Hello there, my name is Mr. Forbes.

Welcome to this lesson from the Hidden Forces Unit, which is called Hooke's Law.

Hooke's law is a rule that describes the relationship between the force acting on a spring and the extension of the spring, how much it stretches.

So we're gonna see what that relationship is.

By the end of this lesson, you're going to be able to describe the relationship between the force acting on a spring and how much it's extended, how much longer it's got.

And you're going to do that by processing some data that's collected during an experiment, drawing a graph using that data, and seeing the connection between the two variables.

So when you're ready, let's begin.

This is a set of the keywords that you'll need to understand for this lesson.

The first of them is line of best fit, and a line of best fit is a line we draw on a graph using the data points to try and analyse the relationship between the two variables.

Directly proportional is the relationship between the variables where one is a constant multiple of the other.

Elastic is the behaviour of a material that once you've removed the forces acting on it, it'll return to its original shape and size.

And Hooke's Law is the law that describes the relationship between the force and the extension on the spring.

And here's a set of explanations for those keywords that you can refer back to during the lesson if you need to.

This lesson's in three parts.

And in the first part I'm gonna lead you through the process of plotting a graph using the data from an experiment in putting forces on a spring.

So we'll end up with a force extension graph.

In the second part of the lesson, we'll look at the patterns in that graph by using a line of best fit.

And that'll also help us to identify anomalous results, results that don't fit the pattern.

And in the third part, we'll look at the relationship between the two and something called Hooke's Law, which is a description of what happens to one variable when the other one changes.

So when you're ready, let's start.

So before we get to plotting the graph, let's outline the experiment that was carried out to generate that data.

What I did was I had a spring and I placed downward forces on that spring to cause it to get longer.

So I hung the masses on the end of it from a mass holder that produces forces that are downwards.

And as a result of that, the spring extends or gets longer like that.

We call that increase in length the extension of the spring.

And that's the piece of information that we're gonna compare to the force acting on the spring.

So that extension there.

Now during the experiment, I added weights to the spring to make it get longer, but I also took them off again towards the end and remeasured the extension so that I could make a comparison between the spring extending and contracting back again, going back to its original length.

And I did that so I could check the accuracy of the readings and to make sure I was making the measurements correctly.

So I added the weights like that and then I took them back off again to get that second set of results.

As I've got two sets of results, I need to calculate a mean value for those measurements.

And to do that, add them both together and then divide by two.

I've already started that process with the results in this table here.

As you can see, I've filled in the first three values for the mean extension.

Let's have a look at how to calculate the next two.

So for three Newtons acting on the spring, I need to find the mean extension by adding the two length measurements, that's 12.

4 centimetres and 12.

6 centimetres, and then divide it by two.

And that'll give me the mean value, which is 12.

5 centimetres.

Similarly, for four Newtons acting on the spring, I calculate the mean extension by adding those two length measurements, then divide it by two, which gives me a value of 16.

6 centimetres.

Okay, just to check if you can calculate means, here's another results table.

And what I'd like you to do is calculate the mean extension for the force of two Newtons using the data on the table.

So pause the video, do the calculation, and then we'll look at the answer when you're ready.

Okay, the answer you should have got there was 8.

0.

So the mean value missing from the table is 8.

0.

And I calculated that by adding the 8.

3 and the 7.

7, then divided by two, and that gives a value of 8.

0.

So 8.

0 centimetres there.

Well done if you've got that.

I'm going to be plotting a graph using the data I've collected for the experiment to see the relationship.

And one of the first stages in plotting a graph is to decide on the scales on the two axes.

So I've got a table of data here that I'll use.

The first thing we're going to do is see what the starting point for the force axis should be.

And as I started with no force acting on the spring, it's gonna start at zero and that's fairly typical for most graphs.

The highest value for the force was six Newtons.

So my axis for that is gonna go up to six Newtons.

On the extension part of the graph, the vertical axis, again, I'm gonna start at zero because there was no extension at first.

And my final value is 24 centimetres.

So what I've decided to do is to go slightly higher than that and go up to 25 centimetres because that'll give me nice even steps.

Okay, to check if you understand how to choose scales properly, I've got a set of data here in the table, and it's slightly different than the data I just showed you.

And what I'd like you to do is to decide on the best scale to use for extension out of that list there.

So pause the video, make your selection, and then restart.

Okay, welcome back.

The best selection there would've been nought to 35 centimetres.

So well done if you selected that.

nought to six Newtons is not the scale for the extension axis.

nought to 30 centimetres, you wouldn't be able to plot the highest value there, and nought to 50 centimetres is probably too large.

Okay, let's go through the stages of plotting a graph carefully.

The first stage is to draw the axes on the paper.

So the horizontal and vertical lines where we're gonna put the scales, and we draw those on the graph paper towards the edges.

Sometimes you leave a little bit of space so you can put the numbers in, but there you go.

I've drawn the axes on my graph.

The next stage is to add the divisions and the scales to the graphs.

So I'm gonna add those in here, starting with the horizontal scale.

I'll write the numbers nought to six which is what I decided on earlier.

And then I'll add the divisions to the vertical ones.

And I'm going up in five centimetre increments there all the way up to 25, which is a little bit above the largest value I've got for my extension.

So that should work well.

Then I'll add my axes labels.

First one is the force one across the bottom, and I add those after the numbers so I can get them nice and neatly on there, and so they don't get in the way of the numbers.

And then I'll put my vertical one there, the extension in centimetres.

And then the next stage is the most important one.

We've gotta very carefully plot each point from the data table using small crosses.

So I'm gonna start plotting those.

And what I like to do is as I do it, I kind of tick them off in my data table to make sure I know where I'm up to.

So I'll plot the first one there, making sure I've plotted it as accurately as I can, a nice sharp small cross.

And then the next one and the next one and so on, adding them one at a time very carefully.

And double checking each point to make sure I put it in exactly the right place.

And there you go, I've completed my graph plotting, I've got all the data there.

The only thing missing from it really is a title.

So I like to write that at the end to make sure it doesn't get in the way of any of the data plotting.

So I'm gonna add the title and put it at the top there.

So there you go, I've completed my graph.

Okay, a fairly simple check for you here.

I'd like you to decide which of those is the best for plotting on a line graph.

Is it hollow circles, diamonds, small crosses, or thick crosses? So pause the video, make your selection, and then restart.

Okay, if you were watching carefully earlier, you should know it was small crosses.

So well done if you've chosen that.

Small crosses allow you to mark the points very precisely on the graph paper.

So well done.

Okay, it's time for your first task now, and obviously it's going to be about plotting graphs.

And so I've got a data table here, data I've collected from an experiment of stretching the spring.

And what I'd like you to do is to plot a graph using it, a force against extension graph please.

So pause the video, plot your graph, and then restart once that's complete.

Welcome back.

If you used the data I showed you there, your graph should look like this one.

So let's have a look at the features I've got.

I've got the title there across the top, I've got an even four scale on the bottom from nought to five newtons.

I've got an even extension scale, nought to 30 centimetres along the side there.

And I've labelled both of the axes there.

And finally, I've plotted all my points as accurate as I could.

And you can see they align up really neatly along there.

Well done if you've got a graph that looks something like that.

Okay, we're ready for the second part of the lesson now.

And in it we're gonna look at the graph we've already produced, and we're gonna add a line of best fit to it.

And that line of best fit will show us the relationship between the two variables, and will also allow us to spot some anomalous results if there are any.

Okay, the first thing I need to do is to explain what a line of best fit is and what it isn't.

A line of best fit is used to show the connection between the variables.

So in this case it's going to show us what the connection between force and extension is.

And it's not a series of lines that join the crosses that we've plotted together.

It's a single straight line, something like this, or perhaps it might be a curve like that.

And in physics, the only two types of line we use on these graphs is a single line drawn with a ruler or we can draw some sort of curve, but we definitely don't try and join those points together.

Most of the lines of best fit we draw in physics are actually straight.

And so you can see the data here.

The pattern looks like a straight line, doesn't it? Just looking at those crosses.

If we're going to draw a straight line, we should use a ruler.

And when we draw the line, we're not trying to connect all the points together, we're just going to draw one single line.

And that one single line should pass between the points.

And we try and get it so that a few of the points are above the line and a few of the points are below the line.

To do that, a transparent ruler is the best because you can see the points beneath the ruler and it allows us to try and judge and get that fit just right.

So I place the transparent ruler and I try and put it so that a couple of the points are above the line and a couple of points are just below it, and you can see those through the ruler there.

And then what I do is I draw the line in a single just straight movement and remove the ruler.

And there I've got a line of best fit.

And as you can see, this is a single straight line of best fit with a couple of points above and a couple of points below.

It doesn't join those crosses together and that's very important.

Okay, to check your understanding of what a line of best fit is, I've drawn three lines on this graph, but only one of them is the correct line of best fit for the points.

So what I'd like you to do is to decide which of them it is.

So pause the video, make your decision, and then restart.

Welcome back.

You should have chosen the solid yellow line there in the middle as the line of best fit, 'cause that passes between the points with a few of them above and a few below.

It doesn't attempt to join them together or anything like that.

You can see the red dash line, that's too steep, that passes above most of the points.

So that's not a very good line of best fit.

And the purple dotted line is too low, that passes beneath most of the points.

So the best line there was the yellow line.

Well done if you chose that one.

Now, when you're plotting a line of best fit, you try and judge which of the data points it goes above and which it goes below.

But there are some data points that we know to be correct, and so we should try and make the line pass closer or even go through those points.

One of those points is when there's no extension of this spring, when I've got no force on it, there's no extension by definition, and therefore the point where zero force and zero extension is a point we know that's definitely correct because we define extension as the increase in length.

So when I'm drawing a graph involving extending the spring, I can use zero zero, the origin of the graph, as one of the fixed points.

So when I try and draw my line of best fit, I always make sure that that line goes through the zero zero here at the bottom.

So that's a fixed point that we should always try and make this type of line of best fit go through.

Although zero zero is a fixed point on this graph and I make the line go through it, some other points might not fit the pattern very well.

You can see from the graph here, most of the crosses I've drawn on the graph are very close to a single line, that line of best fit there.

But one of them is a bit far off it.

That type of point is known as an anomalous result.

And the lines help me identify that one.

Even if I didn't identify it in the data table earlier.

That anomalous result really is probably a measurement error of some kind.

And I shouldn't use it in judging where to draw my line of best fit.

So what I do with it is I identify it, I circle it like I've done here, and then I ignore it.

I don't use it at all in judging where the line of best fit is 'cause it's probably an error.

So draw the line of best fit and ignore the anomalous results, but make sure you've identified them so that whoever's looking at your graph can tell that you've not used it.

So here's a set of points I plotted on a graph from data collected from extending a spring.

I'd like you to look very carefully at this and try and identify which of those points is anomalous, which one doesn't fit in with the line of best fit.

So you might have to look very carefully and select from A, B, C, D, E, or F.

So pause the video and restart when you've made your choice.

Okay, welcome back.

Well, it's rather difficult to actually spot it just from looking at that pattern.

What we need to do is to draw the line of best fit on.

And once the line of best fit is drawn, you can see quite clearly that Point D doesn't belong.

Point D is the anomalous result.

It's slightly off the line, whereas all the others fit very, very well.

Well done if you spotted that.

Okay, it's time for the second task of the lesson now, and I've partially plotted a graph for you.

What I'd like you to do is use the graph provided and the data table shown there to complete plotting.

And then I'd like you to draw a line of best fit.

Once you've drawn that line of best fit, I'd like you to identify and circle any anomalous results that aren't close to the line.

So pause the video and carry out those three steps and then restart.

Okay, welcome back.

Your graph should look like the one here.

You can see that nearly all of the points are in a perfect straight line passing through the origin there, and only one of them seems to be off it there and I've circled that one for you.

So there's my line of best fit and the anomalous result you should have identified was that one, and that was for the force of three Newtons.

It's a little bit out, it's not too far, but it's still far enough off the line for me to decide it's anomalous.

Well done if you did all that.

Right, we're ready to move on to the final part of the lesson now, which is all about Hooke's Law.

And Hooke's Law is the description of the relationship between the force and the extension on the spring.

So let's get on with that.

The first thing that we can look at is the behaviour of the spring when the forces are acting on it and when they're taken off it.

And you can see I've got a data table here showing the extension for different forces.

And what you should notice from it is as I increase the force on it, it gets longer.

And then I decrease, it gets shorter again.

But more than that, you should see that it's actually returned to its original length.

When I've taken all the force back off it, it's returned back to a length, an extension of zero centimetres.

So the spring is returning to its original size.

We call that type of behaviour elastic behaviour.

So the spring is behaving elastically.

That elastic behaviour only applies as long as you don't overstretch the spring.

If you put a very large force on it, then you can permanently change its shape.

So the spring is elastic as long as the force is not too large.

If you put too large a force on it, you end it with something like this.

And you can see I've overstretched that spring and it doesn't return to its original shape.

It's much longer than it was originally even though there's no force on it anymore.

So let's return to looking at graphs of the extension of the spring.

And I've got a graph here of a spring that I've added forces to.

And there's two key pieces of information you should be able to see from it.

The first one is this.

When I draw the line of best fit, it's straight.

So it's an almost perfect fit for those crosses there.

I've got a straight line of best fit.

For every one Newton I add to the spring, it extends by exactly the same amount.

So when I have one Newton, then another Newton, and a third, each time it's extending by exactly the same amount.

The second thing you need to see from the graph is it passes directly through the origin there.

So when there's no force on it, there's no extension.

This type of graph where it's a straight line passing through the origin shows a relationship called direct proportionality.

This graph is showing that the extension is directly proportional to the force.

When there's no force, there's no extension at all.

And that's critical for this type of graph.

And then when I put one Newton on it, the extension of four centimetres, if I double the force to two Newtons, I double the extension.

So the extension's now gone up to eight centimetres.

If I triple the force to three Newtons, I've tripled the extension, it's gone up to 12 centimetres there.

And that pattern continues when I add more force to it.

So it goes up again and again and it's continuing to go up by the same amount every time I increase the force by one Newton.

So as I've said, that relationship is direct proportionality.

The extension is directly proportional to the force.

Okay, let's see if you can identify direct proportionality from a graph.

So I've got three lines on this graph here.

Which of those lines shows direct proportionality? Which one shows that the extension is directly proportional to the force? And I'd like you to describe the features that allowed you to identify which one as well.

So pause the video, then restart once you've got your description.

Okay, welcome back.

You should have selected the black dotted line there, Line C, and you should have selected that one because it's straight.

So it shows that a straight line and it passes through the origin.

Line B, the green line, isn't straight.

And Line A, well, that one doesn't pass through the origin even though it is straight.

So well done if you selected Line C.

Now, when I carry out the experiment with a set of different springs, I find results that look a bit like this.

You can see on my graph I've got the blue line, I've got the red dotted line, I've got the green dash line.

And all three of those show that the extension is proportional to the force.

Even though it takes a greater force to extend the blue line there, it's still a straight line passing through the origin.

So all three of them show direct proportionality.

That relationship is known as Hooke's Law and Hooke's Law states that the extension of a spring is directly proportional to the force acting on it, and it applies to all springs up to a certain limit.

If you put too large a force on it, it stops obeying Hooke's Law.

So Hooke's Law no longer applies.

So Hooke's Law describes the behaviour of all springs.

The extension is directly proportional to the force acting on it.

Okay, let's check if you understood that.

I've got a graph here where I've put much larger forces on the spring and I'd like you to explain, how does that graph show that there's a limit to Hooke's Law? Is it A, the line does not pass through the origin? Is it B, the extension is never directly proportional to the force? Or is it C, the line stops being straight when the force is large? So I'd like you to pause your video and then restart when you're ready.

Okay, welcome back.

Well, you should have selected C there.

You can see from the graph that the line does pass through the origin and there is a straight section of it.

So there are sections where the force, sorry, the extension is directly proportional to the force, but towards the end of the graph you can see it curves.

It stops being a straight line and that means that the extension is no longer directly proportional to the force, and that's the limit to Hooke's Law.

Okay, we've reached the final task of the lesson now and this brings together all of the things we've learned so far.

So what I'd like you to do is this.

You can see I've got a graph there of two separate springs, A and B, showing the extension for different forces.

And I'd like you to explain how the graph shows that both of the springs are obeying Hooke's Law.

Then I'd like you to work out how far each of the springs stretch for each Newton of force acting on them.

And finally, I'd like you to try and make a prediction using the graph.

I'd like to know what the extensions of those springs would be if I use the force of 12 Newtons on each of them.

So pause the video, try and answer those, and then restart when you're done.

Okay, welcome back.

And here's the answer to the first of those.

Both of the lines are straight, and they both pass through the origin.

And that means that the extension is directly proportional to the force, and that's what Hooke's Law says.

So that's why Hooke's Law applies here.

For the second part, we can have a look at the graphs again, and what we could do is look at the force of 10 Newtons and see the extensions.

And that will allow us to look across and calculate the extension per Newton just by dividing those extensions by 10.

And that gives us Spring A, four centimetres per Newton, and Spring B extends two centimetres per Newton.

And for the final part we're gonna make a prediction.

We can either use those values we just calculated or we can extend the graphs like this.

And that will give us the extensions for the springs.

Spring A, 48 centimetres and Spring B would be 24 centimetres extension at 12 Newtons.

Well done if you've got all of those.

Okay, we've reached the end of this lesson now, and let's have a quick summary of what we've learned.

A graph can be used to show that variables are directly proportional to each other.

And to do that we draw a line of best fit.

And if that line of best fit is straight and it passes through the origin, we know that the relationship is one where one is directly proportional to the other.

All springs will obey Hooke's Law where the extension is directly proportional to the force up to a certain limit.

And springs are also elastic.

If you take the forces off them, they'll return to their original length, again, as long as we don't use too large a force on them.

Okay, that's the end of the lesson and I hope to see you in the next one.

Goodbye.