Hooke's Law
Introduction
Introduction
Hooke's law, law of elasticity discovered
by the English scientist Robert Hooke in 1660, which states that, for
relatively small deformation of an object, the displacement or size
of the deformation is directly proportional to the deforming force or load.
Under these conditions the object returns to its original shape and size upon
removal of the load. Elastic behavior of solids according to Hooke’s law
can be explained by the fact that small displacements of their constituent
molecules, atoms and ions from normal positions is also proportional to
the force that causes the displacement.
Figure A
Figure B
An object only obeys
Hooke's Law when its elastic limit is not exceeded. When elastic limit of the
spring is exceeded, the graph will not remain in straight line. The graph will
be curved (as shown in Figure B).The spring is said to undergo plastic deformation
causes the spring to irreversibly change its shape when the elastic limit of a
spring is exceeded.
Experiment
Aim
To study the relationship between the extension and the force applied to an elastic material.
Apparatus
3 different types of elastic materials, ring stand and clamps, meter rule, and slotted weight
Procedure
1. The original length
of Material 1 is recorded.
2. The apparatus is set
up by hanging up Material 1 on the retort stand and clamp.
3. Slotted weight of 1N
is hung at the end of Material 1.
4. The final length of
Material 1 is recorded.
5. The extension of
Material 1 is calculated by the difference between the original length and the
final length of Material 1.
6. The experiment is
repeated using slotted weight of 2N, 3N, 4N, 5N, 6N, 7N, 8N and 9N.
7. The data of Material
2 and 3 is calculated using the formula given.
8. The data is tabulated
in a table.
9. Graph of extension
against force applied for all 3 materials is plotted.
Diagram 1.1
Tabulation of data
The result of
Material 1 and Material 2 are tabulated in Table 1 and Material 3 is tabulated
in table 2.
Given that y1=ax+b and y2=(a+0.5)x+c where c=0.2
Force,x/N
|
Extension of
Material 1,y1/mm
|
Extension of
Material 2, y2/mm
|
1.00
|
3.00
|
2.2583
|
2.00
|
4.50
|
4.3166
|
3.00
|
6.00
|
6.3749
|
4.00
|
7.50
|
8.4332
|
5.00
|
9.00
|
10.4915
|
6.00
|
10.50
|
12.5498
|
7.00
|
13.00
|
14.6081
|
8.00
|
14.00
|
16.6664
|
9.00
|
15.00
|
18.7247
|
Table 1
Given that z= x 3+ b
Force,x/N
|
Extension of
Material 3,z/mm
|
1.00
|
2.375
|
2.00
|
9.375
|
3.00
|
28.375
|
4.00
|
65.375
|
5.00
|
126.375
|
6.00
|
217.375
|
7.00
|
344.375
|
8.00
|
513.375
|
9.00
|
730.375
|
Table
2
Discussion
A
graph of Extension, y/mm against Force, x/N of Material 1&2 is obtained.
Figure 1
-From
the graph 1, we can get the values of a and b with the equation of y1= 1.5583x
+ 1.375 which mean that a=1.5583 and b=1.375.
-We
can solve the equations from the graph using simultaneous equation to find the
intersection point (x,y).
y1= 1.5583x+ 1.375 – (1)
y2= 2.0583x+ 0.2 – (2)
By
putting y2 into equation (1), we get
2.0583x+ 0.2 = 1.5583x+ 1.375
2.0583x- 1.5583x = 1.375- 0.2
0.5x= 1.175
x= 2.35
By
putting x= 2.35 into equation (2), we get
y= 2.0583(2.35) + 0.2
y=5.037005
Thus,
the intersection point of y1 and y2 is at (2.35, 5.037005).
A graph of Extension, z/mm against Force, x/N of
Material 3 is obtained.
Figure 2
From Graph 2, curve graph obtained. This show that
Material 3 does not obey Hooke’s Las as its elastic limit is exceeded. Material
3 is undergoing plastic deformation.
Sources
of Error
- The ruler used to make the measurements may not have been accurate.
- User error when reading the measurements from the ruler, such as a parallax error by reading the measurement from an angle or choosing the closest marker to represent the measurement.
- Inaccurate forces being exerted over the materials, such as the mass not being exactly as stated.
- The overall calibration of the apparatus being used to conduct the experiment may have been inaccurate.
- When resolving the simultaneous equations, the values 1.5583333333 and 2.0583333333 were rounded to four decimal places this would have produced a slight error in the final results of the calculation.
Conclusion
The
results obtained from the experiment confirm that Hooke’s Law is true.
Materials 1 and 2, whose displacement is shown in the
results as y1 and y2, respectively, are in their elastic
region. As such, a linear relationship between the force applied (x) and their displacement (y) is shown in the graph Figure 1
This
indicates that as force is applied, the material is displaced directly
proportionally.
Material 3, shown in the results as x, on graph Figure 2, is indicated to be
in its plastic region. The exponential trend line shows that as force is
applied, the material is permanently displaced.
The steeper trend line in the graph for
material 2 shows that less force
is required to result in a greater displacement of the material than for
material 1, indicating that it is
more elastic.
A comparison of the graphs shows that
for material 3 a
far greater amount of force is required than for materials 1 or 2 to achieve any displacement.
References
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