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Hello there, my name is Mr. Forbes and welcome to this lesson from the Particle Explanations of Density and Pressure Unit.

This lesson's all about calculating density and measuring the volume of different objects.

By the end of this lesson, you're going to be able to define what density is, calculate a value for density for a range of samples, and describe how you can measure the volume of different shaped objects.

Here are the keywords that'll help you through the lesson.

The first is density and that's defined as the mass per cubic metre or cubic centimetre of a material.

Kilogrammes per cubic metre, which is the unit for density.

Grammes per cubic centimetre, which is also a unit for density we're gonna use for smaller samples.

Vernier callipers, which are a device that we can use to measure small lengths precisely.

And a micrometre, which is a device where we can measure even smaller lengths much more precisely again.

You can return to this slide at any point in the lesson.

The lesson's in three parts and in the first part we're going to look at the definition of density and what it means.

In the second part of the lesson, we're gonna look at how you can measure the volume of an object using a displacement can.

And in the third part we're gonna see how you can calculate the density of regularly shaped objects like cubes or spheres.

So when you're ready, let's start by looking at what density is.

Now you should already know that a small block of iron will weigh a lot more than in a similarly sized block of plastic.

So I put the two on balances, let's say put piece of iron on this one here.

I've got a mass of 100.

2 grammes and identically sized of an identical volume of plastic might have a mass of only 12.

8 grammes.

The iron has a lot more mass in the same volume and we say that the iron's got a greater density than the plastic where the density of something is a measure of its mass, which we can measure in grammes or kilogrammes in its unit volume.

The volume could be in cubic metres or in cubic centimetres depending on the size of the sample.

The density of a substance is defined by this equation here.

Density is mass divided by volume.

If we write that in symbols, we get this row, which is that symbol looks a little bit like a p is m divided by V.

Where we've got mass, m, measured in kilogrammes.

We've got volume measured in either cubic metres or cubic centimetres, and that can give us two different units of density.

So the density row is measured in either kilogrammes per cubic metre or grammes per cubic centimetre.

Let's try our first check now please.

So, objects float on water if the density is less than the density of water.

They sink if the density is higher than the density of water, which of those statements describes the density of an object? So pause the video, make a selection, and restart please.

Welcome back, you should have chosen C, the amount of mass it contains in a particular volume.

That's the definition of density.

Well done if you got that.

Now let's try and do a calculation of some density.

I'll do one and then you can have a go.

So I've got a block of wood.

It has a mass of 12.

4 kilogrammes and a volume of 0.

02 metres cubed.

Calculate the density of the wood.

So the first stage is write out the equation for density.

I'm gonna use symbols here, it's a bit quicker.

Then I substitute the two volume values, the mass and the volume.

So the mass is 12.

4 kilogrammes and the volume is 0.

02 metres cubed.

And then I just use my calculator to calculate the density and that's 620 kilogrammes per cubic metre or metre cubed.

Now I'd like you to try one of your own.

So I would like you to calculate the density of the wood using the data shown there.

So pause the video, calculate the density, and restart please.

Welcome back and you should have got a calculation like this and the density of this wood is much less, it's 160 kilogrammes per metre cube.

Well done if you've got that.

Density is the property of a material.

So if I've got an object made of a particular material, it'll have a particular density.

So the density of all pure water samples is 1,000 kilogrammes per metre cube.

The density of all pure gold samples is 19,300 kilogrammes per metre cubed.

And we've got a table here that shows you some other example densities.

You can see the densities of the gases towards the bottom of that table are quite low and the densities of things like the metals are quite high.

The density can change if you've got slightly different conditions.

So you can change the density of a gas, for example, by compressing it and that'll make it more dense.

So let's see if you understood that.

I've got a scientist that measured the density of a 10 gramme sample of pure copper and that density is 8.

96 grammes per centimetre cubed.

What would the density be of a 20 gramme sample of the same copper? So pause the video, make your selection, and restart please.

Welcome back.

Hopefully you selected answer C, 8.

96 grammes per centimetre cubed because it's the same copper.

All of that copper will have the same density because density is the property of a material.

Well done if you've got that.

Now volumes of one metre cubed are much larger than you'd see in a typical laboratory.

If you tried to get a sample that's one cubic metre, it would be very light indeed.

You could just take a metre ruler and try and work out how large that would be for yourself.

So normally we wouldn't use samples of this size and if we did, let's say we had a sample of water, that would have a mass of a thousand kilogrammes and that's a metric tonne, it's a bit large.

So normally in our experiments we use volumes in centimetres cubed and we use masses measured in gramme.

So, one centimetre cubed of water has a mass of one gramme and that allows density to be calculated in grammes per cubic centimetre.

Let's try an example of using grammes and centimetres cubed.

So I've got a sample of plastic, it has a mass of 1.

8 grammes and a volume of 1.

4 centimetres cubed.

Calculate the density of that plastic.

So I just use the same system as I've done before.

I write out the equation, I substitute in the two values and I get my answer.

And that's 1.

3 grammes per centimetres cubed, rounded off to two significant figures.

So now it's your turn.

I'd like you to answer this one please.

I'd like you to calculate the density of copper using the data there.

So pause the video, make your calculation, and restart please.

Welcome back, well your answer should look something like this.

We've substitute in the two values, it's 9.

0 grammes centimetres cubed and that's rounded to two significant figures as well.

Well done if you've got an answer like that.

The density equation can be rearranged to make different variables the subject.

So if we wanted to calculate the mass or the volume, we could rearrange that.

So to find the mass from the density and the volume, we can use this version of the equation, mass equals density, times volume.

And to find the volume from the density and the mass, we can use this version of the equation.

Volume equals mass, divided by density.

We've gotta make sure we use the appropriate unit for mass and volume when we're doing the density.

So if we're dealing with kilogrammes, we've gotta be careful that we'll give an answer in kilogrammes per metre cube if we're using volumes in cubic metres.

Let's try an example of calculating mass here.

So I've got ice and it's got a density of 917 kilogrammes per cubic metre.

A small iceberg has a volume of 60,000 cubic metres.

Calculate the mass of the iceberg.

So what I do is I write out the expression, mass equals density, times volume.

I substitute the two values in from the question there and I put those through my calculator, and I get an answer of 55 million kilogrammes and that's to two significant figures.

Now it's your turn.

So I'd like you to calculate the mass of petrol here in this canister.

So use the data there.

So pause the video, calculate the mass, and then restart please.

Welcome back, well hopefully you've got an answer like this.

You put the equation down, you've substituted in the two values and that gives a mass of 480 grammes because we're using grammes and centimetres cubed there.

So well done if you've got that answer.

Another example here and this time we're gonna calculate some volumes.

So I'll do one and then you can have a go.

A sample of vegetable oil has a mass of 400 grammes and a density of 0.

91 grammes per centimetre cubed.

And I'm gonna calculate the volume of the oil in the sample.

So yeah, again, I write out the equation and it's volume equals mass, divided by density.

I put in the two values and 400 grammes divided by 0.

91 grammes centimetre cubed.

And that gives me an answer of 440 centimetres cubed.

And again, that's two significant figures.

Now I'd like you to try and calculate a volume using the data here please.

So pause the video, work out your answer, and restart.

Welcome back and your answer should look something like this.

We've got a volume of 2.

1 times 10 to the minus three metres cubed to three significant figures.

Well done if you've got that.

Now sometimes you might have to convert density from kilogrammes per metre cubed to grammes per centimetre cubed, or you might have to do that the other way around.

And we can convert that by looking at how many centimetres cubed are in a cubic metre.

So one metre is a hundred centimetres, so that gives us a volume equivalence of one metre cubed is equal to 1 million centimetres cubed.

After that we can also say one kilogramme is a thousand grammes.

So using those two values together, if we needed to convert from kilogrammes metres cubed to grammes metre cubed, we'd have to divide by a thousand, the result of multiplying by a million and then dividing by a thousand.

So, to convert from kilogrammes metres cubed to grammes per centimetre cubed, divide by a thousand.

And if you need to convert the other way around, then you're gonna multiply by a thousand.

So let's see if you can do that.

Which of the following is equivalent to 5.

0 grammes per centimetre cubed? So pause video, work out which one of those four it is, and then restart please.

Welcome back.

Well hopefully you chose 5,000 kilogrammes per metre cubed.

So, to convert from grammes per centimetre cubed to kilogrammes per metre cubed, you multiply by a thousand.

Well done if you've got that one.

And now it's time for this first task of the lesson and what I'd like you to do is to look at question one, and in each of those three figures I've got volume of mass data and I'd like you to find the density of each block please.

And then for question two, I've given you the density of expanded polystyrene and what I'd like you to do is to find the masses of two different volumes of the polystyrene there.

So pause the video, work out the answers to those questions, and restart please.

And welcome back.

Let's have a look at the calculations of the density of each of the blocks.

And if I put the data in the equation, first block is 1.

3 grammes per centimetres cubed, second block, 6.

0 grammes per centimetre cubed, and the third block, 2.

2 grammes per centimetre cubed.

Well done if you've got those three.

And for the second question, we're gonna calculate the mass.

Well the first one's fairly easy.

We write up the equation, mass equals density times volume and just use the density information I've got there and the volume and that gives 3.

75 grammes.

The second one's a bit more challenging 'cause first of all, I've gotta work out the volume of the block and it was 2.

5 cubic metres.

And if I multiply that up, I get the total volume as 2,500,000 centimetres cubed.

And then I can use that to get the mass.

So I can calculate the mass like that and I can give my answer either in grammes or in kilogrammes.

So well done if you've got those answers.

And now it's time for the second part of the lesson and in this part we're going to be looking at how you can measure the volume of something using a displacement can.

So if you place an object in water, it can float or it can sink.

So I've got some water here, I'll place an object in it and it's floating.

As you can see, the water level's risen slightly there as putting the object in it pushes some water upwards out of the way.

Or the object can sink as well and similarly, it'll go to the bottom of the water, but the water level will rise.

So you get an increase in the rise of the water.

So, the level of water in that container rises.

We say that some of the water has been displaced, it's been pushed upwards or displaced upwards.

If we put an object in a container that's completely full, then the water will still rise upwards and it will spill out in this case.

So if I put this object in the water, the water will rise upwards and spill over the edge of the can and end up on the desk there.

So I've got some displaced water that's spilled out.

Now the volume of that displaced water is gonna be equal to the volume of the object that sunk in it.

The volume of that sort of pink shape there is the same as the volume of the displaced water.

If we could collect that water, that would allow us to measure the volume of that shape of that object because those two volumes are equal.

Let's see if you understood that.

I've got a beaker, it's filled through at the very top of it.

I place a stone in it very carefully into that beaker and 50 centimetres cubed of water spills over as the stone sinks in it.

What conclusions can be made about that? So I'd like you to identify the two conclusions.

So pause the video, select the two answers, and then restart please.

Welcome back.

Well, you should have noticed that the stone is more dense than water because it sinks in it and also the stone has a volume of 50 centimetres cubed, it's displaced that amount of water.

So that's equivalent to its volume.

Well done if you spotted both of those.

So a displacement can is a device specifically designed to allow you to measure the volume of something that sinks in water.

So I've got a displacement can shown here.

It's full of water, it's got a spout, it's got a collection beaker to collect some water and I've got an object.

And the reason the spout's there is it allows the displaced water to be come out of that spout and be collected in that measuring cylinder or beaker.

And that will give us the volume of the object that sunk in the water.

So, to use a displacement can, what I do is this, I get the equipment set up like this.

So I'm gonna be collecting the water that escapes.

I'm gonna fill that can up so that the water is exactly up to the spout.

What we usually do is we overfill it and allow some water to come out of that spout.

And then once that stopped happening, we put the beaker in place.

And then I put the object in it.

When I put the object in it, the water level will rise to above the level of the spout and the water will start to come out and fill the beaker, and eventually the water level will fall back down to the bottom of the spout and I can collect the displaced water.

And the displaced water will have the same volume as the object that's been placed in the displacement can.

So let's check if you understood how displacement cans work.

Which displacement can arrangement here is the correct one for measuring the volume of that wall.

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

Welcome back, hopefully you selected the third one there.

That's the correct one because it's full of the spout and the beaker is empty.

And the other ones weren't correct because in the first instance there was some water already in the beaker and that's not good.

And in the third one the water level wasn't high enough, it hadn't not reached the spout.

So well done if you selected C.

We can use a measuring cylinder to measure the volumes of small objects and we can do it like this.

We partially fill the measuring cylinder with water and we record that starting volume.

So we look across making sure our eyes are level with the top of the water and record that volume.

Then we can place the small object in it like a stone and that sinks.

And as you can see, the water level has risen.

So again, we look and see, and record that new end volume of the water and the water level has gone up by the same as the volume of the object.

So the volume of the object is the increase in volume, so we can calculate that.

Let's have a look at an example to show you how that works.

So we've got a measuring cylinder with water and we have a look across and we write down the volume that it's showing, 26.

4 centimetres cubed in this case.

Then we add a stone to it and you can see the water level's gone up.

And we look again and we record the new volume reading and the volume of the stone must be the difference in those two volumes.

So, we calculate the volume of stone by taking the smaller value from the larger value and that gives us the stone volume of 16.

1 centimetres cubed.

Let's see if you can do that.

So, what's the volume of the object placed in this measuring cylinder using the data there? So pause the video, work up the volume, and then restart please.

Welcome back.

Hopefully you selected 8.

4 centimetre cubed.

And to do that we subtracted the smaller volume from the larger volume, the difference being 8.

4 centimetres cubed.

So well done if you've got that.

Now it's time for the second task and I'd like you to think about the advantages and disadvantages of using a displacement can and a measuring cylinder to measure the volume of a range of objects.

And secondly, I'd like you to calculate the density of a marble using the data there.

So pause the video, work out your answers to those two, and then restart please.

Welcome back.

Well, let's have a look at the advantages and disadvantages.

So first for the displacement can, you can use that to measure larger volumes, but its disadvantage is you still have to use a measuring cylinder to measure the volume of displaced water at the end anyway.

So you're using an extra piece of equipment there.

And for the measuring cylinder, its advantage is you can directly calculate the change in volume on it just while looking at it and read directly from the scale.

But its disadvantage is it will only work with small objects.

Well done if you've got advantages and disadvantages something like that.

With the second part we're calculating the density of the marble.

So, we've got the volume so we can calculate the volume from the change in the volume and the measuring cylinder.

And we've got its mass so we can put that data into the equation and that gives us seven grammes per centimetres cubed to two significant figures.

Well done if you've got that.

Now it's time to move on to the third and final part of the lesson and we're gonna look at how you can measure the density of regular objects, objects with a simple shape.

And just as before, to find the density of something, we need to know its mass and its volume.

We can measure the mass quite simply with a top pan balance or something like this.

And most top pan balances can measure to the nearest 0.

1 grammes and some can measure much better than that, 0.

01 grammes or even better.

So, we can measure the mass fairly simply with that top pan balance.

The volume of a regular object can be calculated from some length measurements.

So if we've got a cuboid, something like this, we've got length, width, and height and we can calculate the volume of that cuboid simply by multiplying those three values together.

So the volume is the length times the width, times the height.

If we've got a sphere, then we can calculate the volume of that as well.

So, I put a sphere like this, the volume of a sphere can be calculated from, well its radius value.

So if I know its radius, I can calculate its volume using this equation.

The volume is 4/3 times pi, times the radius cubed.

So I've got two equations there that'll allow me to calculate the volume of simple objects.

Okay, let's see if you understand how to calculate volume.

I've got scientists measuring the diameter of a sphere and that's 0.

20 metres.

Which of these shows how to calculate the volume? So I don't need you to calculate the volume, I just need to know is the correct expression to use to calculate it.

So look carefully, we've got the diameter there of the sphere.

So pause the video, work how to calculate the volume, and then restart please.

Welcome back.

Hopefully you said the bottom of those.

And the reason the bottom one is correct is because the radius of that sphere is 0.

10.

Its diameter and radius are not the same thing.

Diameter is twice the radius.

So the correct equation there was D.

So well done if you've got that.

Let's have a look at calculating some densities using that sort of information.

I've got a sphere of mass 525 grammes and a radius of 4.

00 centimetres, and I'm gonna calculate its density.

And I do that in multiple stages.

The first of them is to calculate the volume.

So I'll write down the expression for the volume of a sphere and I substitute the value in there and that gives me a volume of 268 centimetres cubed.

I'm gonna be careful.

Remember I'm cubing that 4.

0 centimetres.

Now I can calculate the density using our regular expression, density is mass divided by volume.

I've got its mass, I've got the volume, put those into a calculator and that gives me 1.

87 grammes per centimetres cubed.

Now it's your turn.

I'd like you to calculate the density of this cuboid and you can see it's got a mass of 20 kilogrammes and I've got its dimensions there as well.

So pause the video, work out the density, and restart please.

Welcome back.

Well, you should have calculated the volume 'cause this is a cuboid by length, width times height.

That gives a volume of 0.

04 metres cubed and then it can get the density, it's mass divided by volume.

It gives me a density of 500 kilogrammes per metre cube.

Well done if you've got that.

If you want to measure the length of the side of something, we can use a simple rule or ruler and that gives the value to typically the nearest millimetre.

So if you look at your 30 centimetre ruler looks something like this and you can see it's marked up with millimetres.

So, the most precise value you can get is to the nearest millimetre.

In this case, this cube or cuboid is 7.

2 millimetres.

If you wanted to measure the diameter of a sphere or the radius of a sphere, it's a bit more difficult.

It's very hard to actually place a ruler so you can measure from one side to exactly the opposite side.

If I place a ruler like this, it looks reasonable, but actually I'm not passing through the centre.

So, I'm actually getting an inaccurate value for the diameter of that sphere.

So, using a ruler to measure something like a sphere is not very successful at all.

To be more successful at measuring things like spheres, we can use Vernier callipers and these are more precise instruments.

They can measure to the nearest millimetre or even to the nearest 10th of a millimetre or 0.

01 centimetres.

You place the object in the jaws of the callipers, something like this, and you close them so that each of the jaws touch opposite sides and they'll always be touching exactly opposite sides even if the object is a sphere.

So that allows us to measure the diameters quite easily.

But we need to be able to read what those measurements are and to do that we need to understand how to use something called a Vernier scale.

Now some callipers are digital and they're easy to use, they'll just give you a readout of something like 1.

2 millimetres or something like that.

But Vernier scales are a bit more complex.

We're gonna look at how to use those.

Vernier scales have two sets of lines on them, if you look carefully at the picture you can see those.

We've got an upper set and a lower set, and those are the lines we need to understand to measure the sides of an object.

So the first line on the bottom scale, so it's the very first line on that bottom scale as I've shown here, shows you the first two decimal places of our measurement and the alignment of that set of bottom lines with one of the lines on the top gives us a third decimal place.

And that's quite complex, so let's have a look at an example.

So I've got a simplified Vernier scale here or part of it, and I'm gonna try and use it to get a measurement.

And the first thing I need to do is to identify the position of the first line of the bottom scale and it's here.

So that position marker, that position marker shows us at the nearest millimetre.

So if I look, I can see there's the four, that's the four centimetres mark, and then there's a few more subdivisions and each of those represent a millimetre.

So that mark is just beyond the 4.

3 millimetre mark.

The next thing I need to do is to find out one of those bottom scale lines that exactly lines up with the top one.

So, I'm looking very carefully and I can see that this line here lines up exactly with the line directly above it, and that gives me the next digit in my answer.

And that is the fourth line, so it's at.

04.

So what I do then is I add those two values together, the 4.

3 and the.

04.

So the fourth line lined up is that.

Add those two values together and that will give me a size reading of 4.

34 centimetres.

Let's see if you can read off a Vernier scale reading.

So I've got this Vernier reading.

I'd like you to look at the lines very precisely and decide what's the reading on this Vernier scale.

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

Welcome back.

Hopefully you selected 6.

45 centimetres.

If you look carefully, you can see the 6.

4 mark is the one that I'm just beyond, the first line on the bottom scale is just beyond that 6.

4.

And then the line that's lined up most precisely is this one and it's the fifth line at the bottom scale.

So it's a 0.

05.

So it's 6.

4 and 0.

05 added together, and it's 6.

45 centimetres.

Well done if you've got that.

If you need a more precise length measurement, then you can use a micrometre and they can measure to a very high precision.

This micrometre has a Vernier scale, but you can get micrometres with a digital scale, which are much easier to use.

See though, there's the Vernier scale.

They can measure to the nearest hundredth of a millimetre.

So if you need a very precise measurement, a micrometre is the way to go.

Okay, let's see if you understand how you can use different instruments to measure the different precisions.

What I'd like you to do is to draw lines to match the measurements in order of precision.

So we've got three instruments and three different precisions.

Draw lines between those to show which is which.

So pause the video, do that, and restart please.

Welcome back.

Well the answers you should have got something like this.

The 30 centimetre rule can only measure to the nearest millimetre.

A micrometre can measure to the nearest hundredth of a millimetre.

And Vernier callipers are usually measuring a 10th of a millimetre.

Well done if you've got those.

Okay, we've reached the final task of the lesson and what I'd like you to do is to answer these two questions.

The first is about an engineer measuring the density of a metal sphere and the second one is about modelling clay and that's a bit more challenging.

So I'd like you to pause the video, try and answer these two, and restart please.

Welcome back.

Well, hopefully your answers look a bit like this.

The Vernier callipers we used, because they've got a higher resolution, they can measure more precisely and you can lock the sphere in the jaws to measure the diameter properly.

To calculate the density, we've got the radius so we can calculate the volume using that.

21.

69 centimetre cubed and we can use that to get a final density of 4.

40 grammes per centimetres cubed.

Well done if you've got the answer.

And here's the second answer.

The first I did this was to find the volume that we need, the volume is 64 centimetres cubed.

We found that from the mass and the density.

And the size of a cube are all the same length.

So the volume is that length cubed.

So, to find the actual length, we need to take the cube root to the volume.

And we did that and it's 4.

0 centimetres.

Well done if you got the answers to that.

And now we've reached the end of the lesson so here's a quick summary.

The density of a material or object is the mass contained within a certain volume.

And we've got an equation for that.

Density is mass divided by volume or rho equals m, divided V.

And you can see the mass is measured in kilogrammes or grammes, volume in cubic metres or cubic centimetres.

And density is measured in kilogrammes per metre cubed or grammes per cubic centimetre.

Volume can be measured using a displacement can or measuring cylinder, or it can be calculated from length measurements taken with Vernier callipers or a micrometre.

Well done for reaching the end of the lesson.

I'll see you in the next one.