Elastic potential energy = 1/2 * stretching force * extension
(J) (N) (m)
Rearranging: 2Ep / F = Δx
This means we can work out the extension another way and see how much energy there is in the copper wire.
Apparatus
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G-clamp: to hold down the wire on the wooden block
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2 metre rulers in mm: to measure the wire length and the changes
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Level horizontal table: so it doesn’t affect the experiment by adding more force to the load
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Sellotape: make a pointer to see the change in wire size
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Smooth pulley with clamp: so no friction can affect the force, F = μR
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Load of masses: at least 1kg of mass to test the wire
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Copper wire: I will be testing the stretching and so finally the young’s modulus on it
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Micrometer: to accurately measure the diameter of the wire to 0.001mm
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Vernier caliper: to accurately measure the tiny extensions in the length of wire to 0.1mm
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Tri-square: measure the downward movement at a 90° angle
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Goggles: to protect the eyes from the wire
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Gloves: to protect your hands as the thin wire can cut through skin
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Wooden block: to be horizontal with the pulley, so the angle cannot affect the experiment
Stress, strain & young’s modulus
Stress: the load acting on an object per unit cross-sectional area.
This is measured in Pascal’s, and can be though of as pressure acting on an object. This is not the case, this is because stress occurs inside a solid and pressure occurs on a surface.
Stress, σ = force, F (N)
Cross-sectional area, A (m^2)
For e.g. the copper wire, the thinnest part of it would be more likely to stretch, so the thinner the wire the greater the tensile strength.
‘The ultimate tensile stress is the measure of strength.’
This is when the material is likely to break so can no longer stretch, but will not be testing this in my experiment.
Strain: the extension per unit length produced when an object is stretched or squashed. This has no unit because it is a ratio.
Strain, ε = Extension, Δx (m)
Original length, L (m)
For e.g. if there were two wires different lengths, everything else same, the longer wire would be under more strain.
‘It stretches by the same fraction of its original length.’
Young’s modulus: this is the ratio of stress to strain in a material when it is stretched, provided Hooke’s law is obeyed.
Young’s modulus, E = Stress, σ (Pa)
Strain, ε
Method
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Gather all apparatus in one place safely, then setup the apparatus up like diagram shows.
- Measure the length of wire on the metre rulers, ensuring the wire is taught and straight along the rulers. Measure the diameter along the wire, at least in three different places (as the wire may not be the same everywhere). Place the sellotape pointer on the wire, any where as long as it is against the rule and take own these results.
- The wire may have to be long to see a significant change in extension; however the temperature may affect the length. I suggest that a preliminary experiment take place to work this out.
- the weights should be put on at intervals of a 100g = 1 Newton and then the tiny extensions should be measured with the vernier calipers and the other metre used with the tri square to measure the extension also from the original. To get the strain. When placing down weights at the end of the wire’s knot they will have to be removed to check that the elastic limit has not been exceeded.
- The stress can be measured using the micrometer at every 100g intervals, also check that the copper wire isn’t being pulled from the clamp.
- These steps can be repeated at least twice for accuracy, however time is a factor. The wire copper can be changed for testing different wires.
I will be testing the wire with masses of 0N-15N or until the wire breaks before 15N, I will also then take only results of extension and diameter in between these masses.
The affect of a longer wire being tested will mean more expansion can take place due to the extra atoms able to move inward and become thinner, thus expanding the wire. The affect of a smaller wire in length will mean less expansion because there is less area for the wire to become thinner.
This works also for the diameter of the wire, because the wider the wire there is more area for it to become thinner during expansion, for thinner wires there is less area so the wire will break under less mass, compared to the thicker wire.
Yield point
The reason behind the yield point in the copper wire is to do with the crystalline structure or lattice, where the atoms are in a uniform pattern at the original point.
The more atoms in the structure there is then the denser it is.
When there are faults or less atoms in a particular place then it is less dense, this means if pressure were to be placed on these points the wire would get thinner and thinner, until it would be more susceptible to snapping or breaking. This is exactly what happens to the copper wire the more faults in the wire then it is more likely to break.
The diagram shows the atoms in the metal slipping against each other, up or down left or right, or even forward and backward. The middle atoms can edge dislocates, which means they up against each other distorted.
If there are a lot of impurities this will cause unbalance of the atoms up against each other and so the space between means less dense so easier to stretch and snap. These impurities make the metal harder and brittle.
These can affect my results so there may be anomalous results due to the impurities.
Copper can be described as polycrystalline, because it is made up of tiny metal crystal grains. Heating the metal and cooling it rapidly, is called ‘quenching’, so this results in smaller grains. This means the grain cannot move a lot until the ‘grain boundary’, which is the edge of the crystal grain; this would make the metal hard and brittle. So in this experiment if the metal was ‘quenched’ the metal would hardly expand so the young’s modulus would be affected.
If a metal were cooled slowly, large crystal grains would form, this process is called ‘annealing’ so this makes the metal malleable + tough so the copper wire would expand more than normal and so affects the young’s modulus.
Variables & constants
My variables are going to be:
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Length of wire: everything else will be the same when testing this, about 1.35-1.40 m when testing every thing else.
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Masses: “, this will be at least up to 1 kg for every thing else
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Type of copper wire: “, this will be the diameter (0.32 & 0.27) mm
My constants are going to be:
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The resistance of the pulley: this can be controlled by adding grease, this because the more smooth the pulley is then the less resistance from it can be passed to the wire’s expansion.
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The taughtness of the wire: I can not keep this the same because the elastic limit or the yield point maybe changed inadvertently which mean the length of wire will be affected. The more taught something is the straighter it is and so is less length, than some relaxed slightly diagonal material.
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Temperature: I cannot keep the temperature the same or constant, because some regions of heat will be different to others due to movement of the air which I cannot help. Also the room temperature is likely to be from 19°C-21°C. If the temperature were any higher the experiment would be affected because the copper wire can expand, due to the increased movement of atoms in the copper as the kinetic energy transfers to it.
Preliminary experiment
In my preliminary experiment I learned that I needed a wire which could withstand the weight of at least 1kg, but also would effectively extend enough to see the significant change. I decided to use the wires with a diameter of 0.31mm and 0.27mm.
I also decided to use at least 1kg of weights because some parts of the wire may not be all the same so they could be thicker and hold more weight or thinner not take that much weight at all. I think the temperature will be almost the same always, so there is no need to worry about it.
I will need to repeat the experiment twice for each wire for greater accuracy,
Specialist Apparatus
These are equipment, which make the experiment’s results more accurate and in turn the young’s modulus.
Uncertainties
I had uncertainties because, I could not control or keep constant the temperature, so the wire may have extended or contracted. To calculate the strain I had to measure very small extensions of less than 1mm, this was quite difficult, because I did not have specialist equipment to measure length so I had to rely on human decisions, which meant there were always going to be errors. I also had to calculate the stress, therefore I had to use a micrometer to measure the diameter of the wire
Analysis
The table shows that more mass or Newton’s means that there will be more extensions to the original length. To get young’s modulus I used y = mx+c so I could get an average of it for each ruler. I used the equation: stress over strain to get young’s modulus, which was the gradient in the graph.
E.g. 4.11E+03/5.82E-04=7.06E+06
To get the other equations, I used theory from secondary data, which is above in the background information.
To get my results I used specialist equipment; such as the micrometer, which is accurate to 0.01 of a millimeter, the uncertainty is 0.005mm. I also used a metre ruler with mm marking for accuracy, the uncertainty is 0.5mm.
The graphs show that stress is proportional to strain as the points are near the line of best fit, also going through the middle. My graphs show that stress is proportional to strain, also stress is proportional to force and strain is proportional to the extension.
As you can see from my prediction I drew a graph of stress against strain and this is correct because as you can see from the real graph it basically follows the same curve.
From my stress-force and strain-extension graphs I do not need to draw error gradients because they are so accurate and because their R2 value is very close to 1.
Trend line: A graphic representation of trends in data series, such as a line sloping upwards to represent the average. Trend lines are used for the study of problems of predictions, also called regression analysis.
R-squared value: An indicator from 0 to 1 that reveals how closely the estimated values for the trend line correspond to your actual data. A trend line is most reliable when its R-squared value is at 1 or near 1. It is also known as the coefficient of determination.
Evaluation
The uncertainty of the extension is 0.01cm
(0.01/1.7)*100 = 1.7%
The uncertainty of the length of wire is 0.001m
(0.001/1.760)*100 = 0.05%
The uncertainty of the diameter of the wire is 0.01mm
(0.01/0.31)*100 = 3.1%
To ensure I had a safe experiment I wore safety goggles, also setup the experiment in the centre of the table.
I made sure that the clamp stand was firmly placed on the floor so that it wouldn’t wobble and affect the results taken down.
I tried to keep my eye level in line with the marker measurements to rule out parallax error.
I took many results down to have accurate results and averaged them.
The reason for the line of best fit not going through the origin there may have been due to systematic error. This may because there was friction on the pulley, to remedy this problem grease could be used. Also the ruler was not long enough for the whole wire to be measured so the 2 rulers may be disjointed, so to remedy this problem I would need a longer ruler. Also the taught wire may not be horizontal to the pulley when tied to the clamp so the wire is longer than it can be measured, to solve this problem I used a wooden block, but it wasn’t enough.
The main two measurements that contributed to young’s modulus were the diameter and the extensions as they were used to calculate the stress and strain.
