Hooke's Law / Young's Modulus - trying to find out what factors effect the stretching of a spring.

Hooke's Law / Young's Modulus

I am trying to find out what factors effect the stretching of a spring.

Things, which might affect this, are:
· Downward force applied to spring.
· Spring material.
· Length of spring.
· No. of coils in spring.
· Diameter of spring material.
· Cross sectional area of spring.
I have chosen to look at the effect of the weight applied, as it is a continuous variation.
I predict that the greater the weight applied to the spring, the further the spring will stretch. This is because extension is proportional to load and so if load increases so does extension and so stretching distance.

Extension = New length - Original length

To see if my prediction is correct I will experiment, and obtain results using Hookes Law. He found that extension is proportional to the downward force acting on the spring.

Hookes Law
F=ke
F = Force in Newtons
k = Spring constant
e = Extension in Meters

My method of experimentation will be to use a clamp stand and boss clamp to suspend a spring from. A second boss clamp will hold in place a metre rule starting from the bottom of the spring to measure extension in mm. I will then add weights to the spring and measure extension.
Before deciding on the range of experimentation I carried out a preliminary test to find the elastic limit of the springs we had. To do this I added weights to the spring until it did not return to its original shape. This occurred at 12N and so I set a limit of weight to 10N for my experiments. The reason why 11N was not used is that 10 experiments is a more standard number and also that the limit of proportionality is slightly less than the elastic limit of the spring, therefore using 10N ensure I do not exceed the proportional or elastic limit of the spring. I also hope to carry out the experiment 3 times and also take an average to increase the reliability of my results.
Due to the preliminary test no strict safety precautions need to be used, as the only potential danger would be if the spring snapped, however this will not happen if there is no more than the maximum load on the spring of 10N at any one time. This will also remove the problem of the stand on which the experiment is taking place from falling over.

Experiment Diagram

The weights available are 1N masses and so I will take 10 extension measurements starting at 1N up to 10N of force on the spring.

Results Table
Force in N Test 1 Test 2 Test 3 Average
1 33 31 35 33
2 71 75 70 72
3 114 112 115 113
4 150 148 146 148
5 185 183 180 183
6 215 214 212 214
7 250 249 251 250
8 280 280 285 282
9 320 310 315 315
10 355 350 345 350

All measurements in mm

These results are also plotted on a graph on the next page

Now my graph is plotted I can ...

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Experiment Diagram

The weights available are 1N masses and so I will take 10 extension measurements starting at 1N up to 10N of force on the spring.

All measurements in mm

These results are also plotted on a graph on the next page

Now my graph is plotted I can work out the spring constant. To do this firstly I will need to work out the gradient of my line on the graph.

Gradient (m) = Y2 - Y1
X2 - X1

To find Y2, Y1 etc. I will need to select two points as shown on the graph I can then substitute these points and create the sum:

Gradient = 350 - 72
10 - 2

However in Hookes Law the extension is measured in metres where as I measured in mm therefore I must convert my results from mm to m to do this I need to divide my results by 1000, therefore the sum I need for gradient is

Gradient = 0.35 - 0.072
10 - 2

Gradient = 0.278 = 0.03475
8
The gradient is 0.03475 to convert to spring constant I need to use the following formulas:

y=mx+c <THORN> e = mf <THORN> e = f
m

Hookes Law

F=ke

By comparison

K = 1 <THORN> 1 = 28.7769
m 0.03475

This shows the spring constant is 28.7769 N/m
To check that my spring constant from hooks law was correct I am going t experiment again using Simple Harmonic Motion. If both experiments are correct the spring constant from both experiments will be almost identical.
The Simple Harmonic Motion experiment requires measurements of time over an oscillation of the spring being acted on by different forces.
My method of experimentation for simple harmonic motion will be to suspend the spring as in the first experiment and place on the required weight. I will then stretch the spring a further 15mm and then release the spring and take a time measurement for an oscillation in the spring. This however will be inaccurate due to the response time of starting and stopping the timer, because of this I have decided to time 10 oscillations and then divide the result by 10 this will reduce the experimental error. Only one test is needed to find the spring constant but I will take 5 tests each with a different weight. The equation used uses a weight measurement in Kg so the 1N masses I have available will each be 0.1Kg.

Simple Harmonic Motion

T = 2p m
T = Time in seconds k
M = Mass in Kg
K = Spring constant

We want K for the spring constant so:

T = 4p m
K

KT = 4p m

K = 4p m
T

I will now carry out the tests and use the above formula to work out the spring constant and compare it to my results from the Hookes law experiment.

Simple Harmonic Motion Test Results.

Mass Kg Time Time / 10 T
0.1 3.79 0.379 0.143641
0.2 5.47 0.547 0.299209
0.3 6.48 0.648 0.419904
0.4 7.58 0.758 0.574564
0.5 8.46 0.846 0.715716

I can now use T and work out the spring constant.

K = 4 x 3.14 x 3.14 x mass = 39.4384 x mass
T T

39.4384 x 0.1 = 27.456
0.143641

39.4384 x 0.2 = 26.361
0.299209

39.4384 x 0.3 = 28.176
0.419904

39.4384 x 0.4 = 27.456
0.574564

39.4384 x 0.5 = 27.551
0.715716

I will now take an average result from these results

All added = 137 = 27.4 = 27.4 N/m
5

Compared with the Hookes Law result (28.7769) is quite close I shall now calculate the percentage of deviation between the two sets of results.
28.7769 - 27.4 = 1.3769 = 0.047 x 100 = 4.7
28.7769

The difference between the two sets of results is 4.7 %

From my results from both experiments I can see that as load increases so does extension of a spring. This extension is measured in N/m (Newtons per Metre) the figure for this is the amount of Newtons that would have to be added to the spring in order for it to have an extension of 1metre.
The spring constant that my experiments gave were 28.7769 N/m for the Hookes Law test and 27.4 N/m for the simple harmonic motion test. These findings are quite close with a 4.7% deviation between the two.
The principle of Hookes Law can be seen in a graph of spring extension.

1. At this point the limit of proportionality has been reached this is where hookes law is no longer accurate.
2. This point is the spring´s elastic limit if the force is removed from the spring it will no longer return to its original shape.

Beyond this point the atoms in the spring material begin to break their bonds until eventually the spring yields and breaks.

This shows that Hookes Law does have a limit as a spring does and does have limitations and if kept within these boundaries will provide reliable accurate results.

Hooke´s law can also be explained by the molecular structure of the substance the force is acting on:
F

Sep

1.
As molecules are pushed together, the larger the push the stronger the resistance.

2. At this point Hooke´s law of proportionality can be applied.

3. At this stage the effect of force becomes less until the spring yields and separates (breaks).

The results I have gathered are reasonably reliable and accurate and as can be seen in the first experiment graph all of the results were very close to the line of best fit. The deviation between the two tests of 4.7% is higher than I would have hoped but I have recently realised that for p I used 3.14 for my calculating and using the proper amount for p would have given my a higher therefore smaller deviation between the two tests. However I believe the main reason for the difference was because a different spring was used in the two tests and no two springs will be identical. In the Hookes Law test there were no significant anomalous results. In terms of reliability the Hookes Law experiment was carried out 3 times then an average was taken and all results were very close together. However it occurred to me as the extension increased the difference between results rose slightly this could be as the spring was nearing its limit of proportionality. This shows that my results for this test support Hookes Law well, which proves my prediction to be correct.
The preliminary test, which I had carried out also, helped in this area as if the elastic limit had been included in my results they would have been inaccurate for taking an average result. The tests also helped with any safety concerns, as there was very little danger of the springs breaking if the elastic limit was not breached.
In conclusion I am pleased with my results and feel they support each other as well as the laws they were based on. However if the experiment were to be carried out again changes could be made to reduce experimental error. In the first test the ruler suspended by the spring could have been set vertically by using a plumb line. Also a pointer on the spring would help in gathering the information more accurately. If the measurements were to be taken even more accurately an ultra sonic measuring device to measure the extension to a very accurate degree.
In the Simple Harmonic Motion test the timing of the oscillations could be taken over a greater number to help reduce experimental error.
Further investigations could use Young´s Modulus This was created by Thomas Young and gives a number representing (in pounds per square inch or dynes per square centimetre) the ratio of stress to strain for a wire or bar of a given substance. According to Hooke´s law the strain is proportional to stress, and therefore the ratio of the two is a constant that is commonly used to indicate the elasticity of the substance. Young´s modulus is the elastic modulus for tension, or tensile stress, and is the force per unit cross section of the material divided by the fractional increase in length resulting from the stretching of a standard rod or wire of the material. This would be useful as after the limit of proportionality had been reached using Hooke´s law; Young´s modulus would give results until the spring became straight until eventually breaking. However the change in length at this stage is very small and special measuring equipment would be needed in order to conduct a reasonable experiment.