1. Materials
AQA Physics A

2. An old riddle!
Which is heavier, a pound of lead or a pound of feathers?
People who are confused by this riddle do so because they do not understand the difference between mass and density,

3. Density.
Density depends upon the mass of the atoms inside a substance and how closely packed together they are.
We can calculate density using:
Density = mass
volume
ρ = m V
S.I. units are kgm-3 but density can also be given in gcm-3

4. Examples The density of water is 1000kgm-3
This means that a volume of 1m3 has a mass of 1000kg
The density of ice is 917kgm-3. Ice floats on water because its density is less.
The densities of solids and liquids overlap but gases have much lower values of density. This is due to the larger spacing of the molecules
e.g. Air has a density of just 1.2kgm-3

5. Calculations
Q Calculate the mass of air in a typical lab measuring 8 x 6 x 3m
A Volume = 144m3
mass = density x volume
= 1.2 x 144
= 172.8kg

6. Measuring density
A regular shaped object: Calculate its volume after measuring its dimensions using a ruler, micrometer or vernier callipers.
Measure its mass using an electric balance and then calculate its density using ρ = m V

7. An irregular shaped object:
Find the volume of the object using the displacement of water.
(This method was first used by Archimedes when asked to check
whether the King’s crown was
made of pure gold)
Measure the mass using an electric balance and then calculate the density

8. A liquid: Measure its volume using a measuring cylinder.
Find the mass of the liquid and the cylinder and then subtract the mass of the empty cylinder to find the mass of the liquid.
Calculate the density of the liquid using
ρ = m V

9. Further example
Q The crown that Archimedes had to investigate contained 12.5 x 10-5m3 of gold and 2.2 x 10-5m3 of silver. Calculate its mass and density. (Density of gold is 19300kgm-3 and density of silver is 10500kgm-3).
A Mass of gold= ρ x V = x12.5 x 10-5
= 2.41kg
Mass of silver = ρ x V = x 2.2.x 10-5
= 0.23kg
Total mass = 2.64kg
Total volume = 14.7 x 10-5m3
Density of alloy = m = = 18000kgm-3
V x 10-5

10. Stretching Springs Measure the original length of the spring
Set up the apparatus and apply a force of 1N. This gives a tension of 1N in the spring.
Record the new length and find the extension by subtracting the original length.
Repeat until you have a total force of 8N
Plot a graph of Force (F) against extension (ΔL).

11. Hooke’s law
The graph should be a straight line through the origin showing direct proportion.
Hooke’s law states that
the force applied to a spring
is directly proportional to its
extension provided it does
not pass the elastic limit.
Force = constant x extension
F = k(ΔL)
(Springs can be used to measure forces using a Newton meter)

12. Spring Constant
k = spring constant or stiffness constant and is found from the gradient of the graph of F against ΔL
A typical spring used in school has k = 50Nm-1
The stiffer the spring,
the greater the value of k
Spring X is stiffer than
spring Y
F/ N
Extension / m X Y

13. Example
Q A spring has a length of 20mm and stretches to a length of 60mm with a load of 2N. Calculate its spring constant.
F = kΔL
k = F = = 50Nm-1
ΔL 40 x 10-3
Q Calculate the length of the spring when a load of 7N is added to it
ΔL = F = 7 = 0.14m k
New length = = 0.16m

14. Elastic limit
If the spring is overstretched it does not return to its original length. It is permanently distorted.
The elastic limit is the point beyond which the material is no longer elastic and will not return to its original length when the load is removed. F
x x = elastic limit ΔL

15. Springs in series and parallel

16. Springs in parallel
Repeat the experiment for stretching a spring but use 2 springs (P and Q) in parallel.
Calculate the gradient of the graph to find the combined spring constant (k).
Theory (see P165) shows that:
k = kp + kQ where kp and kQ are the spring constants for springs P and Q ( the springs are stiffer and more difficult to stretch)
Find the percentage difference between your value of k and the theoretical value

17. Springs in series
Repeat the experiment for stretching a spring but use 2 springs (P and Q) in series.
Calculate the gradient of the graph to find the combined spring constant (k).
Theory (see P165) shows that:
1 = 1 + 1
k kP kQ
where kp and kQ are the spring constants for springs P and Q (the springs are now easier to stretch)
Find the percentage difference between your value of k and the theoretical value

18. Energy stored Work is done when stretching a spring and this equals
the elastic potential energy stored in it.
Since work = force x distance
= area under graph
= 1 F ΔL = 1 kΔL2
Elastic potential energy
stored in a stretched spring = 1 kΔL2 2
F/N ΔL
Extension/m

19. Example
Q How much energy is stored in a spring which is 32cm long and stretches to 54cm when a force of 6.0N is applied to it.
A ΔL = 22cm = 0.22m
Ep = 1 FΔL 2
= 1 x 6.0 x 0.22
= 0.66J

20. Tensile Stress Tensile stress = tension in wire
x-sectional area of wire
σ = T
A Units:Nm-2 or Pa
Stress is a property of the material rather than the individual object being stretched
Q Calculate the stress in a wire of x-sectional area 4 x 10-8m2 when it is stretched with a force of 200N
A 5 x 109 Pa

21. Tensile strain Tensile strain = extension of wire original length
Since it is a ratio it has no units and is once again a property of the material.
Q Calculate the strain in a wire of length 2.3m which stretches by 1.5mm
A 6.5 x 10-4

22. Stretching different materials
A rubber band and a polythene strip can be stretched using the same apparatus as used with the spring.
A copper wire can be stretched along a bench.
In all cases plot a graph of stress against strain.
The shape of a stress-strain graph is the same as the shape of a force-extension graph but is common to the material used and not the individual wires.

23. Examples Copper is an example of a ductile material.
Brittle
Stress
Strain
Copper is an example of
a ductile material.
Initially it obeys Hooke’s law
and returns to its original length
if the load is removed.
There is then a large increase in its extension and if the load is removed it does not return to its original length.
Glass is an example of a brittle material

24. More definitions
Limit of proportionality: Point beyond which Hooke’s law is no longer obeyed.
Elastic limit: Point beyond which the material is no longer elastic and will not return to its original length when the load is removed.
Yield point: Point at which there is marked increase in extension

25. Elastic deformation: The material will return to its original shape when the load is removed.
Plastic deformation: The material is permanently stretched when the load is removed.
Ductile materials like copper can be drawn into wires and undergo large plastic deformation.
Brittle materials fracture suddenly without any noticeable yield e.g. glass
Strong materials require a high stress to break them
Ultimate tensile stress: maximum stress that can be applied. This is sometimes called the Breaking stress

26. Typical graph E = elastic limit Y = yield point
P = limit of proportionality
UTS = ultimate tensile stress
S = wire snaps

27. A is a brittle material
which is also strong
e.g glass
B is a strong material
which is not ductile
e.g steel
C is a ductile material
e.g copper
D is plastic material

28. Young modulus of elasticity
Young modulus = stress
strain
E = σ ε
= T/ΔL
A L TL
AΔL
E is measured in Pa, the same as stress.
It is a measure of how difficult it is to change the shape of the material and can be found from the gradient of a stress-strain graph

29. Examples
Q1 The Young modulus for cast iron is 2.1 x 1011Pa and is snaps when the strain reaches What is the stress needed to break it?
A 1.1 x 108Pa
Q2 A wire made of a particular material is loaded with a load of 500 N.  The diameter of the wire is 1.0 mm.  The length of the wire is 2.5 m, and it stretches 8 mm when under load.  What is the Young Modulus of this material?
A 2.0 x 1011Pa

30. Experiment to measure Young modulus
Measure the original length of the wire
Measure the diameter of the wire at different points along its length using a micrometer and find is average diameter. Calculate the x=sectional area of the wire.
Load the wire as before and measure its extension.
Plot a stress-stain graph and measure its gradient = Young modulus

31. Area under stress-strain graph
Area under graph =
average stress x strain
= F x ΔL
A L
= work done
volume
The area gives us the energy stored per unit volume and is a measure of the toughness of the material. A tough material needs a large amount of energy to break it.

32. Loading and unloading materials
A material can be stretched as before by increasing the load on it and the extension measured.
It can the unloaded gradually and the extension measured again.
A graph can then be plotted showing both the loading and unloading.
The shape of this graph will depend upon the type of material used and the size of the load.

33. Metal wire
The loading and unloading give the same straight line provided the elastic limit has not been passed.
The atoms in the metal are pulled slightly further apart and then return to their original positions when the load is removed.
If taken beyond this point the unloading will give a line parallel to the first but leaving a permanent extension.

35. Rubber band The rubber band returns to its original length when
the load is removed but
the unloading curve is
below the loading curve.
The molecules are like long
strands of tangled spaghetti that become untangled when the rubber is stretched and the become tangled again when the load is removed.
Shape.  B C A
Strain
Stress/ N

36. internal (heat) energy by the rubber band when it is unloaded.
The shaded area
represents the energy
stored in the form of
internal (heat) energy
by the rubber band when
it is unloaded.
The rubber used in car types should give a small area to reduce the energy lost by heat.
Shape.  B C A
Strain
Stress/ N

38. Polythene
Polythene does not return to its original length when the load is removed.
Like rubber, polythene is a polymer, but when it is stretched new cross links form which prevent the molecules from returning to their original state.