Experiment to determine the Young’s modulus of an aluminium cantilever beam and the uncertainties in its measurement 1. Abstarct: The young’s modulus E, is a measure of the stiffness and is therefore one of the most important properties in engineering design. It is a materials ratio between stress and strain: E=σε

Young’s modulus is a unique value for each material and indicates the strength of that material as well as how it will deform when a load is applied.

2. Introduction: The Young’s Modulus can only be derived experimentally, there are no theoretical methods by which the young’s Modulus of a material can be calculated therefore in this experiment our aims were: * To calculate the Young’s modulus ,E of Aluminium from measurement of the end deflection of cantilever beam of aluminium loaded at its free end * To assess the accuracy and precision of this method by comparing the calculated value of E to the known value Eal=72.6 GPa * To measure the deflected shape of the aluminium beam for one loading condition (15N) and to compare this with the theoretical prediction of the beam bending theory for deflection of a cantilever

yx= PL32EIxL2-13xL3

3. Materials and Methods

The apparatus shown below was set up and the following equipments were used: * Dial gauge was used to measure deflection of the beam

* Magnetic clamp stand (not to affect the bending of the beam) * Solid Aluminium Beam

* 15 Weights(1N each)

* Clamp to keep it still at one end.

* Steel base

4.Results

1. Load/Deflection behaviour :Equation to calculate the young’s Modulus from the slope of deflection vs. load graph E= 4L3bd3(slope)

We measured the length (L) the width (d) and the breadth (b) of the beam 5 times and then calculated the average: Average Length/mm| Average width/mm| Average breadth/mm| 998| 25.28| 15.73|

The uncertainties of the slope, length, breadth and width were estimated using the rage method: δx=xmax-xminN

The fractional uncertainty of the length was estimated using: Fractional uncertainty=δxx, but in this experiment we will be using the standard error which is δx. The uncertainty of the meter ruler used to measure L is ±0.5 mm, the uncertainty of the vernier calliper used to measure the width and breadth is ± 0.005 mm. Then using the range method we calculated the uncertainty for L: ΔL=Lmax-Lmin5(number of measurements)= ±0.4mm, using the same method we calculated Δd = ±0.004 mm and Δb=± 0.01 mm. We decided to use the highest uncertainties of the measurements, hence obtaining the values L= (998±0.5) mm, d= (25.3±0.01) mm, b= (15.73±0.005)mm The results we obtained are the following:

Load(N)| D(avg)| Error in D(avg)|

0| 0| 0|

1| 0.57| 0.01|

2| 1.13| 0.01|

3| 1.68| 0.01|

4| 2.24| 0.01|

5| 2.79| 0.01|

6| 3.41| 0.02|

7| 3.95| 0.01|

8| 4.53| 0.003|

9| 5.09| 0.01|

10| 5.66| 0.03|

11| 6.27| 0.02|

12| 6.8| 0.01|

13| 7.38| 0.01|

14| 7.96| 0.01|

15| 8.53| 0.01|

Where Davg is the average deflection calculated using 3 different readings. The error was calculated using the range method. The values used for the minimum and maximum tredlines used to calculate the average gradient: Load/N| max| Min|

1| 0.56| 0.58|

15| 8.54| 8.52|

Slope= 0.57+0.56712=0.569

Uncertainty in slope=0.57-0.5671=±0.002

The following graph was obtained by plotting the average deflection against the load

The average gradient=0.569±0.002

E= 4L3bd3(slope)=69.8GPa

Uncertainty in E using partial differentiation:

∆E= L2slope. bd312∆L+ 4Lb∆b+ 12Ld∆d+ 4Lslope∆slope= 0.20GPa Hence E= (69.8±0.20)GPa , the actual value of Eal=72.6GPa Comment: The error bars due to uncertainties in the measured deflection are very small as we can see both from the graph and the uncertainty of the slope.

2. Deflected shape of beam: Deflection measurements were taken...

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