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# An investigation into the bending of a cantilever beam and an attempt to determine the mathematical concepts involved.

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Introduction

An investigation into the bending of a cantilever beam and an attempt to determine the mathematical concepts involved.

## Aim

• To determine the mathematical constants involved, in this case the values of (k) and (n).

## Apparatus

• String
• Weights
• Three one meter rulers

## Mathematical theory

• The figure shows the basic structure of the cantilever bean that we used for the experiment. There are forces that act on the beam these have been called F1, F2, F3.  The depression of the cantilever is given by:

x = Kln

• Where x is the value of depression. l is the normal straight length of the beam, and k is the proportionality constant. The following equation can be obtained from the above one.

ln1 = lnx + lnk

## Procedure

• A depression of 20cm is required, which is why you have to find a weight which will cause this successfully. The values of l and x should be measured carefully. The beam should be reduced until a reasonable amount of reading have been achieved. The values for l and x should be recorded each time new readings are found out.

Middle

50

3.91

7

1.95

28

40

3.69

4

1.39

28

30

3.40

2.5

0.916

28

## This is the reading for the first beam

Gradient of the best fit line         = 0.49/1.25

= 0.392

Error in best fit line                 = 3.69- 4.25

1.39- 3.00

= 0.348

Conclusion

colspan="1" rowspan="1">

2.67

9

50

3.91

8.5

2.14

9

40

3.69

5

1.61

9

30

3.40

2.5

0.916

9

## These are the results for the second beam

Gradient of the best fit line        = 1.25/3.25

= 0.385

Error in the best-fit line        = 3.69- 4.03

1.61-2.67

= 0.377

Total error in graph= 0.007

The average value of n                = 0.385 + 0.392

2

= 0.389

The average value of k                = 3.05 + 3.10

=3.075

## Conclusion

• The value that we obtained for n was ≈ 0.389 ± 0.051, and the value of k was ≈3.08 ± 0.025. These two values are directly proportional to the load increase.

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