Calculating the young’s modulus of copper wire

Metallic bonding The young’s modulus measures how stiff a material is, I will be calculating the young’s modulus of copper wire. I will initially do a preliminary experiment, then an improved experiment in which I will try to reduce sources of error that affected my initial results. I will aim to calibrate equipment and reduce uncertainty in the experiment.
Physics behind experiment: http://www.bbc.co.uk/schools/gcsebitesize/science/images/gcsechem_60.gif
Young’s elastic modulus is related to the stiffness of a materials individual bonds. Stronger bonds generally result in higher stiffness; copper has relatively strong bonds due to its metallic bonding, (strong interactions between the metal ions and sea of delocalised electrons) so it has a fairly high young’s modulus value of 110-128 GPa. I will be calculating the stiffness of a copper wire, the calculation for Young’s modulus (stiffness) is: stress/strain or force x cross-sectional area/extension x length. The force per unit area is called stress and the fractional change in length is called strain.
Independent variable- force http://www.cyberphysics.co.uk/graphics/equations/young_modulus2.gif
Dependent variable-extension.
Control variables-
Cross-sectional area
Temperature
Material of wire
Equipment

G clamp, 10 cm jaw
2 wooden blocks
single pulley on a bench clamp
metre rule(s) or tape measures
micrometres
adhesive tape to make a marker
cardboard bridges
hanger with slotted weights, 1 N and possibly 0.1 N wire samples
safety goggles

Method

Straighten the metal wire and fix it horizontally along the bench
Attach a marker to the wire 2m from the clamp and about 50cm from the pulley. The marker will line up with zero on the metre rule, so the extension can be measured.
Wrap the wire several times tightly around the weight hanger and bind it to itself
Measure the original diameter of the wire or line with a micrometre. Find the cross-sectional area using A= π(d/2)^2. Estimate the percentage uncertainty in diameter.
Now with a small load (1N) put the wire under gentle tension to straighten out any kinks that are present in the wire.
Increase the load gradually in steps of 1 N, until the wire or line snaps. Record the load and the corresponding extension in each case.

How I reduced uncertainty
Metre sticks
The metre sticks used may not be completely accurate, so to reduce error I compared the metre sticks. I did this by comparing the millimetres (spacing) in the middle of the metre stick as the ends are sometimes worn. I did this with several metre sticks and used the most consistent one this improves the reliability and accuracy in the experiment.
Weights
We used weights with a mass of 100g. However the weights were either damaged or inaccurate so to reduce uncertainty I weighed them on a balance, I found all of the weights we inaccurate so I recorded the results and used their actual value to avoid affecting my final results.
Parallax error: There may have been a reading error due to the angle viewing extension; to avoid this I positioned myself directly above the wire.
Area of wire: I used a micrometre to calculate the diameter then I used this value to calculate the area (A= π (d/2)^2), this reduces uncertainty as it can measure much more accurate on a much smaller scale than a ruler. It may have been the wire was damaged in places so I took several readings along different parts of the wire and took an average to reduce the uncertainty in area which would have affected my overall results.
Hazard risk Control measures
Emergency action

When wire breaks it may hit someone.
Low
Safety goggles must be worn.
Make sure people are spaced out.
Medical attention if needed

Explanation and expectation of results
This graph shows the Young’s modulus of a ductile metal e.g. copper and how it is expected to behave. The initial straight line illustrates the elastic region, this is because strain is proportional to stress up to a limit. The elastic limit means once the load is removed from the material, it will return to its original shape this is because the material hasn’t been permanently stretched. In this part of the graph the ratio of stress: strain is constant and equal to young’s modulus of the material. Then the graph starts to curve illustrating the yield point, the bonds between molecular layers break and layers flow over each other. After the yield point plastic deformation occurs, usually in metals this is due to the dislocations, the stress then increases because of strain hardening; this is when a metal is strained beyond the yield point, to produce additional plastic deformation, the metal becomes stronger and more difficult to deform. The reason for strain hardening is: the average distance between dislocations is increased and they starts blocking the motion of each other. Finally the graph reaches its ultimate tensile strength (the material fractures).
I will only be using the proportional part of my graph, as this is used to calculate the young’s modulus. Once the metal has reached the elastic limit (the graph starts to curve) the metal has been permanently changed so this part of the graph would not give a true value of the young’s modulus since the gradient is not constant. http://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Stress_Strain_Ductile_Material.png/450px-Stress_Strain_Ductile_Material.png
This is what I am expecting to see in results, I will analyse my graphs and comment on whether they follow this trend for the young’s modulus of a ductile metal.
Analysis of Preliminary experiment (Experiment 1)
Strain is proportional to stress up to a limit, this is the initial straight section of my graph this obeys Hooke’s law. In this part of the graph, the ratio stress/strain is constant and equal to the young’s modulus of the material, here the material behaves elastically. It behaves elastically at this point because there is stretching between the individual bonds of the material so once the load is removed the material will return to its original shape. Once the material has exceeded the yield point plastic deformation occurs which I have illustrated on my graph; this involves the breaking of a limited number of atomic bonds by the movement of dislocations. The movement of dislocations allows the atoms in crystal planes to slip past one another. When the load is removed the material will not go back to its original shape because the material has been permanently changed. Finally my graph reached its ultimate tensile strength (the material fractures).
Analysis of secondary graph (experiment one)
This part of the graph is only the elastic region, I used this graph to obtain a young’s modulus value for copper as the gradient remains constant as the relationship between stress and strain at this point is directly proportional. I obtained a value for young’s modulus of 90 GPa, the actual value is between 110-128 GPa.
Improvements made to reduce uncertainty
After completing the preliminary experiment I had a better insight of where sources of error may have been coming from, therefore I made the following improvements to my second experiment.
Markers-I had a suspicion that the wire was slipping between the wooden blocks, so in my second experiment I placed a marker at the end of the wire adjacent to the wooden blocks therefore if the wire slipped I could record this, by doing this uncertainty in the extension is reduced.
Length- In my second experiment I increased the length of the wire, the young’s modulus would theoretically be the same as it is a property of the material. By increasing the length the percentage uncertainty is decreased therefore the uncertainty in length is decreased.
Wire: There may be kinks in the wire, to avoid error I tried to reduce the number of kinks without actually damaging the wire. To do this I straightened the wire out first by adding a very small mass of weights to the wire until the kinks were removed. I did this before measuring extension as I wouldn’t have been measuring the actual wires extension; this helped to reduce uncertainty in the extension.
Knot- in the first experiment I tied a knot at the end of the wire, it may be the knot weakened the wire and caused the wire to snap prematurely, to reduce uncertainty I will Wrap the wire several times tightly around the weight hanger and bind it to itself.
I also used the same methods to reduce uncertainty in the initial experiment.
Analysis of (improved experiment)
There is an elastic region which is illustrated by the initial straight line this is because strain is proportional to stress up to a limit. In this part of the graph the ratio stress/strain is constant and equal to young’s modulus of the material. Plastic deformation occurs when the graph begins to curve; I have made it clear on my graph where this process is occurring. This is due to the movement of dislocation of which I have mentioned above. The stress then increases because of strain hardening; which I have also explain above. Finally my graph reached its ultimate tensile strength (the material fractures).

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Analysis of secondary graph (improved experiment)
Stress is proportional to strain in this graph as only the elastic region is used, this is before the material has been permanently changed. I obtained a value of 100 GPa for the young’s modulus of copper. This result is better than the one I obtained in my initial experiment as the value is closer to the true young’s modulus of copper (110-128 GPa).
Conclusion
My first result titled graph 1 follows the trend of a typical metal, showing an initial straight line (elastic region). Then the graph starts to curve and the extension begins to increase rapidly, the material also shows signs of necking (the cross-sectional area decreases). There were no anomaly results. The secondary graph of my first results shows the elastic region of a stress/strain graph. From this graph I obtained a young’s modulus value of 90 GPa, my value was relatively lower than the actual value of the young’s modulus of copper. I think the main reason for the error in my value is because of the knot I tied to attach the weight hanger, when the wire snapped it snapped at the knot and so may have snapped prematurely. However only the initial elastic region is needed to calculate a value for young’s modulus so my results shouldn’t have been affected, unless the wire showed more extension due to the weakness of the wire and thus obtaining a lower value. In my first experiment there was little error in force and area, the main source of error seemed to be coming from the extension. There was little that could be done to reduce uncertainty in extension as the smallest measurement I could obtain was ½ of 1 mm. There was also no sign of systematic error.
In my second experiment my graph titled graph 2, again shows an initial straight line, however it increases at a much more proportional rate that my original experiment. This shows there is less error in my results as it follows the trend for the young’s modulus of a metal more accurately. The graph then starts to curve and there is a much larger values of extension recorded due to the movement of atoms via dislocations through the lattice. There is less extension in my second result, this may have been due to the fact I bound the wire to itself instead of tying a knot at the end of the wire to attach the weight hanger (wire wasn’t weakened). This may be why I obtained a higher value for the young’s modulus of copper (100GPa) which is closer to the true value. I also removed kinks out of the wire, and attached a marker to the end of the wire to see if it had slipped, this would have significantly reduced uncertainty in the extension. I also used a longer wire which would have reduced the percentage uncertainty in length, overall the uncertainty in strain (e/L) was significantly reduced as is shown in my results table. Strain was the main source of error in my experiment, so I focused mainly on trying to reduce error in this area, I think this was accomplished due to a more validated value for the young’s modulus and the lower values of uncertainty recorded in strain.
Sources
http://en.wikipedia.org/wiki/Deformation_(engineering) – image of youngs modulus graph
http://www.revisesmart.co.uk/physics/materials/stress-strain-and-the-young-modulus.html – young modulus graph.
http://www.bbc.co.uk/schools/gcsebitesize/science/add_aqa_pre_2011/atomic/differentsubrev5.shtml – diagram of metallic bonding