# 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.

Spring Constant of Springs in Series and Parallel

AS Physics Coursework

By Malcolm Davis

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 Lovett which 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:

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 ...