Method
The method I have chosen for this investigation involves first obtaining an elastic limit and basic spring constant for the exact spring type and then running 2 independent experiments measuring in the each case the magnitude of stretch of the spring(s) at incremental points when placed under the strain of varying loads and then comparing those results, a spring constant (which I shall refer to using the letter k) will be found.
As illustrated above, the first objective is to determine the elastic limit of the spring. This will be done by setting up apparatus in the standard way which will be described below, with a single spring of identical nature to those used throughout the investigation.
A 0.1kg mass (leading to a force/weight of 0.1kg times earth’s gravitational field strength which throughout this investigation will be taken to be 9.81N/kg) will then be added to the bottom of the spring and left to hang still (i.e. ensuring there is no bounce or oscillation in the system) for approximately 2 seconds and then removed. The length of the spring will then be measured and compared to that of the original length of the spring. If no increase in length is measured, then an additional 100g weight will be attached to the bottom of the spring and the system left to hang still for another 2 seconds before the masses (now 0.2kg) removed and the spring again measured for any increase in length. This will be done repeatedly in 0.1kg increments until a noticeable increase in the length of the spring is measured indicating the passing of the spring’s elastic limit.
Next the spring constant for a single spring of identical type to those which will be used in combination. This will be done by first measuring the length of the spring while unweighted and then incrementally applying weights, again 0.1 kg masses will be incrementally attached to the bottom of the spring. 10 readings will be taken (presuming the elastic limit is not surpassed during this time, in which case the readings will be damaged)
In the first experiment, a single spring will be examined to find the basic spring constant for the common spring type, in the second experiment 2 springs in parallel will be examined and in the third 2 springs in series will be used.
Equipment
For these experiments to be conducted the following equipment will be required:
 A Retort stand and accompanying boss and clamps (long enough so that when fully set up a distance of at least 1 metre between the of the top of the spring(s) and the base
 A workbench
 10 springs of identical nature
 10 Standard Weights all of 100g mass
The setup consists of a Stand resting on a work surface with a boss attaching a clamp to the stand. From the teeth of the clamp, the spring or spring sets will be suspended with the weights being attached to the end of the spring(s).
An important consideration is that there is enough space between the work surface and the clamp to allow for the required extension of the spring(s). Ideally this would be done by using a stand tall enough to support such requirements however to overcome limitations of available equipment, a bench could be used with the stand arranged so that the springs can drop down over the edge of the bench towards the floor providing extra extension length.
Measuring of the spring’s extension is done by placing a long ruler near the spring so that relatively accurate readings can be obtained, one important issue however is that equipment such as the ruler, bench, stand etc. does not actually touch the spring(s) or the weights attached to them as this could effect the results, this issue is one of the main possible causes of inaccuracy within the experiment and I will comment more on it later.
During the experiments, something that must be kept constant is the portion of the spring which is being measured. I have decided to measure the entire spring, including the loops at the top and bottom when taking measurements.
Originally, I was going to use the point below the top and bottom hoops as the start and end points however I have decided to use the entire spring due primarily to the difficulty and potential inaccuarcy in determining the exact location each time of other points on the spring when they change due to the stretching of the spring.
I have decided to take 10 readings during each experiment, each reading will be taken when a 0.1 kg weight is added to the bottom of the spring. I have chosen to take 10 readings as it should be sufficient for the purposes of the investigation, while my initial trial experiments show that the final force of gN will not cause the extension of the spring to approach the elastic limit and damage the results.
During the experiments, I will not actually measure the extension however, I will instead measure the total length of the spring(s) and then subtract the original length of spring(s) to obtain a measurement of the extension.
Due to the lowtech nature of the equipment available, each of the experiments will be run three times and the results of each experiment run averaged together at each increment to reduce experimental error. The averaged results at each interval will then be used to plot a graph of the extension in relation to the force applied. Through this graph I will try to establish a line of best fit which, (experimental inaccuracies aside), should pass extremely close to all points due to the expected linear nature of the results, such a straight line will help to prove my hypothesis that simple multiplication and division by 2 of the spring constant for a single spring will provide the spring constant for parallel and series spring sets respectively.
As stated above, Hooke’s Law says that the spring constant of a material can be found by dividing the force applied (in Newtons) by the extension observed (in metres). This means that the from a graph plotting force applied vertically and extension observed horizontally, the spring constant is simply the gradient of the line. Due to possible experimental error, it will be best to take the sum of all forces and divide them by the sum of all extensions.
k = ΣF
Σx
This will give a mean value for k which will be numerically equal to the gradient of the line of best fit.
Safety:
One important aspect that must be given consideration when dealing with springs is dealing with the potential rapid release of the energy contained within them due to either equipment failure or human error. Even small springs such as the ones which will be used may store quite large amounts of energy if placed under a lot of strain.
The amount of energy stored in the spring can be calculated by adapting the formulae for Work Done. In general, Work Done = force x displacement (W = Fs). As the force and displacement of a spring increase from 0 to the maximum, the total energy stored by the spring (i.e. the Elastic strain energy) can be found by averaging the minimum force x extension (which is 0) with the maximum force x extension (which I will call Fe).
Therefor:
W = Fe
2
This means that the Energy stored by the spring equals half the Force applied multiplied by the extension produced. This makes sense under the conservation of energy principle as Hooke’s Law (k = F/x) which can be rearranged to F = kx and x = F/k, means that the spring constant equalises out the ratio of Force and extension so that a greater force applied will produce more elastic strain energy regardless of if a stiff spring is used or a stretchy spring. It is also the area under the graph of a Force/extension graph.
In preliminary tests, the maximum F x e values were approximately (1 x g)N x (0.85)m, which equals 8.34Nm (where g = approx. 9.81m/s²). Therefor an approximate value for the maximum elastic strain energy possessed by the spring would be 4.17J
This energy would not really be sufficient to cause injury to humans with the exception of contact with eyes. It may therefor be advisable for participants to wear eye protection in order to minimise the already small chance of injury.
Another aspect of the experiment which could cause injury arises from the weights being used in the experiment. As the experiment is to be conducted on a bench surface approximately 1.2 metres high involving masses which at the maximum will reach 1 kg in mass the only real plausible danger would be one of the maximum weights being dropped on someone’s foot for example. By using the equation p = mv where v = square root of 2as, the momentum of the mass at the point of contact with the persons foot (presuming a foot height of about 5 cm), could be shown as p = 1 x √(2 x 9.81 x 1.15) = 4.750 kgms1. This momentum would probably not cause to much damage however would cause some pain and so caution when handling the weights should be exercised.
Results
Analysis of Results
As mentioned in the plan, the results above do not show the actual extension of the springs (which is what is to be looked it during the analysis), however they show the total length of the spring(s), including the original spring length.
I have therefor completed tables below of the actual extension of the spring in each case. These extension values have been found by subtracting the original length of the spring (found before conducting each experiment) from the total length shown in the above tables. As the analysis only uses the average results of the 3 runs for each spring combination, there is no point in calculating the actual extension for any of the other results.
These results tables have been used to plot several graphs which are included.
From these tables and graphs it is clear that in each of the 3 spring combinations an effect on the overall extension is produced, also it must be noted that the 3 spring combinations do share many similarities.
The main noticeable similarity is that, as predicted, in each case the extension of the spring(s) increases as more force is applied to the system. Even in the case of the parallel spring combination (which displays the least extension), the extension after the application of the force produced by the 1kg of mass is over 5 times as much as the extension produced by the application of the force produced by 0.1kg of mass.
Therefor it can be said that the graphs and tables show that in each case the extension produced is proportional to the force applied and that this proportionality is linear and obeys Hooke’s law (as was predicted in the experiment plan).
To prove this mathematically, we can apply the results to the standard equation for a linear Cartesian graph (i.e. y = mx + c):
From this substitution of the results into the mathematical form of a Cartesian graph it can be clearly seen that the extension produced equals the force applied multiplied by the inverse of the systems spring constant plus another constant. The constant (c) is the weight of the spring itself, as the springs used in the experiments only weigh a matter of grams however, this is approximated to 0. This means that the extension graph can now be mapped to the shortened form of a linear Cartesian graph so that y(force) = m(spring constant) . x(extension).
As the aim of this investigation is to identify the relationship between the different spring combinations and their spring constant’s, the next step I have taken is to produce a table detailing the spring constant of each system. This spring constant (which due to the experimental nature of this investigation is not an exact value but a close approximation) has been found by taking 2 points and calculate mathematically the graph’s gradient using the formula:
m = y2  y1
x2  x1
As the line for which the gradient is being found is a line of best fit, individual points do not nessicarily lie exactly on the line, in order to increase accuracy therefor, I have decided to find the mean value of the first 5 results for x1 and the mean value of the second 5 results for x2 . To calculate the mean I have used the formula:
_
X = Σx
n
Where ∑x is the sum of the 5 values of x and n is the amount of force between those 2 points (which in this case is always 5). Applying this method gives approximate values for the 2.5th and the 7.5th value which hopefully lie very close to the line of best fit (looking at the graphs show that this is quite an accurate approximation).
The 2 averages are then substituted into the above gradient equation with the 2.5th value being x1 and the 7.5th value being x2. As the gap between x1 and x2 is therefor 0.5g, this value is then substituted in as y. Applying this to the above equation produces a fairly accurate approximation for the gradient of the line of best fit and as stated earlier, this result is also the system’s spring constant. These values have been compiled to form the following table:
From this table we can see that the spring constant’s produced tie in extremely well with the predictions made earlier in the investigation plan. This is especially evident if the spring constants are approximated, as doing this cuts out some of the experimental error (due mainly to the difficulty of reading results as will be discussed later) and ‘real world’ factors which always effect any experiment.
As was predicted it can be seen that when springs were placed in a series combination, the combined spring constant was half that of a single spring, however when springs were placed in a parallel combination, the combined spring constant was double that of a single spring of identical type.
In mathematical terms therefor, it can be said that if the spring constant of a spring = k, then when the springs are in a series formation, the spring constant = k/2. When the springs are in a parallel combination, the spring constant is 2k.
The formula for the spring constant (ks) of a series spring set can also be proved mathematically. As the total extension produced was approximately 2 times the extension produced y the single spring, therefor the extension can be put as 2x. as Hooke’s law states that:
ks= F
x
It can now be said that for springs in series:
ks = F
2x
As rearranged, Hooke’s law states F = kx:
ks = kx
2x
If we cancel out the x’s this produces:
ks = k
2
Similar set of mathematical equations can be used to prove that the spring constant (kp), as the total extension produced was approximately halve that of a single spring, the formula:
ks= F
x
Becomes:
ks= F
x/2
If x is substituted for a rearranged version of Hooke’s law (i.e. x = F/k) then:
ks= F
F/2k
Which, rearranged algebraically equals:
ks= 2k
Evaluation
The experiments that were used in this investigation were all conducted in a relatively short period of time and using relatively lowtech instruments. Yet despite this, fairly accurate results, displaying the predicted effects on the spring constant of parallel and series arranged spring sets, were obtained.
Due to this fact it can be said that the experiments conducted were highly suitable to completing the investigation efficiently. If the resources were available, it would ofcourse been possible to obtain more accurate results with less chance of errors and anomalous results occurring.
The main source of errors and inaccuracy in the experiments most probably came from the use of a ruler to measure the increasingly length of the spring system. As measuring the full length of each spring set at any time required the springs and masses to hang completely unrestricted, it was nessicary to have the measuring ruler located at a distance of atleast a couple of centimetre’s from the spring(s). This may often have given way to wrongly judged spring lengths.
This problem was compounded by the difficulty in judging the required eyelevel at which to look at the spring(s) in comparison to the ruler as looking slightly down or up at the spring and ruler would lead to further inaccurate results.
While conducting the experiments, I noticed several difficulties and limitations which were likely to have had negative effects on the results:
 The distance of the ruler from the spring’s often made it difficult to accurately measure the springs extension.
 In order for the results taken at different instances to be relative, it was nessicary that the system be at a complete rest before taking measurements, however when masses were incrementally added to the system, some bounce always inevitably occurred. This bounce added to the time taken for each reading to be taken.
 As the method used required me to estimate the position of the bottom of the spring relative to a ruler located a couple of centimetres away, it was nessicary that my eye level was the same as that of the bottom of the spring, or else the measurement would occur at an angle and be inaccurate, determining at what level my eyesight should be at often proved slightly hard to determine.
 Due to the method of attaching the parallel springs to the clamp and masses, it is possible that at times, 1 spring would take more load than the other. As by definition, parallel springs should take equal load, this would lead to the spring systems not truly being in parallel and therefor inaccurate results could be produced.
These limitations all had possible negative effects on the results and in future experiments it would be advisable that modifications be made to minimise them. Some possible solutions to overcome the above mentioned problems could be respectively:

A distance measuring radar device could be placed underneath the spring set and masses so to judge more accurately the extension of the springs as so:
 Stiffer springs (i.e. ones with a higher spring constant) could be used as the oscillations would decay far quicker, also if the springs are stretched to their estimated new position before being released, the vertical oscillation potential would be decreased. It is also beneficial that the experiments be carried out in an environment free from air movement (i.e. a vacuum) so that further oscillations can not occur.
 As this limitation also came about due to using a ruler located a couple of centimetre’s away from the spring, the limitation would be solved if the same device used to eliminate limitation 1 was used.
 This problem could be improved by using more symmetrical or specialised equipment by which it is better insured that the load is spread evenly over the 2 springs, such pieces of equipment could include something such as:
Summary
In conclusion, this investigation was able to prove experimentally what the effect placing springs in series and parallel combinations had on their overall spring constant. It showed conclusively that the spring constant of a spring halves when 2 springs of that type are placed in series and doubles when 2 springs of that type are placed parallel.
The experiments used in this investigation could be improved by using the modifications stated above however in general the experiments used proved satisfactory for obtaining results which were accurate enough to high light the described relationship.