# Spring Constant of Springs in Series and Parallel

Planning The aim of this investigation is to examine the effect on the spring constant placing 2 identical springs in parallel and series combination has and how the resultant spring constants of the parallel and series spring sets compare to that of a lone spring with identical spring constant. Hypothesis -???- Hooke?s Law states that "The magnitude of the spring constant (k) is equal to the stretching force applied (F) divided by the resultant extension (x)", it should be possible to determine a spring constant for each spring set. Due to existing knowledge of springs I propose that the series spring set will have a lower spring constant (and hence due to Hooke?s Law display a greater extension) than the parallel spring set. Also, as Hooke?s Law is a linear function, the spring constant of the series spring set should be exactly half that of a single spring, whereas the spring constant of the parallel set should be exactly double that of the single spring. This also means that if the resulting extension or spring length of the spring sets are graphed along a y axis with the increasing force mapped to the x axis (so that the results can be displayed in a traditional scientific graph fashion), the gradient will be the inverse of the spring constant. This hypothesis is backed up by many sources, one such source is ?Physics? by Ken Dobson, David Grace and David Lovettwhich in the 2000 edition states on page 90 that the spring constant of 2 springs in series is k = k/2 and for 2 springs in parallel k = 2k This hypothesis will probably only hold true however while the spring extends at a directly proportional rate to the increase in force on the spring

This is because every material has an elastic limit which is the percentage of extension a piece of material can be stretched to and still return to its original form. As the magnitude of extension of the string approaches this elastic limit, the extension will gradually cease to obey Hooke?s law. At this elastic limit, several changes in the composition of the spring can be observed. Whereas any stretching of a material that occurs below up until this limit is referred to as elastic deformation, stretching the material beyond this limit will result in permanent deformation of the material. Stretching that occurs beyond the elastic limit is referred to as plastic deformation. Once a spring has been stretched beyond its elastic limit, its molecular composition is permanently altered, meaning that the molecules that comprise the material have permanently re-arranged themselves as a result of energy transferred during the stretching process. Removing the force that is causing the stretch will not result in the material returning to its original state. Instead it will return to a semi-stretched state. When being stretched, the properties that a material displays falls into 1 of 3 categories, it will be either:

Ductile

The material has a period of uniform elastic stretching, then a

period of non-uniform plastic stretching until breaking point is

reached. Examples: Iron, Copper

Brittle

The material displays uniform elastic stretching until it reaches

its elastic limit at which it breaks. Examples: Stone, Glass

Polymetric

The material has a short period of uniform elastic stretching, it

then has a long period of parabolic plastic stretching until

breaking point. Examples: Rubber, Petroleum based materials

The springs that will be used for this investigation are of a ductile nature, the stretching displayed as force is increased can be graphed as follows: [image] 0 ? a: During this period, the spring displays uniform elastic stretching, a relatively high rate of force must be applied to stretch the spring. During this time, the spring obeys Hooke?s Law, meaning that the extensions is equal to the stretch force applied divided by the spring?s ?Spring Constant? (which will be discussed later). a ? b: After point a there is a small section of the graph where the spring is still within its elastic limit, meaning that it will still return to its original form once the stretching force is removed, however during this time the spring gradually ceases to conform to the linear principle?s of Hooke?s Law. b ? d: As b is the elastic limit for this spring, any stretching that occurs beyond b results in permanent deformation. If the stretching force was removed (say at point c), the spring would no longer return to its original state, instead it would return to a semi-stretched state due to the molecules now being arranged in a less tight formation. The force required to stretch the material during this period is less per unit of stretch than before as the molecular bonds holding the material together have been weakened. The point d on this graph is the breaking point of the spring, often referred to as the Ultimate Tensile Stress (u.t.s.). At this point the molecular bonds are broken, (in reality this occurs at a point of least bond strength, due to the probable existence of minute quantities of ?lower bond strength? impurities in the material). 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

[image] 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. [image] 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 low-tech 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 = SF Sx 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 Ö(2 × 9.81 × 1.15) = 4.750 kgms-1. 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 Single Spring Mass / 100g Length 1 /mm Length 2 /mm Length 3 /mm Average /mm 1 65.1 68 63 65.37 2 105 114 110 109.67 3 154.2 154 155 154.40 4 190 188.8 189 189.27 5 235.2 234 235 234.73 6 269 270.2 271 270.07 7 299.2 299.3 300.5 299.67 8 347.2 347 346.5 346.90 9 382 385.5 384 383.83 10 401 423 410 411.33 Series Mass / 100g Length 1 /mm Length 2 /mm Length 3 /mm Average /mm 1 129.5 134.3 136.1 133.30 2 206 210.5 224 213.50 3 283 283 307.5 291.17 4 358.5 358 386 367.50 5 433 434 466 444.33 6 511 507 546 521.33 7 591 580.5 627.5 599.67 8 663 654 704 673.67 9 746 728 786 753.33 10 826 795 864 828.33 Parallel Mass / 100g Length 1 /mm Length 2 /mm Length 3 /mm Average /mm 1 37.5 37 36 36.83 2 58 57 58 57.67 3 79 78 77.8 77.93 4 99 97.9 99.4 98.77 5 118.5 117 116.8 117.43 6 138 140 139 139.00 7 159 158 160.2 159.07 8 178 178 179 178.33 9 197.2 198 198.4 197.87 10 219 218 220.2 219.07 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. Single Spring Series Springs Parallel Springs Length of Spring Length of Springs Length of Springs 21.5 mm 45 mm 21.5 mm Force /N Length /mm Ext. /mm Length /mm Ext. /mm Length /mm Ext. /mm 0.1g 65.37 43.87 133.30 88.30 36.83 15.33 0.2g 109.67 88.17 213.50 168.50 57.67 36.17 0.3g 154.40 132.90 291.17 246.17 77.93 56.43 0.4g 189.27 167.77 367.50 322.50 98.77 77.27 0.5g 234.73 213.23 444.33 399.33 117.43 95.93 0.6g 270.07 248.57 521.33 476.33 139.00 117.50 0.7g 299.67 278.17 599.67 554.67 159.07 137.57 0.8g 346.90 325.40 673.67 628.67 178.33 156.83 0.9g 383.83 362.33 753.33 708.33 197.87 176.37 1.0g 411.33 389.83 828.33 783.33 219.07 197.57 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): [image] 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 ? 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 = Sx n Where åx 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: Single Series Parallel Avg of 1st 5 ext?s (x1) /m 0.12919 0.24496 0.05623 Avg of 2nd 5 ext?s (x2) /m 0.32086 0.63027 0.15717 Force unit increment (y2 ? y1) /N 4.905 4.905 4.905 Gradient and Spring Constant / Nm-1 25.59 12.73 48.6 Approx Spring Constant / Nm-1 26 13 49 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 2×. 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 low-tech 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 in-accuracy 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 eye-level 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: 1. The distance of the ruler from the spring?s often made it difficult to accurately measure the springs extension. 2. 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. 3. 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 in-accurate, determining at what level my eyesight should be at often proved slightly hard to determine. 4. 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: 1. [image] 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: 2. 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. 3. 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. 4. 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: [image] 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.

Spring Constant of Springs in Series and Parallel
9.6 of
10
on the basis of
1905 Review.