- State that density is mass per unit volume.
- Carry out calculations involving density, mass and volume.
- Describe the principles of a method to determine the density of air.
- State and explain the relative magnitudes of the densities of solids, liquids and gases.
- State that pressure is the force per unit area, when the force acts normal to the surface.
- State that one pascal is one newton per square metre.
- Carry out calculations involving pressure, force and area.
- State that the pressure at a point in a fluid at rest depends on the depth, density of liquid and gravitational field strength.
- Carry out calculations involving pressure, density and depth.
- Explain buoyancy force (upthrust) in terms of the pressure difference between the top and bottom of an object.

Aids to Understanding

Density issues

If we have two objects of equal volume but one body is twice as heavy, then the heavier object is twice as dense. The particles that make up the body are more tightly packed. The property of density applies to liquids, solids and gases. It should be clear that volume, but not mass, changes with temperature so when we state the density of a solid we understand it is the density at room temperature. The same applies for a gas but the pressure of the gas is important. The density of a gas like air is usually quoted at room temperature and atmospheric pressure.

Conversion Factors

The density of a substance is the mass of
the substance divided by its volume. It can
be stated in units of

kg m^{-3} or

g cm^{-3} or

g (or kg) l^{-1}

Converting is easy but it is also something that
has to be learned.

Example

The element Osmium has the highest density of any

substance on Earth, 22550 kg m^{-3}.
State its density in units of g cm^{-3}.

1 kg = 1000g

So 22550 kg m^{-3} becomes

22550000 g m^{-3}

1 m^{3} = 10^{6} cm^{3} so

1 m^{-3} = 10^{-6}cm^{-3}

so
22550000 g m^{-3}

becomes
22.55 g cm^{-3}

Example

The density of gold is 19.3 g cm^{-3}, express this
in units of kg m^{-3}. Multiply the figure by 1000
to get the answer but make sure you know what you are doing.

Specific Gravity

On the table of common densities, water is highlighted. It is no accident
water has a density of exactly 1000 kg m^{-3}. It is defined as such.
The densities are sometimes quoted with reference to this number. For example
we might say Aluminium has a specific gravity of 2.7, this simply means its density is 2.7
times greater than water. It would be better if it was called specific density, but
the name has stuck for historical reasons. You are invited to find out.

Density and flotation

You might already know that if a body has a specific gravity greater than 1 it will
sink in water, but less than 1 it will float. This is good working knowledge but it does not
explain the phenomenon. However, after more work on this webpage, we will soon know why.

Resolving Force vectors

We resolved resulant forces in to their horizontal and vertical components in
1.1. It might be useful to revisit this.
If the force is acting at an angle to the surface then for the purposes
of pressure calculations we must work out the vector normal
to the surface. In the diagram below if the force is 60 N and the
angle is 30^{o} then the component Of force at right angles to the surface
is 60sin30 = 30 N. (the component parallel to the slope that we found a use for
in a previous section is simply 30 cos 30 = 15√ 3 N

Pressure in liquids

In this course the liquids we study are liquids at rest. In a liquid at rest the unbalanced force acting on all molecules (or any cross section
of the liquid), is zero. It follows that the pressure at a point (any point) in a liquid is acting in all directions.
The diagram below helps to illustrate this important point.

The red vectors can represent either force or pressure vectors. Remember P = F/A so if F is a vector so is P.

The Buoyancy force

The buoyancy force is all around us. For example the lift experienced by a helium filled balloon is due to the natural buoyancy of the air.
Because the upper air pressure on the balloon is less than the lower air pressure, the balloon experiences a lift. If the buoyancy force
is greater than the weight of the ballon, the balloon rises (floats). The same point applies to floating bodies in water. Try to show for
yourself that if the density of the floating object is less than the density of the fluid surrounding it, then the body floats, otherwise it sinks.

For a given fluid and gravitational field strength, the buoyancy force on a body depends only on the dimensions of the body. If the body is fully submerged
then the buoyancy force acting on it does not change with depth. Although the deeper the body becomes the greater the pressure it experiences and becomes more squashed.
For example divers cannot dive to any more than 300m. The world record for a scuba dive is 330 m.

...a misconception...

You may have heard that air inside bodies makes the body float and the more air you pump into it the more likely it is that this air makes it float. This is not quite right. If air is pumped into a body and the body is allowed to expand then its volume increases with no appreciable gain in mass. Therefore the buoyancy due to its environment increases.

Archimedes Principle

This is for interest only.

The following problem yeilds a new way to think about the buoyancy force.

Ex;

Suppose a 1 m^{3} solid cube is completely submerged in water.
i) What is the volume of water displaced by this cube?

ii) Calculate the buoyancy force on the cube.

iii) Calculate the weight of water displaced by this cube.

iv) Compare your answers to ii) and iii)

i) The volume of water displaced is equal to the volume of the cube, 1 m^{3}.

ii) The buoyancy force equals

F = DPA = rgh_{1} - (h_{2})A =
1000x9.8x1x1^{2} = 9800 N.

iii) Weight of water displaced by cube = m_{water}g = rVg

= 1000x9.8x1^{3} = 9800 N

iv) The values are the same.

This can be interpreted as follows. The buoyancy force on a body is equal to the weight of the water displaced by the
body. This is known as Archimedes Principle
Archimedes principle is useful when we are studying irregularly shaped bodies. All we need to know is the volume of the body
and working out the bouyancy force acting on it is a simple matter. In our course though we will always be given shapes whose
volumes are easilly calculated so you will be expected to use the principles discussed in the main body of this web page.

Example:

The shape in the diagram is weighed by a Newton balance and has a weight of exactly 10 N. Suppose you were asked to estimate
what the reading on the Newton Balance would be if it was immersed in;

i) Water

ii)Olive oil

Outline the steps you would take and suggest a possible answer. Look carefully at all values given and also suggest what the metal might be.
Any answer you give can be posted on this site so others can discuss it. Any modifications or additions are welcome and acknowledgement
will be given.

Suggestion

Using first principles, estimate the buoyancy force on a Helium filled balloon and verify using Archimedes Principle.

The Buoyancy Force and Notation

It is all very well and convenient to label the Buoyancy force on a body as B, but alwways keep in mind
that the force on the body is due to the surrounding fluid and can be more formally stated (and this can make
things clearer) as B = F_{BF} ie the buoyancy force on a body is the force on the **B**ody due to the **F**luid.

Solid, liquid or gas | Density (kg m^{-3}) |
Density (g cm^{-3}) |
Solid, liquid or gas |
---|---|---|---|

Cork | 250.0 | 0.250 | Solid |

Olive oil | 920.0 | 0.92 | Liquid |

Water | 1000.0 | 1.0 | Liquid |

Air | 1.29 | 0.00129 | Gas |

Carbon Dioxide | 1.98 | 0.00198 | Gas |

Aluminium | 2700.0 | 2.7 | Solid |

Copper | 8960.0 | 8.96 | Solid |

Perspex | 1190.0 | 1.19 | Solid |

Lead | 11400.0 | 11.4 | Solid |

Nitrogen | 1.25 | 0.00125 | Gas |

Gold | 19300.0 | 19.3 | Solid |

If you look at the relative densities of gases and solids then you will see that, at standard temperature and pressure, 1 kg of gas occupies round about 1000 more volume than 1 kg of a solid. This is means the average spacing of of particles in the gaseous state is ten times greater than when in the solid state. The diagram below illustrates this.

Independent Variable | Dependent Variable | Controlled variables | Result in Graphical form |
---|---|---|---|

Depth | Pressure | Density and Gravitational field | |

Density | Pressure | Depth and Gravitational field | |

Gravitational field | Pressure | Depth and Density |

The results of the experiments can be summarised by one simple expression p = rgh. When we use standard units for our measurements the constant of proportionality (gradient) for each graph above can be taken as 1.

Point Number | Principle or Point | Graphical Illustration |
---|---|---|

1. | The Pressure in a liquid is proprtional to depth | |

2. | The pressure on the lower surface of a body is necessarilly greater than the pressure on the upper surface. | |

3. | The differences in pressure leads to a net pressure, DP, that is directed upwards | |

4. | Since F = DP.A, where A is area of the lower surface a net upwards pressure results in a net upwards force. This force is called the buoyancy force. |