Finding Young's Modulus

This sample essay on Finding Young’s Modulus provides important aspects of the issue and arguments for and against as well as the needed facts. Read on this essay’s introduction, body paragraphs, and conclusion.

Introduction Constanton is a copper-nickel alloy mainly used in the for its electrical resistance properties. It has a high resistance which is constant over a wide range of temperatures. I am going to find out the Young’s modulus of this wire and observe its behaviour.

Apparatus  Constanton Wire  G-Clamp x2  Pulley Hanging weights  Ruler  Micrometer  Small marker flag Wooden end blocks  Sponge Blocks Underlying Theory When a sample is deformed by a force, the deformation is proportional to the magnitude of the force. This is shown by Hooke’s Law where: Force is equal to a stiffness constant (k) times the extension (e).

The force is proportional to the extension. For a sample we can also calculate stress and strain: Where stress is equal to force (F) divided by area (A) and strain is equal to extension (e) divided by original length (l).

When you plot these on a Stress-strain graph it proves Hooke’s law when it is straight line but as soon as the graph curves, the sample is showing plastic deformation as it is past the elastic limit. Using this graph we can work out the Young’s Modulus of a sample which is: This is also measured in Nm-2 or Pascal’s (Pa). It can also be calculated by working out the gradient on the stress-strain graph.

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Young’s Modulus Practice Problems

When a wire obeys Hooke’s Law it deforms elastically. This means that when the load is removed, the wire returns back to its initial length. The atoms in the wire move small distances from their equilibrium positions but then return. After the elastic limit the wire starts to deform plastically. The atoms move within the structure so they cannot return when the load is removed. Measurements Throughout the experiment these measurements will need to be taken and observed:  Stress – Force and surface area  Strain – Initial length and the extension  Young’s modulus  Percentage error – error of each piece of equipment.

Hooke’s law (F=ke) Method To measure the Young’s modulus of constanton I will: 1) Set up the equipment as shown. 2) Choose a suitable section of wire from the real that doesn’t appear bent, twisted or deformed. Measure the diameter of the wire with a micrometer before attaching it to the weights. 3) Attach a marker flag so the extension can be measured. 4) Start the experiment by measuring the initial length of wire and adding the 100g weights and measuring the new length each time. 5) Record your results in a table and plot a stress-strain graph using these results. Weight (g) Mass (N).

Length (mm) Stress (Nm-2/Pa) Strain 6) Repeat the experiment three times or until you get a set of similar results. Results Experiment 1 In the first attempt at calculating the youngs modulus of constanton i used 0. 44mm diameter wire with an initial length of 500mm. I measured both in millimetres because this would avoid converting units when calculating the strain of the wire (e/l). The wire only extended by 1mm when 1700g were added to it so I abandoned the experiment and changed my method slightly to get more extension for mass. Experiment 2 I changed the diameter of wire used to 0.

23mm which is almost half the thickness than before. By using thinner wire we should see more extension for the amount of weight added so we can measure it with a ruler more easily. The initial length of wire was also 500mm. When i carried out the experiment the wire proved to be too thin because as only 500g was added the wire started to show rapid plastic deformation and continued to extend by roughly 6% (30mm) of its original length before the wire broke. Experiment 3 I changed the diameter again so I could record more conclusive results. I used a diameter of wire in between the diameters of the first two experiment (0.31mm) and an initial length of 500mm.

I still couldn’t record too accurate results as the wire didn’t extend enough so I could only plot three points on a graph before it showed plastic behaviour. Further experimental changes were needed. Experiment 4 This time I changed the initial length of wire used to 800mm from 500mm. This would amplify the extension so I could measure it with the ruler because the rate of extension would increase and also the amount of extension would increase. By increasing the initial length of wire it would also decrease the percentage error in the measurement of the wire with the ruler.

The percentage error goes from 0. 1% to 0. 063%. Experiment 5 This was a repeat to check the accuracy of experiment 4. In this experiment i encountered a few problems. The knot holding the weight hangers on kept slipping and the results found did not match the pervious pattern. Experiment 6 This was my third repeat of experiment 4. This gave me a fairly similar set of results to experiment 4. Due to time restrictions, no more experiments could be carried out to do a third repeat. Calculations Using the diameter to work out the surface area.

Let x = diameter X 10-3 = to change from millimetres to metresi? 2 = to change diameter into radius Then substitute it into the formula for the area of a circle. Change grams into Newtons for force. Which is equivalent to i? 10  Changing Pascals (Pa) into Megapascals (MPa)  Working out gradient to find the Young’s Modulus. Graphs To plot the graphs i only plotted points where the wire extended by a millimetre because the wire was extending between those points but I could not take sensitive enough measurements with a ruler. To plot the graphs i also changed Stress from Pascals (Pa) to Megapascals (MPa) to make it easier to plot on the graph.

I also used the graphs to work out the Young’s Modulus of the Constanton by finding the gradient of the graph before it reached the elastic limit. Inaccuracies Here are some factors that may have caused some inaccuracies in my measurements: * The wire may contain impurities that change the way the wire behaves. This cannot be helped. * By attaching a pointer you can affect the sample by restricting the way it behaves. To avoid causing too many inaccuracies use as thin a pointer as possible so there is as little as possible touching the sample.

The pulley wheel may cause friction but this is the most sensible way of converting horizontal movement into vertical.  There also may be bends or variation in cross sectional area in the wire. To minimise the risk of this, don’t use the first few metres of wire until you find a section that looks roughly undamaged.

Percentage Errors The main source of percentage error is in the measurement of the diameter taken by the micrometer even though the micrometer is accurate to i? 0. 005mm and the ruler is only accurate to i? 0. 5mm. In experiments 4, 5, and 6: % error of diameter = [ i?0. 005 / 0. 31] x 100 = 1. 6% % error of length = [ i? 0. 5 / 800 ] x 100 = 0. 06% Other sources of percentage error are: – Diameter of the wire which is an example of uncertainty in the measurements. – Actual mass of the weights which is an example of systematic error.

Conclusion Using experiments 4 and 6 I was able to work out my young’s modulus of Constanton by finding the gradient of the initial straight part of my graph. Experiment 4 = 280GPa Experiment 6 = 240GPa The real value of the young’s modulus is 162GPa so I am out by approximately a factor of two.

This is not too far away from the true value considering the huge uncertainties involved with my measurement technique. To improve my accuracy I would either have to improve my measurement techniques or change my method completely. In conclusion, the method was affective for demonstrating the affects of Hooke’s law but not for measuring accurately the young’s modulus of constanton. Modifications in the Method  Attaching the pointer to the pulley stops the pointer coming into contact with the sample of wire which could obstruct deformation but if the wire extends more than the pulley can measure then the experiment will not work.

Illuminate the pointer to produce a magnified shadow of the movement. This makes it easier to see movement and allows for more accurate measurement however you need to calculate and calibrate magnification.  Use wire that isn’t wound round a real because it distorted the start point of my curve. A typical young’s modulus curve starts at the origin but mine doesn’t because first few hundred grams was used to apply tension to the wire to bend out the curves.