Making Sense of Data: Young’s Modulus Of a Metal and An Alloy

Results: - COPPER

FORCE (N)

Extension (mm)

Extension (mm)

Extension (mm)

Average Extension (mm)

0

0

0

0

0

2

0

0

0

0

4

2

.33

6

3

2

3

2.67

8

4

3

3

3.33

0

5

4

4

4.33

2

6

5

5

5.33

4

7

5

5

5.67

6

9

6

6

7.00

8

1

7

0

9.33

20

6

0

2

2.7

22

22

5

45

27.3

24

96

32

40

89.3

26

BROKE

43

BROKE

28

BROKE

CONSTANTAN

FORCE (N)

Extension (mm)

Extension (mm)

Extension (mm)

Average Extension (mm)

0

0

0

0

0

2

0

0

0

0

4

2

.33

6

3

.67

8

4

2

2

2.67

0

4

3

3

3.33

2

5

3

4

4.00

4

5

4

5

4.67

6

6

4

5

5.00

8

8

4

5

5.67

20

8

4

6

6.00

22

8

5

7

6.67

24

8

6

9

7.67

26

9

7

9

8.33

28

0

7

1

9.33

30

1

9

2

0.7

32

4

0

4

2.7

34

8

5

7

6.7

36

38

5

20

24.3

38

47

20

30

32.3

40

78

34

67

59.7

42

82

94

92

89.3

44

37

07

BROKE

46

51

67

48

BROKE

BROKE

The average extension will be divided by 1000 to convert the units into (m).

Average Extension taken to 3 significant figures.

CALCULATIONS: -

FORCE (N)

STRESS (Pa)

0

0

2

.86x107

4

3.72x107

6

5.58x107

8

7.44x107

0

9.30x107

2

.12x108

4

.30x108

6

.49x108

8

.67x108

20

.86x108

22

2.05x108

24

2.23x108

COPPER: Diameter of Wire = 0.37mm = 0.00037m

Radius of Wire = 0.00037/2=0.000185m

Area of Cross Section of wire = ?rxr

0.000185x0.000185x? = 1.08x10-7 m

Stress is calculated by the formula: Force/Area

The stress in the table on the left is done by the force given divided by the area calculated above: (1.08x10-7 m )

The units used for Stress are left to 3 significant figures as previous measurements are to the same magnitude.

FORCE (N)

EXTENSION (m)

ORIGINAL LENGTH (m)

STRAIN (ratio)

0

0

2.1

0

2

0

2.1

0

4

0.00133

2.1

6.35x10-4

6

0.00267

2.1

.27x10-3

8

0.00333

2.1

.59x10-3

0

0.00433

2.1

2.06x10-3

2

0.00533

2.1

2.54x10-3

4

0.00567

2.1

2.70x10-3

6

0.00700

2.1

3.33x10-3

8

0.00933

2.1

4.44x10-3

20

0.01267

2.1

6.03x10-3

22

0.02733

2.1

.30x10-2

24

0.08930

2.1

4.25x10-2

The Strain is calculated by doing Extension / Original length

The Young's Modulus is calculated by finding the gradient of the preceding graph.

The extension given above was converted to meters by dividing the average extension for copper by 1000.

CONSTANTAN:

FORCE (N)

STRESS (Pa )

0

0

2

2.08x107

4

4.16x107

6

6.24x107

8

8.32x107

0

.04x108

2

.25x108

4

.46x108

6

.66x108

8

.87x108

20

2.08x108

22

2.29x108

24

2.49x108

26

2.70x108

28

2.91x108

30

3.12x108

32

3.33x108

34

3.53x108

36

3.74x108

38

3.95x108

40

4.16x108

42

4.37x108

Diameter of wire = 0.35mm = 0.0035m

Radius of wire = 0.0035/2 = 0.00175 m

Area of cross section of wire = ?*r*r

?x0.00175x0.00175 = 9.62x10-6 m

Stress = Force/ Area

The Stress was achieved by using the formula above. I used the corresponding force and divided it by the cross sectional area (9.62x10-6 m ).

The units are to be taken to 3 significant figures again.

BELOW is a table calculating the strain of Constantan. The formula required to obtain the strain:

Extension / Original Length

FORCE (N)

EXTENSION (m)

ORIGINAL LENGTH (m)

STRAIN (ratio)

0

0

2.1

0

2

0

2.1

0

4

0.00133

2.1

6.35x10-4

6

0.00167

2.1

7.94x10-4

8

0.00267

2.1

.27x10-3

0

0.00333

2.1

.59x10-3

2

0.00400

2.1

.90-3

4

0.00467

2.1

2.22x10-3

6

0.00500

2.1

2.38x10-3

8

0.00567

2.1

2.7x10-3

20

0.00600

2.1

2.86x10-3

22

0.00667

2.1

3.17x10-3

24

0.00767

2.1

3.65x10-3

26

0.00833

2.1

3.97x10-3

28

0.00933

2.1

4.44x10-3

30

0.01067

2.1

5.08x10-3

32

0.01267

2.1

6.03x10-3

34

0.01667

2.1

2.65x10-3

36

0.02433

2.1

.16x10-2

38

0.03233

2.1

.54x10-2

40

0.05967

2.1

2.84x10-2

42

0.08933

2.1

4.25x10-2

Graphs:

Analysis:

Evaluation: The accuracy of the measurements obtained will now be examined. I have set myself a 3% maximum in accuracy. The uncertainty of the measurements taken should be within 3% and below are the calculations to question whether the readings satisfy the requirements.

STRESS:

Area of Wire: Smallest measurement = 0.35x10-3m.

Micrometer measures to 0.005x10-3m

0.000005/0.00035x100= 1.43%

Error multiplied by 2 as error is squared. 1.43*2= 2.86%

Error = 3% Satisfactory

N.B:- The smallest measurement was taken to allow me to gain a figure of the largest possible error.

Another problem would be that the area across the wire could easily vary across length.

Measurement of force: Each 100g mass, they are ? 3g.

3x2 = 6g as 200g was used.

Therefore, for an assumed 2N force = 200-6 = 194g

Force = 0.194 (kg) x 10 = 1.94N

2-1.94 = 0.06N 0.06/2*100 = 3% error. Which is acceptable, however gravitational acceleration is assumed to be 10ms-2. A more accurate figure would be 9.8ms-2 ?

0.194*9.8= 1.90g 2.00-1.90 = 0.1g

0.1/2*100 = 5% error, which is not an acceptable range of error for an A-level experiment.

STRAIN:

Error of Extension: - The unit measurement of extension is 0.5mm.

Smallest extension: Largest extension:

1mm 167mm

0.5/2x100 = 25% 0.5/167x100 = 0.30% - negligible

As you can see, the range of accuracy varies tremendously. I will now calculate the measurement where there is a 3% maximum error.

3/100 = 0.03 0.5/0.03 = 16.7mm ? 17mm

More than 1/2 the measurements taken were under 17mm therefore another error was found in the experiment undertaken.

Error in Original length: -

Measurement taken is 2.10m

Unit measurement is 0.005m

0.005/2.10*100 = 0.24% - Negligible.

The errors tackled above are on reflection to the equation: Stress / Strain. Now I will look upon the errors made in processing the results.

The average was taken of the results and were put to the appropriate degree of accuracy (3 significant figures). The only possible source of error would be the actual graphs:

The experiment as a whole was reliable to a satisfactory standard. There were uncertainties in the experiment, however I believe they were compensated by other measurements that had errors, which were negligible. The experiment was repeated 3 times and results and measurements were averaged. There could have been many unconsidered factors such as kinks in the wire. They are like notches in the wire, when a force is exerted and the wire is extended, the notches are like anomalies in the wire, as they do not extend. The wire should uniformly extend however the force will untangle the notches rather than extend the wire. See diagram:

Another problem was the fact that the area of the cross section could vary across the length; this however could be a minor error. It could be eliminated, as when taking the average diameter reading of the wire, it should be taken three times at both ends of the wire. A 2-meter wire was used rather than 1 despite making the calculations slightly more difficult. When measuring the extension, it will now be easier to read as the extension should be larger due to the fact the wire is longer. The longer the wire, the more accurate the measurement. However this would have made the experiment impractical. Another problem would be the scale of the graphs sketched,