ABOUT HOOKE'S LAW:
Hooke's Law is a scientific law that explains the theory of spring and elasticity. This principle was originally theorized by the 17th century British physicist Robert Hooke. According to Hooke's law, the extension of the spring is directly proportional to the force applied on the spring as long as it is within the elastic region.The Hooke's Law can be explained with the below formula:
F = -kx
This equation explains that F(Force Applied) is directly proportional to the x(Extension or Deformation of Spring).The value of k is the spring constant or also defined as the rate at which the spring is displaced. The spring constant can also be defined as the stiffness.The negative sign here shows the restoring force (Upward Force) of the spring after the force applied (Downward Force) is removed. Restoring force basically explains how the spring returns back to it's original position after the force applied is detached.
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| FIGURE 1 |
https://www.futurelearn.com/courses/maths-power-laws/0/steps/12143
- Figure 1 is a graphical representation of the explanation.As you can see, the extension or displacement of the spring is doubled when the force applied is doubled.This shows that the extension of the spring is directly proportional to the force applied.
- But the Hooke's Law only applies at certain region known as the elastic region.
Now what about materials that are not in the elastic region?
STRESS VS STRAIN GRAPH
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| FIGURE 2 |
https://www.meritnation.com/ask-answer/question/please-explain-stress-strain-curve-with-the-help-of-a-graph/mechanical-properties-of-solids/499828
- The stress vs strain graph explains the behavior of material in the elastic and plastic region.
- The graph increases linearly at the elastic region at A where it obeys the Hooke's Law until the elastic limit B.
- The material is considered permanently deformed after point B where the material cannot return back to it's original position.
- D is the ultimate tensile stress where the maximum stress a material can withstand before it breaks.
- E is the breaking point of the material or also where the failure of material takes place.
FUN FACTS ON ELASTICITY
The bungee jump is an example in daily life on how spring works and the concept of elasticity.
- According to the Guinness Book of Records, the highest bungee jump in the world, is 764ft (233m), from the Macau Tower in China. Jumpers experience four to five seconds of free fall at up to 125mph.
- Bungee jumping is so named after the strong elasticated cords that are normally used to strap down luggage.
SOURCE :
Adams, S., 2008. Bungee jumping: 20 facts. [online] The Telegraph. Available at: <http:/
.co.uk/news
https://www.tripadvisor.com.my/Attraction_Review-g1221358-d10023037-Reviews- Colombia_Bungee_Jumping-San_Gil_Santander_Department.html
INTRODUCTION
An experiment is conducted to show the behavior of three material where material 1 &2 in it's elastic region and material 3 in it's plastic region. The results are recorded and analysed in Microsoft Excel software in the form of graphical representation. A conclusion is made and the errors are also stated.
EXPERIMENTAL METHOD
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FIGURE 3
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- The experiment is set up as shown in Figure 3.
- The material to be tested is hung on the end of the clamp and a ruler is placed parallel to it.
- Initial length of the material is recorded.
- A known mass is placed on the spring and the displacement of the spring is recorded.
- Step 3 to 4 is repeated with a greater force and the displacement of the spring is noted.
- The above steps are repeated for material 2 & material 3.
pic source : http://danieelgohh.blogspot.my/2016/11/hookes-law-experiment.html
RESULTS
The results of the experiment obtained is recorded in the following table 1.
X (Force Applied),
N
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y1 (Deformation of Material 1), mm
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y2 (Deformation of Material 2), mm
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Z (Deformation of Material 3), mm
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1.00
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3.00
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2.2583
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2.375
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2.00
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4.50
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4.3166
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9.375
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3.00
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6.00
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6.3749
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28.375
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4.00
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7.50
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8.4332
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65.375
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5.00
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9.00
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10.4915
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126.375
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6.00
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10.50
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12.5498
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217.375
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7.00
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13.00
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14.6081
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344.375
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8.00
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14.00
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16.6664
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513.375
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9.00
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15.00
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18.7247
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730.375
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TABLE 1 shows the result of x, y1, y2 and z.
The values of x ( Force Applied) and the deformation y1 are already provided. But the values of deformation y2 and deformation z are obtained by using the formulas in Microsoft Excel.
VALUES
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MATH
FUNCTION
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EXCEL
FORMULA
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y2
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y2 = (a + 0.5) x + c
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=2.0583*(A2:A10) +0.2
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Z
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Z = x^3 + b
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= (A2:A10) ^3+1.375
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TABLE 2 shows the values of y2 and z.
Note : The value of c given is 0.2
The value of a and b is obtained through equation of the graph for values deformation y1 and y2.
a = 2.0583 ( The gradient of the graph y2 obtained from the trend-line equation)
b = 1.375 ( The y-intercept of the graph y1 obtained from the trend-line equation)
ANALYSIS :
GRAPH 1 :
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| GRAPH 1 |
Graph 1 shows the graph of deformation of material 1 ( y1) and material 2 ( y2) against the force applied (x). The inverse of the gradient of the equation is the actual spring constant for material 1&2.
Spring constant (y1) = 1/1.5583 = 0.642 Nmm^-1
Spring constant (y2) = 1/2.0583 = 0.486 Nmm^-1
Spring constant (y1) = 1/1.5583 = 0.642 Nmm^-1
Spring constant (y2) = 1/2.0583 = 0.486 Nmm^-1
Both the the graph intersects at one point which is the intersection point. The intersection point can be obtained in Microsoft Excel and finding the values manualy through simultaneous equation.
SIMULTANEOUS EQUATION:
y1 = 1.5583x + 1.375
y2 = 2.0583x + 0.2
Substituting y1 into y2
1.5583x + 1.375 = 2.0583x + 0.2
x = 2.35
y = ( a + 0.5)x + c
y= ( 1.5583 + 0.5)(2.35) + 0.2
y = 5.0371
Intersection point is at (2.35, 5.0371). This intersection point states that when the force applied is 2.35 N the deformation of the material 1 and material 2 is the same which is 5.0371 mm.
EXCEL FORMULA :
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| TABLE 3 |
GRAPH 2:
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| GRAPH 2 |
Graph 2 shows the deformation of material 3 (z) against the force applied (x). The graph is an exponential graph that gives the equation of y = 2.6257 e^0.6853x.
DISCUSSION :
Graph 1 of deformation y1 & y2 against the force applied(x) showed that the material 1 and material 2 is in it's elastic region. The graph is increasing linearly where the deformation of the material 1 and material 2 increases as the force applied is increased. This linear relationship implies that the material 1 and material 2 obeys the Hooke's Law and within the elastic region. Trend line of y2 is also more steeper than y1 in graph 1 where the gradient value is higher in material 2 compared to material 1. But still this value cannot be taken account as the spring constant.
Now try remembering:
We know that F = kx according to Hooke's law .
Then k = F/x but in this case the graph is plotted x/F. Thus to estimate the actual value of spring constant we should find the value of (1/gradient).
The spring constant of material 2 is 0.486 Nmm^-1 and spring constant of material 1 is 0.642 Nmm^-1. Thus,indicating that the material 2 is less stiffer and more elastic than material 1. Practically this means a greater amount of force is needed to produce the same deformation for material 1 compared with material 2. The intersection point indicates that a same value of deformation is obtained which is 5.0371 mm when the force applied is 2.35 N in both material 1& 2.
Now try remembering:
We know that F = kx according to Hooke's law .
Then k = F/x but in this case the graph is plotted x/F. Thus to estimate the actual value of spring constant we should find the value of (1/gradient).
The spring constant of material 2 is 0.486 Nmm^-1 and spring constant of material 1 is 0.642 Nmm^-1. Thus,indicating that the material 2 is less stiffer and more elastic than material 1. Practically this means a greater amount of force is needed to produce the same deformation for material 1 compared with material 2. The intersection point indicates that a same value of deformation is obtained which is 5.0371 mm when the force applied is 2.35 N in both material 1& 2.
Graph 2 of deformation (z) against the force applied (x) explains the behavior of material 3 which is in the plastic region.The exponential graph 2 no longer obeys the linear relationship like graph 1. Unlike material 1 and material 2 , material 3 cannot return back to it's original length which indicates that the material is permanently deformed. Furthermore , only a small amount of force is needed to produce the same value of deformation for material 3 compared with material 1 and material 2. This indicates that for even a small amount of force the deformation will be large. Moreover, this could also mean that the material 3 cannot return back to it's original length and permanently deformed.
ERRORS :
There are possibilities that errors can occur during representing the data. There are some possible errors that can be seen from the graph:
The values plotted is not perfectly accurate. This is because:
- The values taken account mostly have been round-off to 3 or 4 decimal places. This could affect slightly the final values obtained. (E.g the spring constant)
- The sensitivity of the measuring instrument used may be slightly less sensitive.
- Human error that occurs can affect the measurements recorded.
In order to minimize the errors and improvise the accuracy of the measurements, it is advisable to follow some of these steps:
- Human error can be reduced by ensuring the eyesight is perpendicular to the ruler when readings are taken.
- Experiment is repeated to obtain the average value to minimize errors.
CONCLUSION :
Graph 1 explains how both the material 1 &2 obeys the Hooke's law. A linear graph indicates that the materials are in elastic region and obeys Hooke's Law. As the force applied is increased the deformation of the material increases as well.
Graph 2 explains the behaviour of material 3 in it's plastic region. The graph is no longer a linear straight line graph thus it doesn't obey the Hooke's Law. This means that the material 3 can no longer return back to it's original length and is permanently deformed.
REFERENCING LIST :
- FutureLearn, n.d. Hooke's law and the stiffness of springs - Maths for Humans: Inverse Relations and Power Laws - UNSW Sydney. [online] FutureLearn. Available at: <https:
//www.futurelearn .com/courses /maths-power-laws /0/steps/12143> - Anon, n.d. What is Hooke's Law? (article). [online] Khan Academy. Available at: <https:
//www.khanacademy .org/science /physics/work-and-energy /hookes-law /a/what-is-hookes-law> - Anon, n.d. Spring Constant Formula. [online] Math. Available at: <http:/
/www.softschools .com/formulas /physics/spring_constant_formula /60/> - The Editors of Encyclopædia Britannica, 2016. Hooke's law. [online] Encyclopædia Britannica. Available at: <https:
//www.britannica .com/science /Hookes-law> - Afsar, J., 2014. Stress Strain Curve Explanation | Stages | Mild Steel. [online] Engineering Intro. Available at: <http:/
/www.engineeringintro .com/mechanics-of-structures /stress-strain-curve-explanation /> - Mishra, P., 2017. Stress Strain Curve – Relationship, Diagram and Explanation. [online] Mechanical Booster. Available at: <http:/
/www.mechanicalbooster .com/2016/09 /stress-strain-curve-relationship-diagram-explanation .html>
