big heavy crate and so you decide to attach a rope to the top and pull it. You’ve only had 2 weetabix for breakfast that morning and so there’s only so much ‘pull’ you can exert. The question is whether you’re better pulling the crate ‘straight’, or at an angle?
It’s best seeing it in a diagram :
The pull you apply (we call this a force) has a strength, but it also has a direction. The question here is about the difference the direction makes.
Obviously if you tried to just pull vertically upwards, you aren’t going to move the crate along at all (you might not even be able to lift it).
The answer is that you’re better off (in terms of getting the object to move faster) if you pull at a slight angle.
Intuitively we can see that when the rope is at an angle some of the pulling effect (transmitted via tension in the rope) is going to be trying to ‘lift’ the crate, and some is going to be trying to pull it along. This small lifting force has the effect of reducing the amount of friction between the ground and the crate - and this is enough to make it a better option to pull at an angle.
You can even work out (theoretically) what the best angle to choose is - but I’m not going to do that here. The point is that, intuitively, we’ve done something that is technically known as resolving the force into its components.
Although I’ve not shown the analysis, the take-away is that by doing this ‘resolving’ we can arrive at a useful conclusion - it’s better to pull at a slight angle.
Let’s think of another example; this time with no friction. So we might imagine a crate on an ice rink where there is no friction (or only a very tiny amount of friction) between the crate and the ice. This time we’re going to imagine 2 people trying to push this crate.
You can, I hope, see why the crate will move in a different direction to either of the directions in which it is being pushed. Furthermore, I hope you can also see that we could replace the two men with a single man pushing in the direction of the green arrow and get the same movement of the crate2.
If you’ve grokked these two examples, you’re on your way to understanding vectors.
There are certain properties we observe that we’d like to be able to describe in terms of maths so that we can analyse situations like those of the two examples above. You can see we’re going to need some kind of maths that takes account of two things
(a) how much force is being applied
(b) the direction the force is applied in
The mathematical object we need is known as a vector.
One of the best ways, for most people, to first get their heads round vectors is to think of arrows drawn on a piece of paper3.
The length of the arrow is going to represent how much, and the direction the arrow is pointing in is going to indicate, well, obviously, the direction.
If we’re told you’re travelling at 70mph on the M1 motorway in the UK all we know is how ‘much’. We know how fast you’re travelling, but have no idea whether you’re heading for Leeds or London.
We can encapsulate the speed and the direction by using something called the velocity vector.
We can encapsulate where something is by taking some nominal start point and drawing an arrow from the start point to the object; this is known as the position vector. The ‘amount’ of walking we’d have to do to get from A (the start) to B (the position of the object) is the length of the vector and the angle it makes with some agreed baseline gives us the direction we have to set off in.
From the examples above we can begin to see two very important properties of vectors (which are really just the same property)
They can be added together to give another vector (the two men can be replaced with one)
Any given vector can be thought of as being ‘made up’ from the addition of other vectors (the rope at an angle had some force ‘up’ and some force ‘along’)
These things give us a clue as to one reason why vectors are so useful when analysing physical situations. If we’re looking to figure out the effect of a force - represented by a single vector - we can ‘split’ that up into two (or more) forces acting in different directions. We can look at the effect of each of these ‘component’ forces separately on the motion.
In the case of the rope at an angle we can look at the ‘bit’ of the force that’s trying to lift the crate up, and also the ‘bit’ of the force that’s trying to pull the crate along.
What we’re saying is that OK, we really have just one guy pulling at an angle, but it’s the same as if we had 2 guys; one trying to pull upwards a little bit, and one trying to pull along. In this case we’d say, technically, that we’d resolved the force into an up component and an along component.
These are the ideas that we want to capture. The only thing left now is to figure out how to represent all of that with the maths.
The maths I’m going to leave to part II.
What I’ve tried to do here is to give you (some of) the reasons why we need to do the maths.
In quantum mechanics it turns out that we have to describe things using something that’s known as a state vector. Which is why I need to go into the properties of vectors a bit.
When you hear about something like Schrödinger’s Cat this is (mathematically) nothing more than ‘resolving’ the actual state vector of the cat into a ‘dead’ and ‘alive’ component - just like we resolved the force with the rope example into an ‘up’ and ‘along’ component.
This causes no end of confusion because the addition of vectors in the QM context (adding up ‘dead’ and ‘alive’) often gets taken to mean the cat is in some ‘superposition’ of actually being dead or alive. It’s a nonsense, of course.
Just as it’s a nonsense to say that the guy pulling the rope at an angle is in some ‘superposition’ of two guys; one pulling up and the other pulling along.
The Sacred and Holy Month of Pride
He would have to be pushing with a greater force than either of the two men, but we’ll come to that elsewhere (probably part II)
It’s, perhaps, surprising that when you start getting a bit more technical with vectors (maybe in a Linear Algebra course) you can go a long way to understanding some of the more sophisticated theorems about vectors just by drawing arrows.
The maths is all beyond me but on a practical level I would have got myself a large pointy stick and a couple of logs (rollers) to move the crate along!
Is SHMOP pronounced as in mop or mope? I can see either working.