# Physics Investigation into the bending of a Cantilever

Categories: Physics

==> Although safety in this investigation is not a paramount concern, one should still be careful as rulers can cause eye injuries and damage breakable equipment in the lab.

Introduction

==> This is an investigation into the bending of a cantilever, conducted by changing a variable effecting the deflection of the cantilever when clamped to a table. I have chosen to change the deflecting force, and investigate how a change in this will effect the deflection of the cantilever gradually by adding increasing force to the cantilever.

This variable was chosen as it has been shown to be the most reliable and give the most scientifically viable results that can be easily analysed and have sensible conclusions drawn from them, further explanation is also provided in the ‘variables’ section.

Prediction

==> I predict my results will show with an increasing deflecting force the deflection will increase. I predict that proportionality will also occur between the independent and dependant variables, in the way that if the deflecting force doubles, deflection will also.

One way in which this can be shown is using the following formula:

y = 4Fl3

bd3E

Deflection is shown as y on the left of the equation, and F the deflecting force is on the right with the other variables.

==> On the presumption all other variables are kept constant, F and should be proportional to y and double when it is doubled and so on. Using this formula a quantitative prediction can be made, which will strengthen my original prediction.

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F = 2 F = 4 F = 8

y = 4 x 2 x 1 y = 4 x 4 x 1 y = 4 x 8 x 1

1 x 1 x 1 1 x 1 x 1 1 x 1 x 1

= 8 = 16 = 32

1 1 1

y = 8 y = 16 y = 32

Doubling F therefore shows y to be proportional as it also doubles proving my prediction to be correct.

==> This prediction was based on scientific knowledge and theories such as Hooke’s law. Hooke’s law states that the length of a spring stretches is directly proportional to the force stretching it and therefore if the length doubles the force must double also. This theory has been applied to the cantilever and therefore the deflection such as a spring is proportional to the deflecting force such as a force that would be stretching the spring. The deflection is also related to scientific knowledge that when materials are bent atoms on the top of a material are in tension and being pulled apart, and atoms on the bottom are being compressed and pushed together. Providing neither tension nor compression is too great the elasticity of the ruler should withhold the deflecting force obey Hooke’s law.

Variables

y = 4Fl3

bd3E

y – Deflection

F – Deflecting force

l – Length of cantilever

b – Width of cantilever

d – depth or thickness of cantilever

E – A constant called the Young’s modulus this is the fixed for any material

==> The dependent variable that I’m investigating is deflection and I will measure this in centimetres, as it’s convenient and is the appropriate scale for the deflection in this situation. The independent variable I’ve chosen is the deflecting force, chosen because it will give the best results, which can be analysed easily. This variable will be measured in Newton’s, as this is the most appropriate measure of force in this case. F has the simplest relationship with y, as y is simply proportional to F and therefore will give results in whole numbers and have a reasonable scale also. All other variables must therefore be kept constant in order to conduct a fair test and ensure results are accurate.

==> A variable such as l was not chosen as it has a more complex relationship with y and therefore if this were a variable there wouldn’t be many results one could use, as y is proportional to l3. This relationship would result in too much deflection occurring in the cantilever meaning the cantilever would bend too much as tension and compression in the cantilever would be too great causing it to snap, and therefore not obeying Hooke’s law. The workings below show this:

y = 23 y = 43

= 8 = 64

This variable can be kept constant by ensuring the cantilever is clamped securely to the desk in the same place, and not moved during the course of the experiment. This will ensure that l is kept constant and will therefore not interfere with deflection and aid in obtaining accurate results. This must be done so that results are not incorrect and will abide by the prediction.

==> Another variable is the depth or thickness of the cantilever that also has a complex relationship with y, as y is proportional to 1/d3. This relationship would cause too little deflection and would be impractical to use and difficult to measure. The workings below show there would be too little deflection:

y= 1/23 y = 1/43

= 1/8 = 1/64

This variable can kept the same very simply by using the same ruler, and not altering the depth or width in any way. Keeping d the same will allow the deflection to be purely based on F and therefore fair results will be obtained. If the depth or width were altered this could cause inaccurate results to be obtained and affect the analysis of them, which would hinder the conclusion overall.

==> Precise measurements of all variables will be made using the most accurate equipment available to me. The deflection will be measured using a meter rule and measured to the nearest tenth of a centimetre, and taken at eyelevel so I can get a correct reading. This will be done by eye, yet should give accurate results that are suitable for the investigation. The meter rule will also be stuck securely to the floor using blue tack to ensure it doesn’t move and provide inaccurate results. Deflection will be measured by seeing the difference from the original height of the cantilever and the height of the cantilever when various weights are added to it. This Difference is the deflection and will be recorded to the nearest tenth of a centimetre. The deflecting force is measured easily using pre-weighed Newtons and simply adding them to the ruler using a hook and string securely taped the ruler, which prevents movement that may hinder results slightly. The length of the ruler from the table is measured by simply clamping it to the table at an appropriate length, which is decided in the preliminary trials. This will be done to the exact centimetre. The width of the cantilever is kept constant easily by simply not changing it, and therefore no measurements need to be taken.

Preliminary Work

==> These tests were necessary in order to aid in deciding which values for variables were most suitable. The preliminary tests also helped to prove my theory, as they followed the basic pattern I had predicted. The preliminary tests were conducted under fair conditions and only one variable was changed at a time to ensure correct results were obtained. The preliminary testing was conducted as the final experiment would be using the same method and equipment, which can be seen later in this report. The deflecting forces of 3, 6 and 10 Newtons were chosen to show proportionality as 3 doubled is 6, and 10 was chosen to ensure this weight would be suitable to use as the heaviest and wouldn’t break the cantilever. The choices of 50 and 60 centimetres for the length of the cantilevers were chosen as too large a length would cause too much deflection and a smaller length would result in little deflection, and would not give resulted which could be analysed easily.

Deflecting Force (N)

Length of Cantilever and Deflection (cm)

60

50

3

2.5

2.2

6

5

3

10

10

6

==> I chose to use the length of 60cm as this showed the strongest relationship of proportionality, which proves my theory as when 3N doubled to 6N the deflection did also from 2.5cm to 5cm. Deflection doubled also to 10cm when the deflecting force increased to 10N, which still shows a rough proportionality. The cantilever also withstood the force of 10N, which gives evidence that using forces from 1N to 10N would be suitable for the investigation.

==> I received some information that helped me conduct this investigation and decided on various variables form my textbook (PHYSICS for OCR A), ‘The Times – GCSE Physics’ CD-rom and the website www.bbc.com/bitesize from where I was helped with the explanation of Hookes law and about compression and tension.

Method

==> Apparatus used

2 x Meter Rule

Table

Blue Tack

String

Tape

Newtons

Clamp

==> Diagram

==> Fair Test

I shall use a fair set of measurements by ensuring that the deflection is recorded accurately to the nearest tenth of a centimetre. I’ve decided on a suitable range of measurements which include 1N to 10N with intervals of 1N each time, e.g. 1N, 2N, 3N etc, each time the deflection will be recorded. This should give suitable results and allow them to be analysed easily, hopefully showing proportionality between deflection and the deflecting force. A range of 10 measurements should also allow me to draw a graph accurately and show a further pattern between results. This range of measurements will be repeated twice in order for an average to be reached will which provide more precise results. Repeating the tests should also help account for any anomalous results, which can be redone.

==> Step-by-Step

1. Firstly we set up the apparatus as shown in the above diagram, with tape to secure the string to the cantilever to ensure the weights stayed still. A meter rule was attached to the ground also using blue tack.

2. The height of the cantilever was then recorded, in order for the deflection to be later calculated.

3. The weights were then added one by one on a hook connected to the cantilever by string, and the deflection calculated and recorded each time.

4. Step three was repeated, in order to gain two sets of accurate results.

5. The apparatus was then neatly put away and the results were tabulated and averaged.

==> Results

Deflecting Force (N)

Deflection (cm)

1st Test

2nd Test

Average

1

0.8

0.9

0.85

2

1.8

2.0

1.9

3

2.8

3.1

2.95

4

4

4.1

4.05

5

4.9

5.2

5.05

6

6.1

6.7

6.4

7

7.8

7.1

7.45

8

8.9

9

8.95

9

10

10

10.0

10

10.6

10.5

10.55

==> Conclusion

The graph shows that as the deflecting force increases, deflection does also. The Graph also shows that both are proportional to one another, as the line of best fit is a straight line, and goes through the origin. The graph shows as the deflecting force increases from 1N to 2N, the deflection does also from 0.85cm to 1.9cm, which shows deflection approximately doubling, thus proving my predicted proportionality. Further proportionality is shown elsewhere in the graph as when 2N is tripled to 6N, deflection going by the line of best fit also triples from 2.1cm to 6.3cm, this shows that there is a strong relationship of proportionality between these two variables. These results therefore prove my prediction that my chosen variables would be proportional to one another, and abide by Hooke’s law. These results prove Hooke’s law because it states that for a small deformation or displacement to an object, the size of the deformation is directly proportional to the deforming force, this is shown by my results, as Deflection and the deflecting force are approximately equal throughout the tests. The deflecting force also didn’t exceed the elasticity of the cantilever and therefore the results did abide by Hooke’s law.

==> Evaluation

I think my practical went fairly well as I gained accurate results which proved my theory. These results were accurate enough to easily form a scientific conclusion, which was based on scientific knowledge such as Hooke’s law. The accuracy of my measurements was as precise as possible using the instruments and equipment available to me. More accurate equipment would allow more accuracy, yet I felt that the accuracy used was appropriate and gave good results. There weren’t any anomalous results, which reflected the reliability of the method.

I felt the method was reliable as I obtained good results without any anomalous results.

I could improve my method by conducting more repeats of the deflection gained when I increased the deflecting force, this will enable me to gain more accurate averages, and help to identify any anomalous results on the graph. Another improvement I could make to my method is to measure the height of the cantilever more accurately using more precise measuring techniques, or perhaps getting more than one persons judgement. This would help to gain more accurate deflection results.

The results I gained were accurate and did prove my theory and therefore the conclusion I produced was correct, and relevant to the data collected.

To improve this investigation I could use a greater variety of deflecting forces, such as 1.5N and test the elasticity of the cantilever by exceeding 10N. This would give more accurate results and a greater range of results, which would improve my method and investigation.

I could further my investigation by testing another variable, and seeing the affect this has on deflection, and compare this with deflecting force. Although the other variables may be difficult to measure the consistency and accuracy of the equation could be tested by seeing the proportionality of the other variables to deflection.