Greg Honeysett, 26270455
Username: honeysweet7@hotmail.co.uk
Hooke’s law states
that the stress applied to a material is proportional to the strain on
that material (1). When applying a load (or force) to an object such as to one
end of a wire, we find that the object behaves elastically provided the load is
not too great. The change in length of the wire is directly proportional to the
applied load. The mathematical formula for Hooke’s law can be expressed by F=kX
where F is the force applied in Newtons (N), k is the constant or otherwise
known as mechanical stiffness with units of Newtons per meter (N/m), and X is
the change in length of a material in meters (m). When the load or force is
removed, provided the material is still within its elastic limit, it will
return to its original length. Hooke’s law generally holds only up to the
elastic limit of stress for that material. Hooke’s law can be displayed
graphically such as in figure 1 (2) below:
Figure 1 (2)
Figure 1 (2) shows the extension of a material increases as the
force applied increases until the external forces exceed the internal forces of
the material and the elastic limit is surpassed.
Figure 2 (3) below shows the increase in length of the spring
caused by increasing the weight.
Figure 2 (3)
I carried out an experiment to test Hooke’s law. The elastic
properties of three materials were examined. A force was applied to three
different materials and their change in length or deformation was measured. As
the force increased, certain patterns in the elastic behaviour of the materials
became visible. These three material results include y1, y2 and z. Material
results y1 and y2 describe two different elastic materials still in their
linear regions. In other words, results y1 and y2 show two materials not given
a load great enough to exceed their elastic limit. Material result z describes
a material which has been given a load which exceeds the materials elastic
limit. Data x is the force applied in Newtons (N) and y1, y2, and z are
deformation in millimetres (mm).
The results of x, y1, y2 and z are shown below in Table 1:
Table 1
x
|
y1
|
z
|
y2
|
1
|
3
|
2.375
|
2.2583
|
2
|
4.5
|
9.375
|
4.3166
|
3
|
6
|
28.375
|
6.3749
|
4
|
7.5
|
65.375
|
8.4332
|
5
|
9
|
126.375
|
10.4915
|
6
|
10.5
|
217.375
|
12.5498
|
7
|
13
|
344.375
|
14.6081
|
8
|
14
|
513.375
|
16.6664
|
9
|
15
|
730.375
|
18.7247
|
Positive correlation is shown between the different variables such
as force and deformation of x, y1, and y2 in figure 3 below:
Figure 3
Figure 3 shows Hooke’s law in action whereby as the load or force
applied (x) to the two different materials (y1 and y2) within their elastic
limit increases, they experience a proportional increase in length or
deformation. However, the results for y1 show two extreme figures whereby they
do not lie on the line of best fit and are slightly above or below it. These
results include y1=13 and y1=15. The result of y1=14 also does not conform to
the general pattern or conventional gradient of the other results. These
extreme results may be due to an experimental error or the material approaching
its elastic limit.
The differences in deformation of y1 and y2 observed despite the
same levels of force applied and both materials being in their linear regions
is due to different k constants. The mechanical stiffness or ‘k’ depends on the
dimensions of the material such as its cross-sectional area and length. Because
different materials have different mechanical stiffness, their proportional
change in length differs as well.
From Figure 3 I estimated the interception point of the two lines;
y1=1.5583x+1.375 and y2=2.0583x+0.2 to be (2.3, 5). I then calculated the
interception point using simultaneous equations. My calculations and workings
are shown below:
Calculations
From my calculations I obtained an intercept point of (2.35, 5.04)
when rounded to 2 decimal places. This is a very similar result to my estimated
intercept point. If rounded to 1 decimal place, however, the x coordinate would
be 0.1 higher.
Figure 4
Figure 4 above shows the relationship between force and
deformation of a material past its elastic limit. The graph plots variables x
(force, N) against z (deformation, mm) of a third material beyond its elastic
region. As force increases the change in the materials length increases by a
greater proportion. This produces a backwards ‘L’ shaped curve. The external
forces start to exceed the internal forces of the material, causing its elastic
characteristics to deteriorate. At this moment, Hooke’s law no longer applies.
Sources
(1): hhtp://www.thefreedictionary.com/Hook’es+law
(2): http://www.racemath.info/graphics/graphs/hookes_law.gif
