Chapter 7 of 8

All 8 chapters
  1. 1Prisms and cylinders, readunderstood
  2. 2Pyramids and cones, readunderstood
  3. 3Spheres, readunderstood
  4. 4Joined and cut solids, readunderstood
  5. 5Changing units of area and volume, readunderstood
  6. 6Speed, distance and time, readunderstood
  7. 7Density and pressure, readunderstood
  8. 8Rates of flow and rate graphs, readunderstood

M4.5.7 Chapter 7

Density and pressure

Why does a big block of wood float, while a small iron nail sinks?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use density = mass ÷ volume to find any one of the three
  • Use pressure = force ÷ area
  • Find the mass of a block from its size and density

Two blocks of the same size can have very different masses. A block of iron is much heavier than a block of wood of the same size. The iron has more mass packed into each cm³.

Density says how much mass there is in each unit of volume. Density = mass ÷ volume. Its unit is a compound unit: grams per cm³, written g/cm³. Another is kg per m³.

Water has a density of about 1 g/cm³. So 1 cm³ of water has a mass of about 1 g. A litre of water, which is 1 000 cm³, has a mass of about 1 kg.

A table with the head Material, About g/cm³. Rows: Oak wood, 0.7; Ice, 0.92; Water, 1; Aluminium, 2.7; Iron, 7.9; Gold, 19.3.MaterialAbout g/cm³Oak wood0.7Ice0.92Water1Aluminium2.7Iron7.9Gold19.3
Some densities. A solid object made of one material floats in water if its density is less than 1 g/cm³.

The rule density = mass ÷ volume can be turned round, in the same way as speed. Use the table to choose.

A table with the head To find, Use. Rows: density, mass ÷ volume; mass, density × volume; volume, mass ÷ density.To findUsedensitymass ÷ volumemassdensity × volumevolumemass ÷ density
One rule, three ways to use it.
A block, drawn as a box. The front edge is labelled 10 cm and the right edge is labelled 4 cm. The edge going back is labelled 5 cm. Under it is the label mass 540 g.10 cm4 cm5 cmmass 540 g
Yusuf's block of metal: 10 cm by 5 cm by 4 cm, with a mass of 540 g.

Worked example

Yusuf has a block of metal with a mass of 540 g and a volume of 200 cm³. What is its density?

  1. Density = mass ÷ volume.
  2. Density = 540 ÷ 200 g/cm³.
  3. 540 ÷ 200 = 2.7 g/cm³. That matches aluminium in the table.

Answer: 2.7 g/cm³

To find a mass from a size, first find the volume of the block. Then multiply the volume by the density. The mass is in grams if the volume is in cm³ and the density is in g/cm³.

A block, drawn as a box. The front edge is labelled 10 cm and the right edge is labelled 4 cm. The edge going back is labelled 5 cm.10 cm4 cm5 cm
A block of iron, 10 cm by 5 cm by 4 cm.

Worked example

Iron has a density of about 7.9 g/cm³. Find the mass of an iron block 10 cm by 5 cm by 4 cm.

  1. The volume of the block is 10 × 5 × 4 = 200 cm³.
  2. Mass = density × volume = 7.9 × 200.
  3. 7.9 × 200 = 1 580 g.
  4. That is 1.58 kg, because 1 000 g is 1 kg.

Answer: About 1 580 g

Now pressure. A force is a push or a pull, measured in newtons (N). Pressure is how hard a force pushes on each unit of area. Pressure = force ÷ area.

Pressure is measured in newtons per cm² (N/cm²), or in newtons per m² (N/m²). One newton per m² is one pascal (Pa).

The same force can give a small or a big pressure. A block lying on its large face spreads its weight over more area than the same block standing on its small face.

Weight is a force, the pull of the Earth on something. Every 1 kg of mass weighs about 10 N. The iron block has a mass of 1.58 kg, so it weighs about 16 N.

A block lying on its large face. A downward arrow on top is labelled 16 N. The front edge is labelled 10 cm and the right edge is labelled 4 cm. The edge going back is labelled 5 cm.10 cm4 cm16 N5 cm
The block lies on its bottom face, 10 cm by 5 cm, which is 50 cm².
The same block standing on its small face. A downward arrow on top is labelled 16 N. The front edge is labelled 5 cm and the right edge is labelled 10 cm. The edge going back is labelled 4 cm.5 cm10 cm16 N4 cm
The block stands on its bottom face, 5 cm by 4 cm, which is 20 cm².

Worked example

A block weighing about 16 N stands on a face of 5 cm by 4 cm. Find the pressure on the table, first in N/cm² and then in pascals.

  1. The area of the face is 5 × 4 = 20 cm².
  2. Pressure = force ÷ area = 16 ÷ 20 = 0.8 N/cm².
  3. One m² is 10 000 cm², so 1 N/cm² is the same as 10 000 N/m².
  4. So 0.8 N/cm² is 0.8 × 10 000 = 8 000 N/m², which is about 8 000 pascals.

Answer: About 0.8 N/cm², or about 8 000 pascals

Try it

Sink or float

You need: a bowl of water, a stone, a coin, a piece of wood and a candle.

  1. Guess for each thing: will it sink or float?
  2. Put the things in the water one at a time.
  3. Write down what happened.

What you will see: The stone and the coin sink, and the wood and the candle float. It does not depend on how big a thing is. It depends on its density compared with water.

If you cannot try it, this is what happens.
A table with the head Thing, What happens. Rows: Stone, sinks; Coin, sinks; Wood, floats; Candle, floats.ThingWhat happensStonesinksCoinsinksWoodfloatsCandlefloats

A wrong idea many people have

Heavy things sink and light things float.

Why it feels right: Most heavy things we pick up, like stones, do sink, and a feather floats.

The test that breaks it: A big log has a mass of many kilograms, and it floats. A small steel nail has a mass of a few grams, and it sinks. Mass alone does not decide it.

What is true: A solid object made of one material sinks in water if its density is more than 1 g/cm³. It floats if its density is less. Compare the density, not the mass.

A wrong idea many people have

Pressure is the same as force.

Why it feels right: A big force feels like a big pressure when you push hard with your hand.

The test that breaks it: The block weighs about 16 N in both pictures. Lying down, the pressure is 16 ÷ 50 = 0.32 N/cm². Standing, it is 16 ÷ 20 = 0.8 N/cm². The force is the same, but the pressure is 2.5 times as big.

What is true: Pressure is force divided by area. The same force on a smaller area gives a bigger pressure. That is why a sharp knife cuts.

Key words

density
the mass in each unit of volume. Density = mass ÷ volume.
pressure
the force on each unit of area. Pressure = force ÷ area.
pascal
the unit of pressure. One pascal is one newton per m².

Back to your guess

The answer is C: A thing floats when it has less mass in each cm³ than water has.

A big log is heavy, but each cm³ of wood has less mass than each cm³ of water. So it floats. A small nail is light, but each cm³ of iron has much more mass than water. So it sinks. What counts is the mass in each cm³, which is called density.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A stone has a mass of 150 g and a volume of 60 cm³. What is its density?

    Tap your answer.

    1. Not this one. You multiplied the mass by the volume. Density is the mass shared out over the volume, so divide. Go to M4.5.7. Try another.

    2. Yes, that is right. Density = mass ÷ volume = 150 ÷ 60 = 2.5 g/cm³.

    3. Not this one. You divided the volume by the mass. Density = mass ÷ volume, with the mass on the top. Go to M4.5.7. Try another.

    See the answer

    2.5 g/cm³

    Density = mass ÷ volume = 150 ÷ 60 = 2.5 g/cm³.

    • If you chose A: You multiplied the mass by the volume. Density is the mass shared out over the volume, so divide. Go to M4.5.7.
    • If you chose C: You divided the volume by the mass. Density = mass ÷ volume, with the mass on the top. Go to M4.5.7.

  2. Aluminium has a density of 2.7 g/cm³. A block of aluminium is 10 cm by 8 cm by 5 cm. What is its mass?

    Tap your answer.

    1. Yes, that is right. The volume is 10 × 8 × 5 = 400 cm³. Mass = density × volume = 2.7 × 400 = 1 080 g.

    2. Not this one. You divided the volume, 400 cm³, by the density. To find the mass, multiply the density by the volume. Go to M4.5.7. Try another.

    3. Not this one. You used the area of one face, 10 × 8 = 80 cm², in place of the volume. The mass needs the whole volume, 10 × 8 × 5 = 400 cm³. Go to M4.5.7. Try another.

    See the answer

    1 080 g

    The volume is 10 × 8 × 5 = 400 cm³. Mass = density × volume = 2.7 × 400 = 1 080 g.

    • If you chose B: You divided the volume, 400 cm³, by the density. To find the mass, multiply the density by the volume. Go to M4.5.7.
    • If you chose C: You used the area of one face, 10 × 8 = 80 cm², in place of the volume. The mass needs the whole volume, 10 × 8 × 5 = 400 cm³. Go to M4.5.7.

  3. A metal has a density of 8 g/cm³. A piece of it has a mass of 480 g. What is its volume?

    Tap your answer.

    1. Not this one. You multiplied the mass by the density. To find the volume, divide the mass by the density. Go to M4.5.7. Try another.

    2. Not this one. You divided the density by the mass. The volume is the mass divided by the density. Go to M4.5.7. Try another.

    3. Yes, that is right. Volume = mass ÷ density = 480 ÷ 8 = 60 cm³.

    See the answer

    60 cm³

    Volume = mass ÷ density = 480 ÷ 8 = 60 cm³.

    • If you chose A: You multiplied the mass by the density. To find the volume, divide the mass by the density. Go to M4.5.7.
    • If you chose B: You divided the density by the mass. The volume is the mass divided by the density. Go to M4.5.7.

  4. A force of 600 N pushes on an area of 0.05 m². What is the pressure?

    Tap your answer.

    1. Not this one. You multiplied the force by the area. Pressure = force ÷ area. Go to M4.5.7. Try another.

    2. Yes, that is right. Pressure = force ÷ area = 600 ÷ 0.05 = 12 000 N/m², which is 12 000 Pa.

    3. Not this one. You divided the area by the force. Pressure is the force on each m², so the force is on the top. Go to M4.5.7. Try another.

    See the answer

    12 000 Pa

    Pressure = force ÷ area = 600 ÷ 0.05 = 12 000 N/m², which is 12 000 Pa.

    • If you chose A: You multiplied the force by the area. Pressure = force ÷ area. Go to M4.5.7.
    • If you chose C: You divided the area by the force. Pressure is the force on each m², so the force is on the top. Go to M4.5.7.

  5. An iron block is 8 cm tall and stands on a face of 5 cm by 2 cm. It weighs about 6 N. What is the pressure on the table?

    A block standing on its small face. A downward arrow on top is labelled 6 N. The front edge is labelled 5 cm and the right edge is labelled 8 cm. The edge going back is labelled 2 cm.5 cm8 cm6 N2 cm
    Tap your answer.

    1. Not this one. You multiplied 6 by 10. Pressure is the force divided by the area. Go to M4.5.7. Try another.

    2. Not this one. You used a large face, 8 cm by 5 cm, which is 40 cm². The block stands on the small face, 5 cm by 2 cm. It has less area and gives a bigger pressure. Go to M4.5.7. Try another.

    3. Yes, that is right. The area is 5 × 2 = 10 cm². The pressure is about 6 ÷ 10 = 0.6 N/cm².

    See the answer

    About 0.6 N/cm²

    The area is 5 × 2 = 10 cm². The pressure is about 6 ÷ 10 = 0.6 N/cm².

    • If you chose A: You multiplied 6 by 10. Pressure is the force divided by the area. Go to M4.5.7.
    • If you chose B: You used a large face, 8 cm by 5 cm, which is 40 cm². The block stands on the small face, 5 cm by 2 cm. It has less area and gives a bigger pressure. Go to M4.5.7.

  6. From an earlier chapter

    A train goes at 126 km/h. What is its speed in m/s?

    See the answer

    35 m/s

    To change km/h into m/s, divide by 3.6. 126 ÷ 3.6 = 35 m/s.

  7. A block has a mass of 480 g and a volume of 160 cm³. What is its density, in g/cm³?

    See the answer

    3 g/cm³

    Density = mass ÷ volume = 480 ÷ 160 = 3 g/cm³.

  8. A block of a material with density 2.5 g/cm³ is 4 cm by 5 cm by 6 cm. What is its mass, in g?

    See the answer

    300 g

    The volume is 4 × 5 × 6 = 120 cm³. Mass = density × volume = 2.5 × 120 = 300 g.

  9. A force of 90 N pushes on an area of 30 cm². What is the pressure, in N/cm²?

    See the answer

    3 N/cm²

    Pressure = force ÷ area = 90 ÷ 30 = 3 N/cm².

Keep it

Copy this into your exercise book.

Density = mass ÷ volume. Water is about 1 g/cm³. Pressure = force ÷ area. N/m² is the pascal. 1 N/cm² = 10 000 N/m². Smaller area, bigger pressure.

Draw: A block standing on its small face, and the same block lying flat. Put a 16 N arrow on each.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (741 KB). It opens on any phone.