Lesson video

Hello, this lesson is about calculating the energy of a spring.

It's part of the physics unit, energy of moving objects.

And my name is Mr. Fairhurst.

By the end of this lesson, you should be able to calculate the energy a spring has because it's stretched, which is the same thing as the energy a spring has because it's compressed.

These are the keywords that you're going to come across during the lesson.

If at any point you want to just check what they mean, just pause the video and come back to this slide and have a look.

I've split the lesson into three parts.

In the first part of lesson, we're going to look at the energy in the elastic store and what we mean by that.

And then in the second part of the lesson, I'm going to see how energy can be transferred into the elastic store of a spring.

And finally, in the third part of the lesson, we're going to see how we can calculate the amount of energy stored by a spring.

So let's make a start with that first part.

Let's start with one of those toys that you might be familiar with from your childhood.

It's one of those pop-up toys, and when you squash down the figure against the force of the spring, it's got a little rubber sucker on the bottom that sticks to the plastic base.

And when it's stuck down there, you can wait with anticipation as the spring pushes back against the suction and then suddenly, it pops up.

What we've done is we've compressed the spring and we've transferred energy into the elastic store and when it's popped up, the energy in the elastic store has been transferred into the kinetic store, and the amount of energy that's transferred is equal to the energy it had in the elastic store.

That's assuming, of course, that no energy is dissipated, which a little bit will be.

Now, as it rises up into the air, some of that energy it's got in the kinetic store will be transferred into the gravitational store.

And when it reaches the very top of the flight, at that instant that it stops moving, just for a moment, all of the energy in the kinetic store has been transferred into the gravitational store.

Let's put those ideas together and have a go at this question.

Just pause the video whilst you do so and start it again once you're ready.

Okay, so how did you get on? The compressed spring in the pop-up toy has got 0.

15 joules of energy.

How much energy will it then have at the top of its flight when it stops moving? Well, it's going to have exactly 0.

15 joules.

That's assuming that no energy is dissipated.

So well done if you got that right.

We've now got two toys and the orange toy's got a stiffer spring than the green toy and that means that you're going to need to push harder and do more work when you compress the spring in the orange toy.

But when it is compressed, the orange toy is gonna have more energy than the green toy.

Now, if you put some numbers on those springs and let's introduce the blue toy, which has got a spring concept that is two times bigger than the green toy, we now can say that this spring is two times stiffer and that when it's compressed, the spring in that blue toy is gonna have two times the amount of energy than the spring in the green toy.

We're going to need to do two times the amount of work in squashing the spring.

Have a look at this question and see what you think.

Pause the video whilst you do so and start it again once you're ready.

Okay, so how did you get on? You were asked how much energy does the orange toy have compared to the green toy when the springs are fully compressed? And you're given the spring constants for both toys and the spring constant for the orange toy's three times bigger than the one for the green toy, so you're gonna have to push three times harder and do three times the amount of work when you squash that spring.

So the orange toy's gonna have three times more energy when it's compressed.

So well done if you got that answer.

I'd now like you to have a go at this task that's got two different springs.

They're both the same length when they are not stretched.

A weight of three newtons is hung off both springs and they stretch as they're shown in the diagram.

Have a go at both questions.

Pause the video whilst you do so and start it again once you've got your answers ready.

Okay, so how did you get on? Question one said which spring has got the greatest spring constant? Explain your answer.

The spring constant tells you how stiff a spring is and how hard it is to stretch and the spring that's hardest to stretch is spring B, the same size force on both springs and that force does not stretch spring B as much as it stretches spring A.

So it's harder to stretch.

When both springs have got the same extension, which spring will have the most energy? And again, explain your answer.

Now, the answer is again spring B.

It's going to be harder to stretch so you're going to use more force when you're stretching it and do more work to get it to the same length and therefore, more energy will be transferred into the elastic store.

So well done if you got that answer as well.

So let's now move on to part two of the lesson and in this part, we're going to look at how energy is transferred to a spring.

I don't know if you've ever had to go at archery, but when you pull the spring back, you're flexing the wooden bow and you're transferring energy to the elastic store.

And the further back you stretch the string, the harder and harder it is to stretch that bow any further.

And the same's true for our little toy.

When you first start squashing it down, it's quite easy and the closer the toy gets to the base, the harder and harder it is to squash any further and the more force you need to squash it that extra centimetre further.

Now have a look at these images.

Which one shows the most energy transfer to the spring when it's compressed by one further centimetre? Just pause the video whilst you think about it and start again once you're ready.

Okay, so how did you get on? The correct answer is of course, C.

That one needs the most force to squash it down the extra centimetre, so we do most work in squashing it and therefore, we're transferring the most energy.

So well done if you got that answer.

Let's think now about stretching a spring.

Each time we add an equal size weight to the spring, it extends by an equal amount further.

So when we add one newton of force, it extends by four centimetres.

When we added a second newton of force, it extended by a further four centimetres.

And when we added a third newton of force, it extended by yet another four centimetres.

You might recall doing this experiment earlier on in your studies.

Now, when we had a total force of three newtons, we extend the spring by 0.

12 metres, but what we cannot do is to calculate the energy transferred by using the work done equation because when we're doing this extension, the force that we're extending with was not three newtons all of the time.

For the first four centimetres, the force we extended with was one newton.

For the second four centimetres extension, we used two newtons and we only used three newtons of force for that final little stretch.

So we can't use that equation because the force is constantly changing as the spring changes its length.

Have a look at this question and see what you think.

Pause the video whilst you do so and start again once you've got your answers.

Okay, so true or false? The energy transferred to a spring is equal to force times extension.

We've just seen that that's not true and to justify the answer, what we need to say is that the size of the force stretching the spring changes as it extends.

We cannot use work done is force times distance because we don't have a fixed quantity for the force we're using to extend it.

So well done if you got both of those answers correct.

So if you can't measure the amount of energy transferred to a spring by using the work done equation, how can you measure the amount of energy in a spring? Well, what you can do is you can measure the amount of energy transferred from the elastic store into the gravitational store when the pop-up toy jumps up.

And that's what some pupils did when they measured how high the popup toy reached after its spring was compressed by different amounts.

And they found that when it was compressed by one centimetre, it popped up to five centimetres, it popped up to 20 centimetres after being compressed by two centimetres, and after being compressed by three centimetres, it popped up to 45 centimetres.

And when they looked at the results, they found a pattern.

You might be able to see a pattern in these results if you look carefully and think about them, but it's not an obvious one.

What these pupils noticed was that 20 is equal to four times five centimetres and that 45 was equal to nine times five centimetres so that if they doubled the compression, the height increased by four times.

If they made the compression three times bigger, the height increased by nine times, which is three times three.

So in other words, they found that the height was proportional to the compression squared.

What we've said is that when it reaches the top of its flight, all of the energy that was in the elastic store has now been transferred into the gravitational store.

And that the energy in the gravitational store is proportional to the height that the toy reaches.

And that's also going to be proportional to the amount of energy that was in the elastic store before it took off.

And that means that the energy in the elastic store is proportional to the compression of the spring squared.

Have a look at this question and see what you think.

Pause the video whilst you do so and start it again once you've got your answer.

Okay, so what do you think? The catapult was pulled back two centimetres and you were asked how much more energy will be in the elastic store if it's pulled back a further two centimetres.

So it's pulled back two times further, but the amount of energy in the elastic store is proportional to the extension squared.

So two times further squared is four times more energy.

So well done if you said answer C.

I'd now like you to have a go at this task.

The spring constant of spring A is two times greater than the spring constant for spring B and both springs are stretched, but spring B has got two times the extension of spring A.

And you're asked two questions about the situation.

Pause the video whilst you think about your answers and start it again once you've written them down.

Okay, so how did you get on? In question one, you were asked to state which spring has got the most energy, which is spring B, and then in question two, you were asked to explain your answer.

Well, spring A has got twice the spring constant as spring B, so when it's extended by the same amount, it will have twice the amount of energy.

However, spring B is extended by twice as far and by doubling the extension, we increase the energy it has by four times because it's the extension squared.

So overall, spring B will be the one with the most energy.

So well done if you got that answer and explanation correct.

We're now going to go onto part three of the lesson in which we're going to find out how we can calculate the amount of energy in a stretched spring.

Now, we know that when we stretch a spring, we're transferring energy into the elastic store, and we've already seen that the energy that the spring has because it's stretched is proportional to the spring constant.

And we've also seen that the energy of the spring has when it's stretched is proportional to its extension squared, which means if we double or triple the extension, the amount of energy it has goes up by four times or nine times.

We can put those two relationships together to work out the energy a spring has because it's stretched and we're going to call that the elastic potential energy.

And we can show that the elastic potential energy of the spring's equal to 1/2 times the spring constant times the extension squared.

It's proportional to the spring constant and it's proportional to the extension squared.

In symbols, the elastic potential energy is equal to 1/2ke squared where capital E with the subscript e is the elastic potential energy measured in joules.

K is the spring constant in newtons per metre and e is the extension or compression in metres.

Now, before we move on, I want to use this graph to show you where the 1/2 came from in that last equation.

This is a graph of the extension of the spring against force when the spring has been stretched elastically.

And as you can see, as the extension is getting bigger, the force needed to stretch it that little bit further is itself getting bigger.

So as the extension is increasing, the force needed to stretch it further is also increasing, and the force is changing continually as the spring is getting longer.

Now, despite that, I'm going to use this equation of work done is force times distance to calculate the total amount of work done on that spring.

Now, distance is quite straightforward.

Distance is just equal to the extension of the spring, but the force is continually changing.

But because we've got a straight line graph, I know that the average force is going to be exactly halfway along the line.

So the average force is equal to 1/2 the final force and that's the force that I'm able to use to calculate the final work done.

Now, the final force on the spring is equal to the spring constant times the extension.

So 1/2 the final force is equal to 1/2 the spring constant times the extension.

If I substitute first with distance and average force now into the equation for work done, I get work done is 1/2 the spring constant times the extension for the force times the extension again for the distance, which gives us 1/2 times the spring constant times the extension squared.

Have a look at this question.

I just want you to think about what you'd key into your calculator in order to calculate the energy stored in the spring.

Pause the video whilst you do so and start it again once you're ready.

Okay, so how do you correctly calculate energy? We type into your calculator energy is 1/2 times the force times the extension times the extension.

We're squaring just the extension and not the force of the 1/2 in the equation.

So well done if you got that right.

Let's have a look at this question to see how we can use that equation.

A spring of length 30 centimetres has a spring constant of 250 newtons per metre.

Calculate the energy it would gain if it is stretched to 40 centimetres.

Well, first of all, we need to calculate the extension, which is 10 centimetres, and we can use that in metres in the equation.

And that's 0.

10 metres.

We then need to state the equation for energy, which is 1/2 times the spring constant times the extension squared, substituting the values that we've got.

I'm going to multiply the figure in the brackets first and then multiply everything else out just to make sure I'm squaring the right things.

Our answer we should give to two significant figures because in the question, the length of the spring was only given to two significant figures.

And as you know, we can only give our answer to the smaller number of significant figures of any one of the values that we were given in the question.

Okay, have a look with this question yourselves.

Pause the video whilst you do so and start again once you've got your answer with all your workings out.

Okay, so how did you get on? Well, first of all, you needed to work out the extension, which is 30 centimetres and convert that to metres, 0.

30.

Then using the equation for energy stored in the spring, we can substitute the numbers in, multiply the brackets out first and calculate the answer.

And again, we're going to give our answer to two significant figures because that's what we're allowed to do by the advice we were given in the question.

I'd now like you to have a go at these questions, show all your workings out and give you answers to an appropriate number of significant figures.

Pause the video whilst you have a go at these questions and start again once you've got all your answers ready.

Okay, so how did you get on? Question one said a spring of 10 centimetres has got a spring constant of 500 newtons per metre.

Calculate the energy it would gain if it is stretched to 20 centimetres.

So first of all, you need to calculate the extension, which is 10 centimetres and convert that to 0.

10 metres.

Once you've got that, we can write down the equation and substitute the values you've got.

You can then multiply the brackets out and calculate your answer, which is 2.

5 joules, which is given to two significant figures.

It's two significant figures because that was the smallest number of significant figures we had for values in the question, which were for the lengths.

Question two, so calculate the energy transferred to a spring when it's stretched by 0.

25 metres if its spring constant is 2,000 newtons per metre.

Well, we're given the extension this time.

It's 0.

25 metres, so we simply need to write down the equation, substituting the values, multiply the brackets out and calculate the answer, which is 62.

5 joules.

In the question, we had distance to two significant figures.

So we then need to change that answer to 63 joules, which is to two significant figures.

In question three, you're asked to calculate the spring constant of a spring that gains 200 joules of energy when it's stretched by 0.

05 metres.

So we've got the extension, so we simply need to write down the equation and substitute the values that we've got.

This time, I'm going to calculate everything I can on the right-hand side.

So we have 200 joules is 0.

00125 metres squared times the spring constant, divide both sides by the 0.

00125.

So we get the spring constant is equal to 160,000 newtons per metre.

And then question four, calculate the increase in the length of a spring with a spring constant of 1,500 newtons per metre when it gains 500 joules of energy.

Again, we write down the equation first and substitute the values in.

If we multiply what we can on the right-hand side and then divide both sides by 750, we end up with the extension squared is equal to 0.

6667 metres squared.

We need to square root both sides to find the extension on its own.

And if we square root both sides, we get the extension of 0.

81649658 metres.

Looking back at the question, we've got the energy given as 500 joules, which is one significant figure.

So I've given that as 0.

8 metres.

You might suggest that that 500 is three significant figures, which would also be acceptable, in which case, your extension be 0.

816 metres.

So very well done if you got that answer and very well done if you got all of those answers.

So very well done for making it to the end of the lesson.

This is a short summary slide that sums up the main points from the lesson.

And what we found out was that the energy of a spring, because it is stretched, is called the elastic potential energy, and the elastic potential energy is equal to 1/2 the spring constant times the extension squared and in symbols is E for energy with a subscript e, equals 1/2 times k times e squared, where K is the spring constant measured in newtons per metre and e is the extension of the spring, which is measured in metres.

So again, well done to making it to the end of the lesson.

I do hope to see you next time.

Goodbye.