1.
(a)
The Young modulus is defined as the ratio of tensile stress to tensile strain.
Explain what is meant by each of the terms in italics.
tensile stress ...............................................................................................................
.....................................................................................................................................
.....................................................................................................................................
tensile strain ...............................................................................................................
.....................................................................................................................................
(3)
(b)
A long wire is suspended vertically and a load of 10 N is attached to its lower end. The extension
of the wire is measured accurately. In order to obtain a value for the Young modulus of the material
of the wire, two more quantities must be measured. State what these are and in each case indicate
how an accurate measurement might be made.
quantity 1 ....................................................................................................................
method of measurement .............................................................................................
.....................................................................................................................................
quantity 2 ....................................................................................................................
method of measurement .............................................................................................
.....................................................................................................................................
(4)
(c)
Sketch below a graph showing how stress and strain are related for a ductile substance and label
important features.
stress
strain
(2)
(Total 9 marks)
Thornleigh Salesian College, Bolton
1
2.
A strain gauge is made from a constantan wire of original length 25 mm. If the wire stretches its
resistance changes. The gauge is attached to an object that is then placed under tension, which causes the
length of the constantan wire to increase. The resistance, R, was measured for various lengths, l, and the
following results were obtained:
99.96
R/
l/10–2 m
2.500
100.64
2.508
101.76
2.523
102.80
2.536
103.85
2.548
104.71
2.557
When the wire is stretched, it may be assumed that for small extensions:
R  l2
(a)
Complete the table showing the value of l2 for each value of R.
(2)
(b)
Plot a graph of R on the y-axis against l2 on the x-axis.
(4)
(c)
Use your graph and the value for the resistivity of constantan given below to find the diameter of
the wire when its resistance is 103.40 .
resistivity of constantan = 4.7 × 10–7 m
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(5)
(d)
Define tensile strain. Use your graph to determine the strain when the resistance
of the wire is 103.40 .
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 13 marks)
Thornleigh Salesian College, Bolton
2
0.18 m
Y
0.90 kg
A
3.
0.12 m
X
The mass of a retort stand and clamp is 1.6 kg and their combined centre of mass lies along the line XY.
A spring which has a negligible mass is attached to the clamp and supports a mass of 0.90 kg, as shown in
the diagram. The spring requires a force of 6.0 N to stretch it 100 mm.
(a)
Calculate the extension of the spring.
.....................................................................................................................................
.....................................................................................................................................
(2)
(b)
Show that this arrangement will not tip ( i.e. will not rotate about A) when the 0.90 kg mass is at
rest in its equilibrium position.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
Thornleigh Salesian College, Bolton
3
(c)
If the mass is lifted up and released, it will vibrate about the equilibrium position. Explain, without
calculation, why the stand will tip if the amplitude exceeds a certain value.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(3)
(Total 7 marks)
4.
As part of a quality check, a manufacturer of fishing line subjects a sample to a tensile test. The sample of
line is 2.0m long and is of constant circular cross-section of diameter 0.50mm.
Hooke’s law is obeyed up to the point when the line has been extended by 52mm at a tensile stress of 1.8
× l08Pa.
The maximum load the line can support before breaking is 45 N at an extension of 88 mm.
(a)
Calculate
(i)
the value of the Young modulus,
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(ii)
the breaking stress (assuming the cross-sectional area remains constant),
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(iii)
the breaking strain.
...........................................................................................................................
(5)
Thornleigh Salesian College, Bolton
4
(b)
Sketch a graph on the axes below to show how you expect the tensile stress to vary with strain.
Mark the value of stress and corresponding strain at
(i)
the limit of Hooke’s law,
(ii)
the breaking point.
stress
strain
(4)
(Total 9 marks)
Thornleigh Salesian College, Bolton
5
5.
The cross-section of an overhead power cable is shown below. The cable consists of a hard steel core
surrounded by fifteen straight, thick copper wires.
copper wire
diameter 6.5 mm
steel core
diameter 25 mm
100m lengths of cable are suspended between adjacent pylons as part of an electricity distribution system.
Calculate the mass of a 100m length of the cable.
density of copper = 8.93 × 103 kg m–3
density of steel = 7.80 × 103kg m–3
mass of hard steel core .........................................................................................................
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
mass of copper wires .............................................................................................................
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
mass of cable .........................................................................................................................
(Total 4 marks)
Thornleigh Salesian College, Bolton
6
6.
The diagram shows tensile stress-strain curves for three different materials X, Y and Z.
stress
X
Y
Z
strain
For each material named below, state which curve is typical of the material, giving the reasoning behind
your choice.
(a)
copper ..................................
reasoning ....................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(b)
glass ....................................
reasoning ....................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(c)
hard steel ..............................
reasoning ....................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(Total 6 marks)
Thornleigh Salesian College, Bolton
7
7.
(a)
Describe an experiment to determine the Young modulus for a material in the form of a wire. Draw
a labelled diagram and explain how you would make the necessary measurements. Show how you
would use your measurements to calculate the result.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(8)
(b)
rigid support
F
aluminium
copper
A copper wire and an aluminium wire, each of diameter 0.72 mm, are joined end to end as shown
in the diagram with the aluminium wire fixed at right angles to a rigid support. A steadily
increasing force, F, is applied. Use data from the Data booklet to
(i)
explain which wire will yield,
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(ii)
determine the value of F at which yield should occur.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(4)
(Total 12 marks)
Thornleigh Salesian College, Bolton
8
8.
One end of a steel wire of length 1.2m and 2.0mm diameter is attached to a rigid beam.
A 25 g mass is attached to the free end of the steel wire and placed against the underside of the beam as
shown.
rigid beam
fixing point
25 g mass
steel wire
The 25 g mass is released and falls freely until the wire becomes taut. The kinetic energy of the falling
mass is converted to elastic potential energy in the wire as the wire extends to a maximum of 1.0 mm.
Energy converted to other forms is negligible.
For maximum extension of the wire, complete parts (i) to (v).
(i)
Show that the elastic potential energy stored by the extended wire is 0.29 J.
...........................................................................................................................
...........................................................................................................................
(ii)
Calculate the tension in the wire.
...........................................................................................................................
...........................................................................................................................
(iii)
Calculate the stress in the wire.
...........................................................................................................................
...........................................................................................................................
(iv)
Calculate the strain of the wire.
...........................................................................................................................
...........................................................................................................................
(v)
Hence, calculate the Young modulus for the steel of the wire.
...........................................................................................................................
...........................................................................................................................
(9)
(Total 9 marks)
Thornleigh Salesian College, Bolton
9
9.
A student carries out an experiment to investigate how the extension of a steel wire varies with an
increasing tensile force. The results of the experiment are shown plotted on the graph. The initial length
of the wire is 0.50m and its diameter is 0.80 mm. The wire breaks at an extension of 1.46 mm.
force/N
110
100
90
80
70
60
50
40
30
20
10
0
0.20
0.40
Thornleigh Salesian College, Bolton
0.60
0.80
1.00
1.20
1.40
1.60
extension/mm
10
Use information from the graph to determine
the Young modulus for the material,
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
an estimate of the yield stress for the material.
...............................................................................................................................................
...............................................................................................................................................
(Total 6 marks)
10.
(a)
When determining the Young modulus for the material of a wire, a tensile stress is applied to the
wire and the tensile strain produced is measured.
(i)
State the meaning of
tensile stress .....................................................................................................
...........................................................................................................................
tensile strain .....................................................................................................
...........................................................................................................................
(ii)
Define the Young modulus.
...........................................................................................................................
...........................................................................................................................
(3)
Thornleigh Salesian College, Bolton
11
(b)
The graph represents tensile stress - tensile strain curves for two different materials A and B. X
and Y are the respective points at which each material fractures.
tensile stress
X
material A
Y
material B
0
(i)
tensile strain
0
One of the materials is brittle, the other ductile. State which material is brittle.
...........................................................................................................................
(ii)
Making use of the curves in the graph, describe the behaviour of each material as it is
stretched from its original state to breaking point.
material A .........................................................................................................
...........................................................................................................................
material B .........................................................................................................
...........................................................................................................................
(iii)
State, giving a reason, which material has the greater value of the Young modulus.
...........................................................................................................................
...........................................................................................................................
(5)
(c)
A vertical steel piano wire of length 1.5 m and cross-sectional area 1.3 × 10–6 m2 supports a load of
80N.
Given that the Young modulus for steel = 2.10 × 1011Pa, calculate the extension in the wire
produced by this load.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 10 marks)
Thornleigh Salesian College, Bolton
12
11.
(a)
(i)
Draw and label suitable apparatus required for measuring the Young modulus of a material
in the form of a long wire.
(ii)
List the measurements you would make when using the apparatus described in part (i).
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
(iii)
Describe briefly how the measurements listed in part (ii) would be carried out.
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
Thornleigh Salesian College, Bolton
13
(iv)
Explain how you would calculate the Young modulus from your measurements.
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
(13)
(b)
A uniform heavy metal bar of weight 250 N is suspended by two vertical wires, supported at their
upper ends from a horizontal surface, as shown.
T
T
brass
steel
A
B
250 N
One wire is made of brass and the other of steel. The cross-sectional area of each wire is 2.5 ×10–7
m2 and the unstretched length of each wire is 2.0 m.
the Young modulus for brass = 1.0 × 1011 Pa
the Young modulus for steel = 2.0 × 1011 Pa
(i)
If the tension, T, in each wire is 125 N, calculate the extension of the steel wire.
.................................................................................................................
.................................................................................................................
.................................................................................................................
(ii)
Estimate how much lower the end A will be than the end B.
.................................................................................................................
.................................................................................................................
(3)
(Total 16 marks)
Thornleigh Salesian College, Bolton
14
12.
(a)
(i)
Define the Young modulus for a material.
.................................................................................................................
.................................................................................................................
(ii)
Explain what is meant by the elastic limit for a wire.
.................................................................................................................
.................................................................................................................
(2)
(b)
A wire supported at its upper end, hangs vertically. The table shows readings obtained when
stretching the wire by suspending masses from its lower end.
load/N
0
2.0
4.0
6.0
7.0
8.0
9.0
10.0
10.5
extension/mm
0
1.2
2.4
3.6
4.2
4.9
5.7
7.0
8.0
(i)
Plot a graph of load against extension.
(One sheet of graph paper should be provided)
(ii)
Indicate on your graph the region where Hooke’s law is obeyed.
(iii)
The unstretched length of the wire is 1.6m and the area of cross-section
8.0 × 10–8 m2. Calculate the value of the Young modulus of the material.
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
(8)
(c)
(i)
By considering the work done in stretching a wire, show that the energy stored is given by
1
2 Fe, where F is the force producing an extension e.
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
Thornleigh Salesian College, Bolton
15
(ii)
Calculate the energy stored in the wire in part (b) when the extension is 4.0 mm.
.................................................................................................................
.................................................................................................................
.................................................................................................................
(4)
(Total 14 marks)
13.
A material in the form of a wire, 3.0 m long and cross-sectional area = 2.8 × 10–7 m2 is suspended from a
support so that it hangs vertically. Different masses may be suspended from its lower end. The table
shows the extension of the wire when it is subjected to an increasing load and then a decreasing load.
load/N
0
24
52
70
82
88
94
101
71
50
16
0
extension/mm
0
2.2
4.6
6.4
7.4
8.2
9.6
13.0
10.2
8.0
4.8
3.2
(a)
Plot a graph of load (on y axis) against extension (on x axis) both for increasing and decreasing
loads.
(One sheet of graph paper should be provided)
(4)
(b)
Explain what the shape of the graph tells us about the behaviour of the material in the wire. You
may be awarded marks for the quality of written communication in your answer.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(4)
Thornleigh Salesian College, Bolton
16
(c)
Using the graph, determine a value of the Young modulus for the material of the wire.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(3)
(d)
State how the graph can be used to estimate the energy stored during the loading process.
...........................................................................................................................
...........................................................................................................................
(1)
(Total 12 marks)
14.
A uniform wooden beam of mass 35.0 kg and length 5.52 m is supported by two identical vertical steel
cables A and B attached at either end, as shown in Figure 1.
5.52 m
cable A
cable B
beam
Figure 1
(a)
Calculate
(i)
the weight of the beam,
.................................................................................................................
(ii)
the tension in each cable.
.................................................................................................................
.................................................................................................................
(2)
Thornleigh Salesian College, Bolton
17
(b)
Each unstretched cable has a diameter of 8.26 mm and a length 2.50 m. Calculate the extension of
each cable when supporting the beam.
the Young modulus for steel = 2.10 × 1011 Pa
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
(4)
(c)
An object of mass 20.0 kg is hung from the beam 1.00 m from cable A, as shown in Figure 2.
5.52 m
cable A
cable B
1.00 m
20.0 kg
Figure 2
(i)
Show that the new tension in cable A is 332 N.
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
.................................................................................................................
Thornleigh Salesian College, Bolton
18
(ii)
Calculate the new tension in cable B.
.................................................................................................................
.................................................................................................................
.................................................................................................................
(6)
(Total 12 marks)
15.
(a)
When a tensile stress is applied to a wire, a tensile strain is produced in the wire. State the meaning
of
tensile stress, ...........................................................................................................
.................................................................................................................................
tensile strain. ...........................................................................................................
.................................................................................................................................
(2)
(b)
A long thin line metallic wire is suspended from a fixed support and hangs vertically. Weights are
added to increase the load on the free end of the wire until the wire breaks. The graph below shows
how the tensile strain in the wire increases as the tensile stress increases.
C
tensile
stress
B
A
tensile strain
With reference to the graph, describe the behaviour of the wire as the load on the free end is
increased. To assist with your answer refer to the point A, and regions B and C.
You may be awarded marks for the quality of written communication in your answer.
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
(5)
(Total 7 marks)
Thornleigh Salesian College, Bolton
19
16.
(a)
Define the density of a material.
.................................................................................................................................
.................................................................................................................................
(1)
(b)
Brass, an alloy of copper and zinc, consists of 70% by volume of copper and 30% by volume of
zinc.
density of copper
density of zinc
(i)
= 8.9 × 103 kg m–3
= 7.1 × 103 kg m–3
Determine the mass of copper and the mass of zinc required to make a rod of brass of
volume 0.80 × 10–3 m3.
.......................................................................................................................
.......................................................................................................................
.......................................................................................................................
.......................................................................................................................
(ii)
Calculate the density of brass.
.......................................................................................................................
.......................................................................................................................
(4)
(Total 5 marks)
17.
(a)
State Hooke’s law for a material in the form of a wire.
................................................................................................................................…
................................................................................................................................…
(2)
Thornleigh Salesian College, Bolton
20
(b)
A rigid bar AB of negligible mass, is suspended
horizontally from two long, vertical wires as shown in the
diagram. One wire is made of steel and the other of brass.
The wires are fixed at their upper end to a rigid horizontal
surface. Each wire is 2.5 m long but they have different
cross-sectional areas.
When a mass of 16 kg is suspended from the centre of AB,
the bar remains horizontal.
steel
A
brass
B
the Young modulus for steel = 2.0 × 1011 Pa
the Young modulus for brass = 1.0 × 1011 Pa
(i)
What is the tension in each wire?
...........................................................................................................................
(ii)
If the cross-sectional area of the steel wire is 2.8 × 10–7 m2, calculate the extension of the
steel wire.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(iii)
Calculate the cross-sectional area of the brass wire.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(iv)
Calculate the energy stored in the steel wire.
...........................................................................................................................
...........................................................................................................................
(7)
(c)
The brass wire is replaced by a steel wire of the same dimensions as the brass wire.
The same mass is suspended from the midpoint of AB.
(i)
Which end of the bar is lower?
...........................................................................................................................
(ii)
Calculate the vertical distance between the ends of the bar.
...........................................................................................................................
...........................................................................................................................
(2)
(Total 11 marks)
Thornleigh Salesian College, Bolton
21
18.
The diagram below shows how the impact force on the heel of a runner’s foot varies with time during an
impact when the runner is wearing cushioned sports shoes.
force/N
2400
1600
800
0
0
(a)
0.1
0.2
0.3
0.4 time/ms
Estimate the maximum stress on the cartilage pad in the knee joint as a result of this force acting on
the cartilage pad over a contact area of 550 mm2.
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
................................................................................................................................…
(4)
(b)
On the diagram above, sketch the graph of force against time you would expect to see if a sports
shoe with less cushioning had been used.
(3)
(Total 7 marks)
Thornleigh Salesian College, Bolton
22
19.
(a)
When determining the Young modulus for the material of a wire, a tensile stress is applied to the
wire and the tensile strain is measured.
(i)
State the meaning of
tensile stress .......................................................................................................
.............................................................................................................................
tensile strain .......................................................................................................
.............................................................................................................................
(ii)
Define the Young modulus ................................................................................
.............................................................................................................................
(3)
(b)
Figure 1 shows two wires, one made of steel and the other of brass, firmly clamped together at
their ends. The wires have the same unstretched length and the same cross-sectional area.
One of the clamped ends is fixed to a horizontal support and a mass M is suspended from the other
end, so that the wires hang vertically.
brass
steel
M
Figure 1
(i)
Since the wires are clamped together the extension of each wire will be the same.
If ES is the Young modulus for steel and EB the Young modulus for brass, show that
E S FS

,
E B FB
where FS and FB are the respective forces in the steel and brass wire.
.............................................................................................................................
.............................................................................................................................
.............................................................................................................................
Thornleigh Salesian College, Bolton
23
(ii)
The mass M produces a total force of 15 N. Show that the magnitude of the force
FS = 10 N.
the Young modulus for steel = 2.0 × 1011Pa
the Young modulus for brass = 1.0 × 1011Pa
.............................................................................................................................
.............................................................................................................................
.............................................................................................................................
.............................................................................................................................
(iii)
The cross-sectional area of each wire is 1.4 × 106m2 and the unstretched length is 1.5 m.
Determine the extension produced in either wire.
.............................................................................................................................
.............................................................................................................................
.............................................................................................................................
.............................................................................................................................
(6)
(Total 9 marks)
20.
(a)
State Hooke’s law for a material in the form of a wire and state the conditions under which this law
applies.
.....................................................................................................................................
.....................................................................................................................................
(2)
Thornleigh Salesian College, Bolton
24
(b)
A length of steel wire and a length of brass wire are joined together. This combination is suspended
from a fixed support and a force of 80 N is applied at the bottom end, as shown in the figure below.
steel
brass
80 N
Each wire has a cross-sectional area of 2.4 × 10–6 m2.
length of the steel wire = 0.80 m
length of the brass wire = 1.40 m
the Young modulus for steel = 2.0 × 1011 Pa
the Young modulus for brass = 1.0 × 1011 Pa
(i)
Calculate the total extension produced when the force of 80 N is applied.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(ii)
Show that the mass of the combination wire = 4.4 × 10–2 kg.
density of steel = 7.9 × 103 kg m–3
density of brass = 8.5 × 103 kg m–3
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(7)
Thornleigh Salesian College, Bolton
25
(c)
A single brass wire has the same mass and the same cross-sectional area as the combination wire
described in part (b). Calculate its length.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 11 marks)
21.
(a)
When a tensile stress is applied to a wire, a tensile strain is produced in the wire.
State the meaning of
tensile stress, ...............................................................................................................
.....................................................................................................................................
tensile strain. ...............................................................................................................
.....................................................................................................................................
(2)
(b)
A long, thin metal wire is suspended from a fixed support and hangs vertically.
Masses are suspended from its lower end.
As the load on the lower end is increased from zero to a certain value, and then decreased again to
zero, the variation of the resulting tensile strain with the applied tensile stress is shown in the
graph.
tensile
stress
0 A
0
Thornleigh Salesian College, Bolton
B
D
C
tensile strain
26
(i)
Describe the behaviour of the wire during this process. Refer to the points
A, B, C and D in your answer.
You may be awarded marks for the quality of written communication in your answer.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(ii)
State, with a reason, whether the material of the wire is ductile or brittle.
...........................................................................................................................
...........................................................................................................................
(iii)
What does AD represent?
...........................................................................................................................
(iv)
State how the Young modulus for the material may be obtained from the graph.
...........................................................................................................................
(v)
State how the energy per unit volume stored in the wire during the loading
process may be estimated from the graph.
...........................................................................................................................
(9)
(c)
The wire described in part (b) has an unstretched length of 3.0 m and cross-sectional area 2.8 ×
10–7 m2. At a certain stage between the points A and B on the graph, the wire supports a load of 75
N. Calculate the extension produced in the wire by this load.
the Young modulus for the material of the wire = 2.1 × 1011 Pa
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 13 marks)
22.
(a)
(i)
Describe the behaviour of a wire that obeys Hooke’s law.
...........................................................................................................................
Thornleigh Salesian College, Bolton
27
...........................................................................................................................
(ii)
Explain what is meant by the elastic limit of the wire.
...........................................................................................................................
...........................................................................................................................
(iii)
Define the Young modulus of a material and state the unit in which it is measured.
...........................................................................................................................
...........................................................................................................................
(5)
(b)
A student is required to carry out an experiment and draw a suitable graph in order to obtain a
value for the Young modulus of a material in the form of a wire.
A long, uniform wire is suspended vertically and a weight, sufficient to make the wire taut, is fixed
to the free end. The student increases the load gradually by adding known weights. As each weight
is added, the extension of the wire is measured accurately.
(i)
What other quantities must be measured before the value of the Young modulus can be
obtained?
...........................................................................................................................
...........................................................................................................................
(ii)
Explain how the student may obtain a value of the Young modulus.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(iii)
How would a value for the elastic energy stored in the wire be found from the results?
...........................................................................................................................
...........................................................................................................................
(6)
(Total 11 marks)
Thornleigh Salesian College, Bolton
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23.
An aerial system consists of a horizontal copper wire of length 38 m supported between two masts, as
shown in the figure below. The wire transmits electromagnetic waves when an alternating potential is
applied to it at one end.
38 m of copper wire
14.0 m
12.0
P
mast
(a)
Q
mast
The wavelength of the radiation transmitted from the wire is twice the length of the copper wire.
Calculate the frequency of the transmitted radiation.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(1)
(b)
The ends of the copper wire are fixed to masts of height 12.0 m. The masts are held in a vertical
position by cables, labelled P and Q, as shown in the figure above.
(i)
P has a length of 14.0 m and the tension in it is 110 N. Calculate the tension in the copper
wire.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(ii)
The copper wire has a diameter of 4.0 mm. Calculate the stress in the copper wire.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
Thornleigh Salesian College, Bolton
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(iii)
Discuss whether the wire is in danger of breaking if it is stretched further due to movement
of the top of the masts in strong winds.
breaking stress of copper = 3.0 × 108 Pa
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(7)
(Total 8 marks)
Thornleigh Salesian College, Bolton
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Markscheme
1.
(a)
(b)
(tensile) force
(1)
cross – sectional area
extension
tensile strain =
(1)
original length
mention of tensile and original (1)
3
diameter of wire (1)
in several places [or repeated] (1)
using a micrometer (1)
(original) length of wire (1)
using a metre rule (or tape measure) (1)
max 4
tensilestress =
stress
(1) (plastic region)
(c)
2
(1)(linear region)
strain
[9]
2.
(a)
R/
(b)
(c)
(d)
99.96
100.64
101.76
102.80
103.85
104.71
l/10–2 m
2.500
2.508
2.523
2.536
2.548
2.557
l2/10–4 m2
6.250
6.290
6.366
6.432
6.492
6.538
correct values above (1)(1) (deduct one mark for each error)
(If two significant figures only – deduct one mark)
both axes, with units, correctly labelled (1)
six points correctly plotted (1)
best straight line through plotted points (1)
sensible scale (1)
at R = 103.40 , l2 = 6.46 (1) × 10–4 m2 (1)
thus l = 2.542 × 10–2 (1) (m)
constantan resistivity = 47 × 10–8 (m) (1)
= 1.2(1) × 10–5 m (1)
tensile strain =
2
4
5
extension
(1)
original length
(at R = 103.40 , l 2.542 × 10–2 m)
thus strain = (2.542 – 2.5) (1) = 0.017 (1)
2.5
max 2
[13]
3.
(a)
(b)
use of mg = kl (or 0.90 × 9.81 = 60 l)(1)
l = 0.15m (1)
2
no tipping if moment of weight of clamp about A > moment of 0.90 kg (1)
moment of 0.90 kg about A = 0.90 g × 0.18 = 0.16 g
moment of weight of clamp about A = 1.60 g × 0.12 = 0.19 g
 no tipping (1)
2
Thornleigh Salesian College, Bolton
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4.
(c)
as mass vibrates tension changes (1)
maximum tension increases as amplitude increases because maximum length increases (1)
tipping when moment of tension exceeds moment of weight of clamp (1)
3
[7]
(a)
(i)
strain = 0.026 (1)
E = 6.92 × 109 Pa (1)
(ii)
A = 1.96 × 10–7 (m2) (1)
stress = 230 × 108 Pa (1)
(iii)
breaking strain = 0.044 (1)
stress/
108 Pa
(i)
2.5
5
(ii)
2.0
1.5
1.0
(b)
0.5
0
0
0.01 0.02 0.03 0.04 0.05 strain
shape overall (1)
(i)
straight line (1)
0 to (0.026, 1.8) (1)
(ii)
curve (1)
to (0.044, 2.3) (1)
Max 4
[9]
5.
steel
A = 4.91 × 10–4 (m2)
m = 383kg (1)
copper
A = 3.32 × 10–5 (m2) m = 444 kg (1)
one cross-sectional area calculated correctly (1)
use of m = lA (1)
mass of cable = 827 kg (1)
[Max 4]
stress
X
Y
Z
6.
strain
(a)
Y (1)
significant plastic deformation (or Young modulus less than X) (1)
(b)
Z (1)
Thornleigh Salesian College, Bolton
32
no plastic deformation (or smallest value of Young modulus) (1)
(c)
X (1)
small amount of plastic deformation (or Young modulus greater than Y) (1)
[6]
7.
(a)
diagram showing two supported wires and vernier [or long wire and appropriate scale] (1)
one justification of design (1)
measurements:
identified length with ruler (1)
diameter with micrometer (1)
in several places [or in different directions] (1)
add load [mass] and read vernier (1)
repeat for range of loads (1)
within limit of proportionality [allow elastic limit] (1)
calculation of at least one value from readings (1)
graph or calc and average (1)
if apparatus unsuitable, mark to scheme to max 6/8
max 8
Thornleigh Salesian College, Bolton
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(b)
aluminium yields, has smaller yield strength identified from data booklet (1)
use of F(=sA) (1)
= 50 × 106 ×  × (0.36 × 10–3)2 (1)
= 20.3N (1)
4
[15]
8.
(i)
appropriate discussion of energy conservation (1)
 p.e. = 2.5 × 10–2 × 9.8 × 1.2 (1) (= 0.29 J)
(ii)
F=
2Ep
(1) = 590N (1)
e
(iii)
A = 3.1 × 10–6 (m2) (1)
stress = 1.9 × 108 Pa (1)
(iv)
strain =
(v)
E=
e
0.001
=
= 8.3 × 10–4 (1)
L
1.2
8
stress
= 1.9  10 (1) = 2.3 × 1011 Pa (1)
strain
8.3  10 – 4
[9]
9.
uses slope of straight line region (1)
slope = 1.54 × 105 (Nm–1) (1)
l
E = slope ×
(1)
A
A = 5.03 × 10–7 (m2) (1)
E = 1.5 × 1011 Pa (1)
Fy = 87 (N) (1)
yield stress = 1.7 × 108 Pa (1)
[6]
10.
(a)
(b)
(c)
(i)
tensile stress: the force per unit cross-sectional area (1)
tensile strain: extension per unit length (1)
(ii)
the Young modulus = tensile stress/tensile strain (1)
(i)
brittle: material A (1)
(ii)
A, (brittle) obeys Hooke’s law (until it fractures without warning) (1)
B, (ductile) obeys Hooke’s law up to the limit of proportionality (1)
beyond this point wire is permanently stretched (or behaves plastically) (1)
(iii)
A has greatest value of the Young modulus
because of steeper gradient (1)
F l
80
1.5

 gives) 2.10 × 1011 =
(1)
6
e
A e
1.3  10
e = 0.44 × 10–3 m (1)
3
max 5
(Y =
2
[10]
Thornleigh Salesian College, Bolton
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11.
(a)
(i)
diagram to show:
(long) wire fixed at one end (1)
mass/weight at other end (1)
measuring scale (1)
mark on wire, or means to measure extension (1)
max 3
[alternative for two vertical wires:
two wires fixed to rigid support (1)
mass/weight at end of one wire (1)
other wire kept taut (1)
spirit level and micrometer or sliding vernier scale (1)]
(ii)
(iii)
(iv)
measurements:
length of the wire between clamp and mark (1)
diameter of the wire (1)
extension of the wire (1)
for a known mass (1)
max 3
length measured by metre rule (1)
diameter measured by micrometer (1)
at several positions and mean taken (1)
(known) mass added and extension measured
by noting movement of fixed mark against vernier scale
(or any suitable alternative) (1)
repeat readings for increasing (or decreasing) load (1)
max 5
graph of mass added/force against extension (1)
F m
(1)
or
e
e
Fl
correct use of data in E =
where A is cross-sectional area (1)
eA
gradient gives
[if no graph drawn, then mean of readings
and correct use of data to give 2max) (1)
max 2
13
The Quality of Written Communication marks are awarded for the quality of answers to this
question.
(b)
(i)
for steel (use of E =
e=
Fl
Fl
gives) e =
(1)
EA
eA
125  2
(1)
2.0 1011  2.5 10 7
= 5.0 × 10–3 m (1)
(ii)
extension for brass would be 10 × 10–3(m) (or twice that of steel) (1)
end A is lower by 5 mm  (allow C.E. from (b)(i))
max 3
[16]
12.
(a)
(b)
(i)
the Young modulus: tensile stress/tensile strain (1)
(ii)
maximum force or load which can be applied without wire being
permanently deformed
[or point beyond which (when stress removed,) material does not
regain original length] (1)
graph:
suitable scale (1)
correct points (1) (1)
best straight line followed by curve (1)
(i)
Thornleigh Salesian College, Bolton
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35
(ii)
indication of region or range of Hooke’s law (1)
(iii)
(use of E =
Fl
)
Ae
values of F and e within range or correct gradient (1)
to give E =
6.7
16
.

3
4  10
8.0  10 8
(1)
= 3.3(5)  1010 Pa (1)
(c)
(i)
work done = force  distance (1)
= average force  extension (= ½Fe) (1)
[or use work done = area under graph
area = ½ base  height]
(ii)
energy stored =
6.7  4  10 3
2
8
(1)
= 13.(4)  103 J (1)
4
[14]
13.
(a)
(b)
correct plotting of points (1) (1)
increasing load graph (1)
decreasing load graph (1)
(initially) the material/wire obeys Hooke’s law
[or behaves elastically] (1)
up to the limit of proportionality (1)
(beyond this), elastic limit is reached (1)
undergoes plastic deformation (1)
undergoes permanent change (1)
reference to Hooke’s law obeyed as load decreases (1)
4
max 4
The Quality of Written Communication marks were awarded for the quality of answers to this
question (2).
(c)
(d)
Fl
F l
gives E =
)

Ae
e A
46
gradient = (e.g.)
(1) (= 1.095 × 104)
3
4.2  10
3
E = 1.095 × 104 ×
= 1.2 × 1011 (1) Pa (1) (1.17 ×1011 Pa)
2.8  10 7
(E =
3
area under the graph at any given point
[12]
14.
(a)
(b)
(c)
(i)
(35 × 9.81) = 343 N
mg
) = 172 N (1)
2
 (8.26  10 3 ) 2
d 2
area of cross-section (=
)=
= 5.36 × 10–5 (m2)
4
4
(ii)
tension in each cable (=
(i)
moments about T2, (cable B) gives
5.52 T1 (1) = 343 × 2.76 (1) + 196 × 4.52 (1)
 1833 
 (1) (= 332 N)
T1 = 
 5.52 
(ii)
T1 + T2 = 343 + 196 = 539 (N) (1)
T2 = 539  332 = 207 N (1)
Thornleigh Salesian College, Bolton
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36
(allow C.E. for. value of T1, from (i))
[or moments about T1 gives 5.52 T2 = (343 × 2.76) + (196 × 1.) (1)
T2 =1143/5.52 = 207 N (1)
6
[12]
15.
(a)
(b)
tensile stress: (stretching) force (applied) per unit cross-sectional area
tensile strain: extension (produced) per unit length (1)
(1)
2
Hooke’s law (or stress  strain) obeyed up to point A (1)
A is limit of proportionality (1)
elastic limit between A and region B (1)
region C shows plastic behaviour or wire is ductile (1)
region B to C wire will not regain original length (1)
beyond region C necking occurs (and wire breaks) (1)
max 5
QWC
[7]
16.
mass
volume
(a)
density =
(1)
(b)
(i)
volume of copper =
(ii)
mb (= 5.0 +1.7) = 6.7 (kg) (1)
(allow C.E. for values of mc and mz)
6.7
b =
= 8.4 × 103 kg m–3 (1)
0.8  10 –3
(allow C.E. for value of mb)
[or b = (0.7 × 8900) + (0.3 × 7100) (1) = 8.4 × 103 kg m–3
1
70
× 0.8 ×10–3
(= 0.56 × 10–3 m3)
100
(volume of zinc = 0.24 × 10–3 m3)
mc (= cVc) = 8.9 × 103 × 0.56 × 10–3 = 5.0 kg (1)
(4.98 kg)
30
mz =
× 0.8 × 10–3 × 7.1 × 103 = 1.7 (kg) (1)
100
(allow C.E. for incorrect volumes)
(1)]
max 4
[5]
17.
(a)
extension proportional to the applied force (1)
up to the limit of proportionality
[or provided the extension is small] (1)
Thornleigh Salesian College, Bolton
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(b)
(i)
8 × 9.81 = 78 (5) N (1)
(allow C.E. in (ii), (iii) and (iv) for incorrect value)
(c)
(ii)
78.5  2.5
(use of E = F l gives) 2.0  1011 =
e
Ae
2.8  10 7
–3
e = 3.5 × 10 m (1)
(iii)
similar calculation (1)
to give AS = 5.6 × 10–7 m2 (1)
[or AB = 2AS (1) and correct answer (1)]
(iv)
(use of energy stored = ½Fe gives) energy stored = ½ × 78.5 × 3.5 × 10–3 (1)
= 0.14 J (1)
7
(i)
end A is lower (1)
(ii)
= ½ 3.5 × 10–3 = 1.8 × 10–3 m (1)
(1.75 × 10–3 m)
(1)
2
[11]
18.
(a)
(b)
maximum force (from graph) = 1840 (N) (±100 N) (1)
1840 ( N)
force  
max stress  
(1) (for correct denominator) (1)

 contact area  550  10 – 6 (m 2 )
= 3.3 × 106 N m–2 (1)
4
using shoes without cushioning:
impact time would be less (1)
maximum impact force would be greater (1)
area under the curve the same (1)
3
[7]
Thornleigh Salesian College, Bolton
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19.
(a)
F
A
with F and A defined (1)
tensile stress: force/tension per unit cross-sectional area or
tensile strain: extension per unit length or
the Young modulus:
(b)
(i)
(ii)
(iii)
ES =
e
with e and l defined (1)
l
tensile stress
(1)
tensile strain
3
FS l
E
F
F l
(1) and EB = B
(1) hence S = S
A e
A e
EB
FB
ES
= 2 (1)
EB
F = 2FB (1)
FS + FB = 15 N (1) gives FS = 10 N
[or any alternative method]
F l


gives 
E 
Ae


10  1.5
F l 
e= 
(1)
 =
1.4  10 – 6  2.0  1011
 A E
= 5.36 ×10–5m (1)
6
[9]
20.
(a)
(b)
Hooke’s law: the extension is proportional to the force applied (1)
up to the limit of proportionality or elastic limit
[or for small extensions] (1)
(i)
(use of E =
2
80  0.8
F l
gives) es =
(1)
Ae
2.0  1011  2.4  10 6
= 1.3 × 10–4 (m) (1) (1.33 × 10–4 (m))
80  1.4
= 4.7 × 10–4 (m) (1) (4.66 × 10–4 (m))
11
6
1.0  10  2.4  10
total extension = 6.0 × 10–4 m (1)
eb =
(ii)
(c)
m = ρ × V (1)
ms = 7.9 × 103 × 2.4 × 10–6 × 0.8 = 15.2 × 10–3 (kg) (1)
mb = 8.5 × 103 × 2.4 × 10–6 × 1.4 = 28.6 × 10–3 (kg) (1)
(to give total mass of 44 or 43.8 × 10–3 kg)
44  10 3
(1)
8.5  10 3  2.4  10 6
= 2.2 m (1) (2.16 m)
(use of mass = 43.8 × 10–3 kg gives 2.14 m)
7
(use of m = ρAl gives) l =
2
[11]
21.
(a)
tensile stress: (normal) force per unit cross-sectional area (1)
tensile strain: ratio of extension to original length (1)
Thornleigh Salesian College, Bolton
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39
(b)
(c)
(i)
loading:
obeys Hooke’s law from A to B (1)
B is limit of proportionality (1)
beyond/at B elastic limit reached (1)
beyond elastic limit, undergoes plastic deformation (1)
unloading:
at C load is removed
linear relation between stress and strain (1)
does not return to original length (1)
(ii)
ductile (1)
permanently stretched (1)
[or undergoes plastic deformation or does not break]
(iii)
AD: permanent strain (or extension) (1)
(iv)
gradient of the (straight) line AB (or DC) (1)
(v)
area under the graph ABC (1)
E
e
Max 9
Fl
(1)
Ae
75  3.0
3.8(3) mm(1)
2.8  10 – 7  2.1  1011
2
[13]
22.
(a)
(b)
(i)
the extension produced (by a force) in a wire is directly
proportional to the force applied (1)
applies up to the limit of proportionality (1)
(ii)
elastic limit:
(iii)
the Young modulus: ratio of tensile stress to tensile strain (1)
unit: Pa or Nm–2 (1)
the maximum amount that a material can be
stretched (by a force) and still return to its original
length (when the force is removed) (1)
[or correct use of permanent deformation]
(i)
length of wire (1)
diameter (of wire) (1)
(ii)
graph of force vs extension (1)
reference to gradient (1)
gradient = E
5
A
(1)
l
[or graph of stress vs strain, with both defined
reference to gradient
gradient = E]
area under the line of F vs e (1)
[or energy per unit volume = area under graph of stress vs strain]
6
[11]
Thornleigh Salesian College, Bolton
40
23.
(a)
(=2 × 38) = 76(m)
 c 3.0  10 8 
  3.9(4) MHz (1)
f  
76 

(b)
(i)
angle between cable and horizontal
1
12 

=  sin –1   59  (1)
14 

T= 110cos59° = 57N • (56.7N) (1)
(allow C.E. for value of angle)
(ii)
cross-sectional area (= (2.0 × 10–3)2)
=1.3 × 10–5(m2) (1)
(1.26 × 10–5(m2))
57
 tension
stress =  
(1)

–5
 area  1.3  10
= 4.4 × 106Pa (1)
(4.38 × 106Pa)
(use of 56.7 and 1.26 gives 4.5 × 106 Pa)
(allow C.E. for values of T and area)
(iii)
breaking stress is  65 × stress
copper is ductile
copper wire could extend much more before breaking
because of plastic deformation
extension to breaking point unlikely
any three (1)(1)(1)
7
[8]
Thornleigh Salesian College, Bolton
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