Calculating the Young's Modulus of Copper and Constantan Wire

Aim
===

The aim of the investigation is to calculate the young's modulus of
copper and constantan wire, using five different diameters.



Prediction
==========

I predict that the stress Vs strain graph for copper and constantan
will look like: -

* E - the elastic limit, above this point Hooke`s law (where stress
and strain are proportional to each other) can now longer be
applied.

* P - the plastic limit, above this point the material no longer
returns to its original shape once the force is removed. e.g. the
point a will contract online the dotted line parallel to the
original slope.

* D - the breaking stress, where the wire finally snaps.


[IMAGE]

I predict that the copper and constantan wires will follow the same
pattern, as constantan is an alloy of copper, it is normally 60%
copper and 40% nickel.



Background information
======================

Young's modulus is given by tensile stress divided by tensile (the
direction it is in tension) strain. E = tensile stress / tensile
strain Young's modulus E is a constant for the same range over which
Hooke`s law can be applied. Stress is the force per unit cross
sectional area on a wire, and hence has units of N m-2 or Pascal's, it
is a measure of the strength of the material. Strain is the extension
divided by the original length, and hence has nounits, it measures the
length at which the material will stretch.

Young's modulus (E) measures a material's elastic response in tension;
it is normally measured from the stress/strain curve. It establishes
the strength of a material. In metals and ceramics E is constant for
the material, which doesn't change much with alloying.



Method
======

Ceiling support


Apparatus
---------

· Micrometer screw gauge

· Searles apparatus.

· 100g masses

· 1m rulers

· Wire clippers

· Sand tray

· Stop watch

· Pliers



[IMAGE]Diagram
==============

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[IMAGE]

Test weight

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Method
======

Firstly the wire that was being tested was cut out to a length of just
over 1m, two wires of the length were cut out. The reason why they are
cut just over 1m is because the wire has to be attached to the
equipment and this takes up some of the wire. Before the wires are
attached to the equipment the diameter of the wire was measured,
measurements were taken at three different points of the wire and an
average was taken, this was to make sure that throughout the wire was
the same diameter. It is important to take the diameter of the wire as
it is needed to work out the area of the wire. The area of the wire
needs to be calculated to work out the stress of the wire
(Force/area). The reference wire and the test wire was then hung on
the bracket attached to the ceiling, this is where the wire was
measured to 1m, it was important that the test wire was the same
length throughout the experiment, so that the extension of the wires
could be compared to one another and it would also be better for
working out the strain of the wire. As the wires were not entirely
straight test weights were added to the reference wire and the test
wire to help straighten out the wire, the test weights that was added
was 500g (approximately 5N). The test weights were added for 2 minutes
and this was timed using a stopwatch. After these 2 minutes was up the
spirit level was level and the reading on the vernier scale was
recorded. Then 200g were added to the test wire and not the reference
wire, as the reference wire was there to show how much the test wire
had stretched. The 200g was added for 2 minutes, the reason this was
done as the wire may not have reacted to the weight being added
straight away, therefore decided to time the amount of time the
weights were added so that the experiment was kept fair and accurate.
After the 2 minutes were up the spirit level was levelled up again and
then new reading on the vernier scale was recorded and this then gave
us the extension of the wire. This was repeated with the adding of
weights 400g, 600g, 800g and 1000g. To eliminate any inaccuracy each
wire was tested 3 times, this is so any errors where averaged out. The
extension of the wire is needed to calculate the strain of the wire
(extension of wire/original length)

This experiment was repeated with all the different thickness for
copper and constantan wires. As we are using weights and the wires
could snap at any time it is important to take into considerations the
safety aspects. In case the wire does snap goggles should be worn and
also a trap of sand should be place underneath the weights so that if
the weights do fall they fall into the tray of sand instead of on our
toes or on the floor.



Fair Test
=========

To make the experiment as fair and accurate as possible only one
variable should be used. The variables in this experiment: -

-Thickness of wire: the thicker the wire the more force needed to
extend the wire and vice versa

-Temperature: the temperature affects the wire as if the wire is warm
it will extend further where as a colder wire will not make the same
extension and is likely to snap quicker than a warm wire.

-Load: the more weights that are added to the wire the more extension
that occurs

In the experiment I am testing how much the wire extends by varying
the weights, but I am also changing the thickness of the wire. To make
it a fair test I changed one variable at a time.

To make the experiment accurate I am using searles apparatus. This is
where the scale for measuring extensions is on the reference wire. If
the test wire extends as weights are being added to it the scale moves
with it therefore measuring the extension. The extension of the wire
is measured using a vernier scale and this is used to measure tiny
extensions accurately and is accurate to 0.1mm. If the temperature
changes and makes the test wire expand or contract the reference wire
changes in the same way. The constantan and copper wires diameters can
be measured using a micrometer screw gauge. This is accurate to
0.001mm.



Analysis
========

Graph 1: This line graph of stress & strain for copper after a load of
1.96N shows that after a load of 1.96N the plastic limit hasn't been
reached, this region is the point where the material no longer returns
to its original shape once the force is removed. This means that if
the force of 1.96N were to be removed the copper wire would return to
its original length of 1m. The reason that the wire can return back to
its original length is because the no dislocations in the ploystalline
structure have occurred. Also after a load of 3.92N (graph 2) and
5.88N (graph 3) the plastic region hasn't been reached. As the plastic
limit hasn't been reached it shows that Hooke's law can be applied,
this is where stress and strain are proportional to each other.

Graph 4 & 5: These line graphs of stress & strain for copper after a
load of 7.84N & 9.8N shows both the elastic (above his point Hooke's
law can no longer be applied) and a plastic region. We can tell that
the elastic limit of the copper wire was at 81MP on both graphs, this
is found from reading of the graph where the straight line ends and
the curve begins to start. This also shows the yield stress for
copper. The yield point is a point just after the elastic limit this
is where there is a big change in the material and this is due to a
distorting force. The bonds between molecular layers break and layers
flow over each other. After this point the material starts to become
plastic, as it has reached and exceeded its elastic limit. The reason
that the material can no longer move back to its original shape once
the force is lifter is due to dislocations in the polycrystalline
structure of the metal moving, the dislocations move along by a
process called slip. All the atoms are in a crystalline arrangement
and during the elastic area they are stretched and elongated until
they reach their elastic limit when they will begin to slip against
each other and permanently deform. As the wire extends the crystalline
structure stretches and begins to dislocate and the dislocations slip
until the structure is perfect and there are no dislocations left.

The young's modulus of copper can be seen in table 7. It would have
been expected to find the young's modulus of the copper wire of
different wires to be the same however looking at the table it can be
seen that it does not. This therefore shows some errors in the
experiment. By taking an average of the results the I have found the
young's modulus of copper to be 20917.09 Mpa.

Graph 6, 7 & 8: This line graph of stress & strain for constantan
after a load of 1.96N, 3.92N & 5.88N shows that after a load of 1.96N,
3.92N and 5.88N the plastic limit hasn't been reached. As the plastic
limit hasn't been reached it shows that Hooke's law can be applied,
this is where stress and strain are proportional to each other.

Graph 9 & 10: This line graph of stress & strain for constantan after
a load of 7.84N & 9.8N like graph 1 shows both the elastic and a
plastic region We can tell that the elastic limit of the constantan
wire was at 85MP, this is found from reading of the graph where the
straight line ends and the curve begins to start. This also shows the
yield stress for constantan.

It can be seen that the young's modulus varies quite a lot this shows
that there have been some errors in the experiment, which will be
discussed later.

The young's modulus of constantan can be seen in table 14. The young's
modulus has been calculated for each wire separately, the reason for
this is because in theory the young's modulus for copper whatever the
diameter of the wire should be the same, however looking at the table
it can be seen that the youngs' modulus is not the same throughout the
table. Looking at part of the table: -

Young's modulus (Mpa)

1.96

3.92

5.88

7.84

9.8

7699.798978

7699.798978

8662.273851

9726.061867

10043.21606

21276.94528

10638.47264

9118.690836

9118.690836

9386.887625

21460.61282

30402.53483

36483.04179

32429.37048

36483.04179

84181.12692

59421.97195

47351.8839

41231.57237

43541.9622

41917.35127

44383.07781

43529.55708

42210.4796

41917.35127

Taking an average of the young's modulus for constantan we get
30012.63 Mpa.

Looking at the graphs and the elastic limit and young's modulus of the
two wires it can be seen that they are quite similar. This is to be
expected, as constantan is an alloy, which usually mainly consists of
about 60% copper, this would therefore mean that it would have similar
properties. It will not have the same properties as copper as it is
also usually contains 40% nickel, it would then also have some of
nickel properties. Also as mentioned above the young's modulus of a
metal doesn't really change much with alloying. By comparing the two
graphs it can be seen that constantan has a steeper graph and the
steeper the graph to stiffer the material, therefore constantan is
slightly stiffer that copper. Also by looking at the young's modulus
that has been worked out it can be seen that constantan has a bigger
young's modulus then copper. In physics "modulus" means a measure of
the extent that a substance processes some property. Young's modulus
describes how well an abject retains its length when stretched or
compressed. It is also a measure of stiffness. Therefore as copper has
the smaller value of young's modulus it means that copper is more
easily stretched then constantan, since less stress is needed to
achieve a given strain.

On the graphs I have included error bars. The reason that I have
included error bars is because the metre ruler has an error of 0.1mm
and also an error of 0.01mm on the veriner scale and the micrometer
screw gauge. These bars then take this error into account.

Looking at the graphs it can be seen that there are some anomalies
this could have been due to many factors. Errors that could have taken
place include the errors in the equipment and the other random
systematic errors that can occur. Also there could have been parallax
errors (this is an error which occurs when the eye is not placed
directly opposite a scale which a reading is being taken), reading
errors this is when errors come about when guess works is involved in
taking a reading from a scale when the reading lies between the lines.

Another error could have been creep; this is where the wire is
stretching when no more weights have been added. As the wire is
stretched the diameter of the wire decreases and before the wire snaps
plastic deformation takes place. This reason this happens is because
metals such as copper and constantan are ductile and they can have
large plastic deformations without fracturing. It happens because
atoms move as the plastic deformation in the crystal structure move to
a place of lower stress. This causes the wire to become thinner as the
atoms are moving away from the stress part. The stress then increases
because the cross-sectional area has decreased. This increases the
ductile flow and so the metal yields and gets thinner and thinner.
Once plastic deformation starts, atoms will continue to flow without
any increase in stress. .

Another cause for inaccuracies may be due to the fact that the 1m
length of test wire may not have been accurately measured out. As the
vernier scale measures in mms, but the test wire was measured out
using a metre ruler and this is not accurate to 1mm.

Improvements that could be made to the experiment would be use a
longer test wire as it would be easier to measure the strain, for
example, if we used a sample ten metres long we would have a ten times
better resolution, making our results much more accurate. We would
then need to ensure that the weight of the 10M wire was very much
smaller that the weight we added to induce strain.

Also to further improve the experiment I would test more different
diameter of wire as the there is a gaps in-between the points on the
graph where we had to predict what would happen there it would be more
accurate to test more wires with different diameters in-between the
points. I would also add weights at smaller intervals instead of 200g
jumps, this would help in making the graphs more accurate, therefore
allowing us to read of the graph more accurately getting better
readings.

Aspects of the experiment that were good were, the vernier scale is
sensitive and the maximum extension, which could be read off it, was
dependant on how much force is applied. This method is much more
accurate than the clamp and pulley method which could have been used.
The experiment was done in the same classroom and was done at room
temperature as the temperature of the wire reduced the accuracy of the
results as the wire stretches more under warm conditions because the
wire is less stiff then if it was in cold temperatures.