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# The Stiffness Of Springs

Extracts from this document...

Introduction

The Stiffness Of Springs

The spring constant is a measure of the stiffness of an elastic system.

How is the stiffness of a single spring related to the stiffness of springs in series and parallel?

Plan an experiment that will enable you to make a comparison of the stiffness for identical springs in series and parallel from your results.

Plan

The task of this experiment is to determine the relationship between the stiffness of springs in series an in parallel.

The stiffness of a spring can be shown as:

## F = kx

Where F is the force on a spring, x is the extension of the spring and k is the spring constant or the spring’s stiffness.  This means that the force on a spring is proportional to the extension with k being the constant.  Therefore as more force is put onto a spring the more it will extend.  By using this simple formula we can find the spring constant.

## k = F/x

By dividing the force by the spring extension we can find k.  Both the force and the spring extension are easily measurable.  We can show the relationship between the force and the extension in a graph.

x

.                                  F

The graph shows that as the force gets bigger the extension does.  The gradient of the line is the spring constant.  It is a straight line, as the spring constant does not change up to a certain point.  There is a point when a certain force will create a very large extension.  This point is called the elastic limit.  Once a spring reaches this point it becomes permanently deformed.  This means it does not return to its original shape or that its spring constant becomes altered.

The point at which the gradient of the line changes in the elastic limit.

Middle

321

6

355

Springs in Series

 Mass No. Spring I + II (mm) 1 239 2 307 3 376 4 445 5 513 6 580
 Mass No. Spring III + IV (mm) 1 249 2 330 3 395 4 462 5 520 6 601
 Mass No. Spring I + III (mm) 1 249 2 315 3 382 4 448 5 514 6 582
 Mass No. Spring I + II + III (mm) 1 317 2 420 3 523 4 622 5 724 6 824
 Mass No. Spring II + III + IV (mm) 1 329 2 433 3 535 4 637 5 737 6 840
 Mass No. Spring I + II + III + IV (mm) 1 404 2 539 3 674 4 811 5 946 6 1080

Springs in Parallel

 Mass No. Spring I + II (mm) 1 158 2 173 3 192 4 209 5 226 6 243
 Mass No. Spring II + III (mm) 1 162 2 177 3 196 4 213 5 229 6 246
 Mass No. Spring III + IV (mm) 1 167 2 184 3 200 4 218 5 235 6 252
 Mass No. Spring I + II + III (mm) 1 175 2 183 3 193 4 204 5 215 6 226
 Mass No. Spring II +III + IV (mm) 1 176 2 186 3 198 4 209 5 220 6 231
 Mass No. Spring I + II + III + IV (mm) 1 158 2 163 3 170 4 177 5 186 6 195

On the following pages are graphical interpretations of these results.

Analysis

These are the spring constants.  These have been worked out by first drawing the force/extension on a graph.  Then by drawing a line of best fit through these points.  Then the inverse of the gradient of the line gives you the spring constant.  This is because the gradient of a line is y/x.  This would mean that the gradient showed that spring constant was extension/force when it should be force/extension.  To solve this I have to use the inverse function of the gradient as my spring constant.

The lines on these graphs do not pass trough the origin.  This is due to the fact that I have not taken away the original length of the springs from the totals.  This gives every result an offset which will vary from spring systems.  However I found that this is irrelevant, as this would not affect the gradient from which I will obtain the spring constant.

Spring constant should be measured in Nm however as I have measured all my extensions in millimetres I will have to change everything into metres, as it is a standard unit of measurement.

Conclusion

If I had used identical springs instead of similar springs then I would have had more conclusive results.   Also if the springs became slightly deformed during the experiment then my results could be affected.

Where I took the results there might have been a breeze which might have fractionally affected my results which could lead to more errors.  If I did this experiment in an environment where air pressure was normal and there was no breeze then I could achieve more accurate results.

There are several anomalous results in my experiment.  These have either been marked with ** or with a circle.  These results do not fit the pattern which the other results do.  This could be due to many different errors.  If I had repeated these results I might have a got a results that did fit the pattern.

On my graphs I have quantified some of the errors by drawing error boxes around the points I have plotted.  This is because the masses were ± 3 of 100g.  Also I left some space for error for the extension.  This was ± 1mm.  This accounted for if I had incorrectly read any results or if there was a breeze or friction.

To improve my results I also could have repeated the experiment and taken averages for each value or taken more readings.  However I did not have sufficient time for this.

The formulas that I had found out might not be correct as none of the results actually correlated with the formula exactly.  However this is probably due to the many errors that could affect my results.  I based my formulas on rough patterns which seemed to fit with my results.

Another avenue to explore in this experiment would be mixed systems with both series and parallel systems within it.

This student written piece of work is one of many that can be found in our AS and A Level Waves & Cosmology section.

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