Understanding Young's Modulus in Flexible Material

Categories: Data

Essay, Pages 9 (2070 words)

Views

1415

I have picked the first method out of the three options of experiments to conduct based on the flexion of a cantilever. I now have to decide on a method of collecting and processing the data for the first method, taking care to reach a value for Young modulus with some estimate of accuracy attached to it.

Method 1: Wood (metre rule)

Diagram

Apparatus

1. 2 Measuring rulers (1m each)

2. Drawing Pin

3. 9 Weights (50g each, totalling 450g)

4. Approximately 50cm of string

5. G-Clamp

6. Clamp Stand with clamp

7. Screw gauge with a sensitivity of 0.1mm (Micrometer)

8. Vernier Calipers with a sensitivity of 0.2mm

The Micrometer

Vernier Calipers - read the sliding scale along the top and bottom

Variables

The variables that will be kept constant are the length of the overhang of the ruler, the position where the ruler is clamped and the position on the ruler where the weights are hung. The only variable that will change during the experiment is the amount of weight that is hung on one end of the ruler to measure the different deflection of the ruler at different heights.

The weights that are hung on one end of the ruler will vary each time by adding 50g to the previous weight and each time the deflection of the ruler is read until 450g of weights have been added.

Method

Arrange the apparatus as shown in the diagram. As the apparatus is fixed appropriately we can then start the experiment. As the apparatus is fixed we might see the ruler has a slight bend without any weights on it this is due to its own weight, this can be counted as a systematic error.

Now I am going to start adding 50g weights to the ruler. The weights will be added from 50g till it becomes 450g, and every time we add 50g we must measure the depression at the end (deflection) of the ruler. This can be done by measuring the initial position of the ruler and measure the deflection from the initial position to the bend position of the ruler with a ruler, to give the bend or deflected height of the ruler.

As the weights are added to the ruler the side of the ruler where the weights are hung is under tension and the other side is in compression. Now we have to measure the depth of the ruler using a screw gauge (micrometer). We also have to measure the breadth of the ruler using the vernier calipers. Since we have the measure of the deflection of the ruler, width, height, length of the overhang of the ruler we can now work out the Young modulus of the ruler.

The relationship for calculating Young modulus in this experiment is:-

E = 4gL3 * M

BD3 Y

Where

E = young's modulus

L = length of the overhang of the ruler (cantilever)

M = mass

Y = depression at the end (deflection of the ruler)

B = width of the ruler (strip)

D = thickness of the ruler (strip)

g = 9.80 Nkg-1

We repeat the above experiment two more times by measuring again the deflection of the ruler with the same weights and from the same side of the ruler. As we have three sets of results we can take an average to give a result that is more accurate.

Similarly I took 5 readings for the thickness of the ruler at any five different places of the ruler by using a screw gauge to give an average thickness value of the ruler. I also took 5 readings for the width of the ruler using the vernier calipers to give me an average width value of the ruler. An average is taken for almost all the reading to minimise errors while calculating the Young's modulus of the ruler.

Intended Readings

The readings that will be taken for this experiment will be the measuring the deflection of the wooden ruler when different weights ranging from 50g to 450g is added in intervals of 50g are added to the wooden ruler. This process will be repeated 2 times as mentioned in the method above. The thickness of the ruler can be measured using a screw gauge. This will be measure 5 times at different parts of the ruler to give an average thickness. Also the width of the ruler can be measured using the vernier calipers. This will also be done 5 times at different parts of the ruler to give an average width. All these readings are taken a number of times to give an average for all the values for the final calculation of the Young's modulus.

Safety Conditions

When the experiment is conducted, it must be insured that all of the apparatus is attached securely to ensure that nothing comes apart. Also check that all the apparatus are in good condition for the experiment to work well. General laboratory rules must be recognised to ensure safety throughout.

As we are trying to prove that, as Hooke's law says if one side is in tension the other will be in compression hence the same way in the experiment one end of the ruler was clamped and weights were hung on the other end hence one side of the ruler was under tension and the other in compression. Also from the preliminary work done and my knowledge from the text book this method proved to work. Hence I adopted this method.

Analysis

Results - Preliminary

Before we went onto the main experiment, we spent out first lesson conducting a preliminary experiment to get a feel of how everything works.

Below is a graph to show Young's modulus of the ruler:-

Results - Actual

Also the value for the width of the ruler was measured using the vernier calipers and this was done 5 times to give an average value.

There were other values taken as well like the thickness (D) of the ruler was measured with a screw gauge 5 times to give an average value.

The length of the overhand was approximately 90cm.

I also used the value of the gravitational force which is a constant 9.80Nkg -1.

Below is the graph to show the nominal mass against the average depression.

After the experiment had been conducted, each 50g mass was weighed individually to see if there was any error created. The following was found:

Nominal Mass

Actual Mass

(Kg)

0.0500

0.0501

0.1000

0.1005

0.1500

0.1507

0.2000

0.2015

0.2500

0.2515

0.3000

0.3016

0.3500

0.3519

0.4000

0.4019

0.4500

0.4519

As can be seen from the above table, the massed used had slight inaccuracies in their value. This would have affected the results slightly and because of this I now need to find out how much the results were affected. To do this I will simply change the Nominal Mass on the following graph to Actual Mass.

Calculation of Young Modulus

To calculate the gradient of the graph the value of y would need to be divided by m (Y/M).

The gradient as calculated from the graph was 3.2475.

Therefore substitute the value of 3.2475 for M/Y

E = 4gL3 . M

BD3 Y

= 4 x 9.80 x 0.93 x 3.2475

0.03024 x 0.00567183

= 28.5768 x 3.2475

5.51753*109

= 5.179276343*109 x 3.2475

= 1.6819699925*1010

= 16.819699925*109 N/m2

From my research on, I found out that the Young's Modulus value for wood is:-

13*109 N/m2

By looking at the actual value and the value obtained from the experiment, it can clearly be deduced that the experiment was accurate enough to obtain a rough Young's Modulus value for wood. In order to gain a value closer to the actual value, the sources of error need to be considered. These errors are either experimental errors or human errors in measurement and through the general conducting of the experiment.

Sources of Error

The experiment was carried out fairly accurately but still there were errors that were consistent and variable throughout the experiment. As the consistent errors were that the screw gauge that I used to measure the width of the ruler had a zero error of -0.01mm. As the ruler was clamped on one end and on the other end the weights were hung there was a ruler that was clamped along with this ruler so that the deflection could be measured but both of these rulers had a bend initially because of its own weight but these deflections were different hence this slightly affected my accuracy of my final result. The variable error was that, the weights were not hung from a fixed position this also could have affected results.

How to Minimise Errors

As for the zero error in the screw gauge was -0.01mm. I improved my results by adding this error onto the readings that I got for the width of the ruler. As the weights were hung at different positions I took three reading for each weight and its deflection so that I could get a more accurate result. Also for the other readings, like measuring the width and the thickness, they were measured a number of times so that the results could be more accurate.

Errors in Calculating Young's Modulus

When calculating the value for Young's Modulus there were errors, although quite minimal, in the constants used in the original equation which calculates the Young's Modulus value. Below I will show the percentage errors for each constant:-

B = 0.001/5.6718 = 0.017%

D = 0.001/3.024 = 0.033%

D3 = 3 x D = 0.099%

L = 0.1/90 = 0.11%

L3 = 3 x L = 0.33%

Therefore the total percentage error from this experiment is equal to B + D3 + L3

= 0.017 + 0.099 + 0.33

= 0.446%

Conclusion and Evaluation

There could be many errors in the experiment that could have affected the accuracy of the final result. First of all, when measuring the deflection of the ruler with certain weights as they had an initial bend due to their own weight. The screw gauge that I used during my experiment to measure the breadth of my ruler had a zero error of -0.01mm and the weights that were hung on one end of the ruler was not hung at a fixed position which could have affected the value for the deflection.

I used a screw gauge to measure the breadth and the vernier calipers, which I used to measure my depth of the ruler with. The screw gauge I used had a sensitivity of 0.1mm, and the vernier calipers had a sensitivity of 0.2mm.

Systematic errors in the experiment were; the zero error in the screw gauge, the initial bend of the ruler due to its own weight, hence this had an impact on the measurement of the deflection throughout the experiment.

Random errors in the experiment were; the weights hung on different positions at one end of the ruler. The overhang of the ruler slightly varied as different sides of the ruler used to find out the deflection with the same weights to give an average value for the deflection of the ruler.

Measurements taken on the ruler had some errors and they were the initial bend of the ruler due to its own weight affecting the deflection, weights hung on different positions on one end of the ruler affecting the deflection. Also the slight variation in the length of the overhang of the ruler when the experiment was repeated was another error factor.

Screw gauge used had a zero error of -0.01mm and it had a sensitivity of 0.1mm.

Vernier calipers used had a sensitivity of 0.2mm.

The experiment was carried out as accurately as I could and according to my result and the graph I think that there were no anomalous results in my experiment.

Throughout the experiment the same apparatus was used even for the repetitions. The only thing that varied during the repetitions was that I used different sides on the ruler to check that the deflections at each side were almost the same.

The method used for the above experiment was quite reliable. As the apparatus used were the same throughout the experiment.

To improve the method by which I carried out my experiment will be to use a screw gauge, which does not have a zero error. Fix a point on one end of the ruler for the weights to be hung so that there are no errors while reading the deflection of the ruler. Use a fixed length for the overhang. Initially see that the ruler does not have bend, if it does use a similar ruler with the same amount of bend length to place it next to the ruler on which weights are hung to get an accurate reading for the deflection.