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:

. The accuracy is limited by the measurements of the diameter and length of the wire.

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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

Hooke's Law ( -where L is the length of the rod and A is the cross-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

Mass

(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 and

we obtain the equation for the linear fit,

y=mx+c, where y is the distance and x is the mass of the load.

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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.

The final result for the slope and the intercept is m=-0.665 and c=9.985

To quantify how good is the fit, Chi-squared is calculated using the next

formula:

Chi-squared for different number of measurements

Number of measurements Chi-squared

10 15.359

9 13.153

8 11.706

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0 1 2 3 4 5 65

6

7

8

9

10

11Plot 2: mass vs. distance

Mass(kg)

Dis

tanc

e(m

m)

data 2

linear

0 1 2 3 4 5 65.5

6

6.5

7

7.5

8

8.5

9

9.5

10Plot3 mass vs. distance

mass(kg)

dist

ance

(mm

)

data 2

linear

Only the last value for chi-squared is compatible with the rule-of thumb, , where ; 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 using the following method:

We can now calculate the constant k, using the first two data of the first table and knowing that where are the distances calculated from the

floor to the weights. Therefore, and

The date used to determine Young's modulus is listed in the table below:

Mass (kg)

(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

,where m' and c' are the slope and intercept of the linear fit, M represents the mass.

Also,

=> m' is calculated using the same formula as for the previous linear fit and we obtain

m'=0.00067, therefore For the uncertainty we calculate the uncertainty of the slope

and to calculate the uncertainty for E we use the formula of

the propagation of errors with multiple parameters:

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The result of the uncertainty is

Final result:

References:www.wikipedia.orgMATLAB, Version 7

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