AIM
This experiment was designed to analyse one set of given results, showing
deformation in a material, after having numerous forces applied to it. The investigation involved calculating the results of two additional unknown materials, and to determine whether or not they
reach their known elastic limits, under an applied force.
INTRODUCTION
The
definition of Hooke’s Law and Young’s Modulus from the Collins English
Dictionary are as follows;
[Named after Robert Hooke
(1635-1703), English physicist, chemist, and inventor]
Young's
modulus
(Physics / General
Physics) a modulus of elasticity, applicable to the stretching of a wire etc.,
equal to the ratio of the applied load per unit area of cross section to the
increase in length per unit length. Symbol E
[Named after Thomas Young
(1773-1829), English physicist, physician, and Egyptologist]
Hooke’s Law states that a force applied on a material is relative to the strain on that material, and it will often cause deformation. Young’s Modulus is a ratio of stress given by the force applied, over the strain received, and is used as a vital part of this principle. These have been calculated through experiments over the years and have been tabulated to give individual Modulus of Elasticity figures for each material. The Young's Modulus formula may then be used to calculate the strain as shown.
Equation 1.
When calculating results there is a
certain time where the stress is proportional to the strain and when the force
is removed, the material will return to its original state. This is called the elastic region. If the force
applied takes the material passed its own Young’s Modulus; this means it has
passed the elastic point and it can no longer return to its original form. Here it may
stretch in the plastic region, and if enough force is administered it may eventually snap. These ranges depend upon the material tested. In this graph
we can see these different states.
Stress/Strain Curve - Hooke’s Law only used for up to the elastic limit
In practice, using a spring for
example; Hooke’s Law uses the force applied (F) divided by the Young’s
Modulus (E) of the material, to give the new length (extension) of the spring (X) as
in the equation shown:
In this
instance it is common to use the constant K instead of E for
Young’s Modulus. (F = KX)
METHOD
The apparatus is set up as follows;
A metal rod set into a base with an arm attached to the
top. This hangs the spring and a stretch indicator hangs from the bottom of the
spring. The indicator is aligned with the zero on the scale. As a mass
increment (X) is hung from the bottom of the indicator, the spring will
stretch and the new length recorded (Y1).
RESULTS
In this experiment, three materials were tested and
results for any deformations or new lengths were given as shown below.
Table 1.
The x
values show the force applied in Newton's whereby the Y1,
Y2 and
Z values show the deformation in mm.
The
results were given directly for material one in the figures tabulated above and
the second and third materials were calculated by use of two algebraic
simultaneous equations, in excel. These workings are displayed in Fig 2.
Fig 2.
These
results were then plotted onto graphs using excel (shown below).
Graph 1.
Graph 2.
CONCLUSION
Analyses of the results of the new extensions, due to the
X forces being applied, give two straight line graphs and one curved
graph. Not having the materials used for each spring or the
results as each passes the ‘elastic limit’ means certain presumptions must be
made.
Clearly the first two materials (Y1 & Y2)
have shown a straight line, thereby proving that these forces (1-9 Newton's) have
not caused enough stress to push the materials passed their individual elastic
limits.
The third material (Z), does not have a straight
line at all, thereby proving that the strain on this material does not
proportionally coincide with the stress caused by the forces. This means it
does not follow Hooke’s Law. This gives reason to believe that it has
passed the materials own elastic limit, and has ventured into a plastic state.
This means that the material will not return to its original state. Too much
force applied would cause the material to fracture and break.
REFERENCES
Hooke’s Law Definition [Collins English Dictionary – Complete and
Unabridged] HarperCollins Publishers 1991, 1994, 1998,
2000, 2003
Hooke’s
Law Atanackovic, Teodor M.; Guran, Ardéshir
(2000). "Hooke's law". Theory of elasticity for scientists and
engineers. Boston, Mass.: Birkhäuser. p. 85
Young’s
Modulus Definition [Collins English Dictionary – Complete and
Unabridged] HarperCollins Publishers 1991, 1994, 1998,
2000, 2003
Young’s
Modulus Young,
Thomas (1845). Course of Lectures on Natural Philosophy and the Mechanical
Arts. London: Taylor and Walton.
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