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Determining the Young's Modulus of Steel Sonia Baci School of Physics and Astronomy The University of Manchester First Year Laboratory Report Oct 2009
Abstract: Young's modulus for steel was determined by using a steel wire hung to a beam attached to the laboratory ceiling. A series of weights with increasing mass were attached to the wire, causing it to stretch downwards. The final result is: 11 2 2.09 ± 0.03 ×10 N / m . The accuracy is limited by the measurements of the
diameter and length of the wire.
-1-
Introduction: Young's modulus (E) is a measure of the stiffness of an elastic material. It is also known as the Young modulus, modulus of elasticity or elastic modulus. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which F ∗ L -where L is the length of the rod and A is the cross EA
Hooke's Law ( L =
sectional area of the rod that is tensioned by a force F) holds. This can be experimentally determined by hanging different weights to the rod. Description of the experiment: To determine the Young's Modulus for steel, we use a steel wire of length 1m and a diameter of 0.30 mm with an uncertainty of 0.005 mm. We hang the wire by a beam attached to the ceiling of the laboratory and then attach to the wire a series of weights. Each time, the distance from the floor to the weight is measured.
Data obtained in the measurement of Young's Modulus Mas s (kg)
Distance (mm)
0.5
9.70
1.0
9.26
1.5
8.96
2.0
8.64
2.5
8.38
3.0
8.00
3.5
7.62
4.0
7.36
4.5
7.04
5.0
6.60
5.5
6.00
6.0
5.22
When a load of 6.5 kg was attached, it broke the wire. First, we calculate the linear fit to check if there are any incompatible data. ∗−∗ Using the formulas m = and 2 ∗ −∗ 2 ∗ −∗ c= we obtain the equation for the linear fit, 2 ∗ −∗ y=mx+c, where y is the distance and x is the mass of the load.
-2Plot 2: mass vs. distance 11 data 2 linear
10
) m m ( e c n a t s i D
9
We notice that the last two data do not fit to our line, which means they need to be excluded from the measurements used to calculate the Young's modulus.
8
7
6
5
0
1
2
3 Mass(kg)
4
5
6 Plot3 mass vs. distance 10 data 2
9.5
linear
9
The final result for the slope and the intercept is m=-0.665 and c=9.985
8.5
) m m 8 ( e c n 7.5 a t s i d
7
6.5 6 5.5
To quantify how good is the fit, Chi-squared
1
2
X
2
3 mass(kg)
4
is calculated using the next
2
N
formula: X
0
i =∑ i 1
2
Chi-squared
2
X
Number of measurements
for different number of measurements Chi-squared
10
15.359
9
13.153
2
X
5
6
8
11.706 -3-
Only the last value for chi-squared is compatible with the rule-of thumb, 2 0.5 X / v 2 , where v = N − p ; N is the number of measurements while p is the number of parameters (in this case, p=2). Young's modulus: First, we need to determine L using the following method: F = k L m 1∗g= k L 1 ; m2∗g =k L 2
k =
g∗m1 −m2
L 1− L 2
We can now calculate the constant k, using the first two data of the first table and knowing that L1 − L 2 =d 1−d 2 where d 1 and d 2 are the distances calculated from mi∗g the floor to the weights. Therefore, k =11.136 N / m and Li = k
The date used to determine Young's modulus is listed in the table below: Mass (kg)
L (m)
0.5
0.000285
1.0
0.000725
1.5
0.001025
2.0
0.001345
2.5
0.001605
3.0
0.001985
3.5
0.002365
4.0
0.002625
4.5
0.002945
5.0
0.003385
L =m ' ∗ M c ' ,where m' and c' are the slope and intercept of the linear fit, M represents the mass. g∗ L ∗m Also, L = F ∗ L / AE which can be written as L = AE => m ' = g∗ L / AE => E = g∗ L / m ' A m' is calculated using the same formula as for the previous linear fit and we obtain m'=0.00067, therefore E = 2.09 ∗1011 For the uncertainty we calculate the uncertainty of the slope 1
m' =
∗1
N
2
< x
2
> −
2
∗
and to calculate the uncertainty for E we use the formula of
the propagation of errors with multiple parameters:
-4-
E = E ×[ A / A m '/ m' ] 2
2
2
11
The result of the uncertainty is E =0.03 ×10 N / m Final result: E = 2.09 ± 0.03 ×1011 N / m 2
