The spring I am going to use in this experiment are quite cheap springs, therefore they probably wont have the same spring constant so I wont be able to use the same spring constant for each spring. Hopefully the springs will have a similar spring constant so it will be easier to compare. I can overcome this by numbering each spring and then testing each spring individually for its spring constant. Then I can use these values for each spring.
The masses I am going to use are said to be 100g. However when I weighed these masses on some electric scales I found that each mass was not 100g but ± 3g from 100 grams. As the mass is directly proportional to the force and the force is directly proportional to the extension, I think that it would be appropriate if I used a more accurate value for the mass by weighing each mass using electric scales. If I number each of the masses then I can keep track of which masses I place on to the spring(s). I think using this I can find a more accurate value for k (spring constant).
I am going to use 6 masses weighing approximately 100g each. This will give me six extensions. I think that this is a sufficient number of results required to obtain the spring’s stiffness.
I am going to use an ordinary 1m ruler to measure the extension of the springs. This ruler has markings every 1 millimetre. I am going to measure the extension to the nearest millimetre as this is the most accurate I can get using this equipment. I do not think it is necessary to measure the extension any more accurately than that.
I am going to use a piece of mechano with 5 holes in it to allow me to create spring systems in parallel. This will allow the springs to work in a system without interfering with each other. The piece of mechano I will use has a mass of approximately 5g. However I am not going to quantify this in my results as this mass will be constant in my results and therefore wont effect the increases of mass.
To make sure that while I am testing the spring constant I do not exceed the spring constant I am going to test a similar spring to the ones I will be using. I am going to place enough weights on to it to just take it past its elastic limit. This is a test just to make sure that when I am testing my springs that I do not exceed the elastic limit and damage my springs which would ruin the experiment.
Prediction
I predict that 2 springs put in to a parallel spring system will produce a smaller extension then a spring system of 1 spring in series. This will happen because the spring system as a whole will have spring stiffness of the total of the two individual springs used. Therefore a parallel system would have a combined spring stiffness of the individual springs in it. e.g.
If I use two identical springs in a parallel spring system each with the same spring stiffness with a mass of 100g, the extension produced will be half the extension produced by one spring in series (see diagram above). I think this will happen because ass the two springs are in a parallel system, they will share the tension of the mass equally. As I think this will produce half the extension, the system must have a larger spring stiffness as F = kx. The force will stay the same, and if the extension decreases by half then the spring stiffness must double. As the two springs are identical this must mean that the stiffness of the spring system would be the combined stiffness’ of the two springs.
I believe that two springs in series however would produce a larger extension. This would be explained by the fact that the tension created by the force would be acting in full on both springs. They would not share the force. I think that each individual spring would extend the same amount, as it would do if it were just on its own. Therefore the extension would be double compared to one spring on its own. This would mean that the spring stiffness for the system would be halved if two identical springs were used.
Results
These are the results I have obtained.
When I performed this experiment I had to take the following safety precautions. I had to be careful not to drop any of the weights onto my own or other people’s fingers or toes. I also wore safety goggles in case a spring flew and hit me in my eye which was a possibility.
A spring similar to the ones I am using becomes permanently deformed when the force becomes greater than approximately 12 N. Therefore I cannot use that much force onto my springs as it will give me incorrect results. To be on the safe side I am only going to use a maximum of 6 N as this will give me enough results to plot a suitable graph.
A Table showing the actual masses of the masses I used in this experiment, measured on electric scales.
Single Springs
Springs in Series
Springs in Parallel
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.
Already from these results I can see that the spring constant follows a pattern. This means that I should be able to work out a formula for the spring constants in their systems.
The following graphs show how the spring constant changes for each system against the number of springs.
These two graphs show a lot. They show that there is a clear link between the number of springs in a system and the spring constant of that system. However the two graphs look very different to each other. The graph for parallel in series is clearly linear, while the graph for springs in series is not linear but still follows a pattern.
I will now try to find a formula for the spring constant of a series spring system. If we examine the results obtained for this I maybe able to establish a formula.
Springs in Parallel
From these two tables I can see that the spring constants for I +II spring system is very close to the total of the individual values of Spring I and Spring II added together.
For the majority of these results the theoretical constants are quite similar to the actual constants. As my results are not totally accurate there is a large margin for error which would explain my imperfect results.
I believe that for springs in parallel the final spring constant will be the total of all the individual spring constants added together.
This gives me the formula
k1 + k2 + k3…. = kT (kT = total constant of a series spring system)
Springs in Series
The trend in this system is that as more springs are included in the system the spring constant decreases. This is clearly shown in the graph. I estimate that for 5 springs in parallel the spring constant will be approximately 5 Nm. This will get smaller and smaller. The graph for this looks like a graph for a fraction e.g. y = 1/x. if this is true this graph will continue getting smaller and smaller but never reaching the x axis.
If I look at the numbers in the tables above I see that the spring constant for I + II is half the value for spring I. However this is not true with I + II and I + II + III and it does not show any relationship between the individual springs used and the final spring constant.
The formula for springs in parallel does not work with these results. However if I look at the inverse of these figures we can see that there might be a relationship. The inverse is also the same as the gradient as the spring constant is the same as the inverse of the gradient.
Here I can see I more direct relationship. It looks as though the inverse of the springs constants of the individual springs added together give the total of the inverse of the spring constant of the springs in a series system.
These results are very close to what they should be according to my idea. I can put this into a formula.
1/k1 + 1/k2 + 1/k3. …= 1/kT (kT = total constant of a series spring system)
Conclusion
This shows that there is relationship between springs in series and in parallel. I have found out two independent formulas which follow my patterns. The formula which I have found out for springs in parallel is what I had predicted that the values of k would be added together for the total k as the force will be shred by more springs therefore creating a greater stiffness and half as much extension. However for the formula for springs in series I was not entirely correct. I did say that they extension would increase with more springs therefore the spring constant would increase with more springs. I was wrong in saying that the spring constant would just half for two springs when it was actually the inverse of these.
Evaluation
For everything I did in this investigation, there was a degree of error. This explains why I don’t exact, perfect results. I am now going to quantify these errors. I used 10 ms –2 for gravity instead of 9.81 ms –2. I also used 100 grams as the mass, event though I knew that the masses I used were not exactly 100g each. They were ± 3 of 100g. Already I have quite a bit of error in my results as the force relies on both of these quantities. There fore my force should be
F = ma
F = 0.1 ± 0.003kg x 9.81
F = 0.981 ± 0.003 N
However I used 1 N instead.
If I had used a more accurate ruler with smaller gradations then I could have got a more accurate reading from my experiment. I could have measured to the nearest 0.1 mm with a more accurate ruler or other measuring apparatus. However I used a ruler with only 1mm gradations so I could only be accurate up to 1mm.
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.