Hooke’s Law explains the behaviour of the spring and other linearly elastic objects. According to the Hooke’s Law, the stress is always directly proportional to the strain. The aim of the Hooke’s Law experiment was to investigate and verify Hooke’s Law and the acceleration due to gravity with a linearly elastic spring. In the experiment, different masses were used to determine the extension of the spring and time taken for the oscillation and then plot graphs to assess the Hooke’s Law and the acceleration due to gravity, . The spring constant was obtained to be 1.2558. The acceleration due to gravity was found to be 9.714m/s2. Compared to the theoretical value of g, a percent error of 0.979 percent was observed. There minimal errors accrued in the experiment which were both systematic and random, mostly due to the instruments. The errors can be improved by regularly calibrating the instruments, use of digital instruments, and conducting the experiment carefully.
Table of Contents
1. Introduction and Aim.. 6
1.1 Aim: 6
1.2 Objectives: 6
1.3 Background Information: 6
1.4 Experiment Introduction: 7
1.5. Hypothesis. 8
2. Theory and Equations. 8
3. Materials and Methods. 13
3.1 Tools and Materials: 13
3.2 Safety Precautions: 13
3.3 Methodology: 13
4. Results and Calculations. 15
4.1 Experiment Results: 15
4.2 Variables: 16
4.3 Graphs and Calculations: 17
5. Uncertainties and Errors. 20
5.1 Error Analysis: 20
5.2 Random Errors: 20
5.3 Systematic Errors: 21
5.4 Error Calculations: 22
5.5 Average Error Calculation: 24
5.6 Percentage Error: 24
6. Discussion. 24
7. Conclusion. 26
8. References: 26
Table of Figures
Figure 1: Illustration of a mass exerting force to a spring. 7
Figure 2: Mass attached to a vertical spring. 8
Figure 3: Graph of Force against Extension. 9
Figure 4: Frequency of a horizontal spring. 9
Figure 5: Set-Up of the experiment 12
Figure 6: Extension Graph. 16
Figure 7: Period Graph (100g) 16
Figure 8: Period Graph (150g) 17
Figure 9: Period Graph (100g) 20
Figure 10: Period graph (150g) 21
Table of Tables
Table 1: Results of the extension against mass. 14
Table 2: Results of the Oscillation using mass 100g. 14
Table 3: Results of the Oscillation for 150g mass. 15
1. Introduction and Aim
1.1 Aim:
The aim of the Hooke’s Law experiment was to investigate and verify Hooke’s Law and the acceleration due to gravity using a linearly elastic spring. The experiment aims at using different masses to determine the extension of the spring and time taken for the oscillation and then plot graphs to assess the Hooke’s Law and the acceleration due to gravity, .
1.2 Objectives:
This experiment will be guided by the following objectives:
1. To assess Hooke’s Law of a spring using the extension versus mass graph.
2. To obtain the acceleration due to gravity experimentally and compare it with the known value.
3. To obtain the period of oscillation of a spring for a given mass and use the findings to plot the square of period versus mass graph.
1.3 Background Information:
Robert Hooke, an English scientist, discovered Hooke’s Law in 1660 (Davidson, 2010). While working with springs, Hooke noted that when a stress is applied, the shape alteration of the spring was directly proportional to the stress applied. This led to the definition of Hooke’s Law that states, “In an elastic body, the stress is directly proportional to the strain” (Nolan, 2005). Stress is hereby defined as the force applied on a surface per cross-sectional or unit area (Lowe and Rounce, 2002). Increasing the stress on an object causes the object to deform. The response of the material to the applied stress is referred to as the strain (Gerdeen, Lord, and Rorrer, 2006). Borodulina, Kulachenko, Galland and Nygards (2012) state that the stress-strain curve is a straight whose gradient is referred to as the Young’s Modulus and obeys the Hooke’s Law. Spring like materials that obey the Hooke’s Law are referred to as Hookean or linearly elastic material while the materials which do not obey Hooke’s Law are simply referred to as non-Hookean materials (Means, 1991). In Hookean springs, the extension of the spring is directly proportional to the applied mass (Bueche and Hecht, 1997). As such, a spring can be assessed on whether it obeys Hooke’s Law by observing the gradient of the extension versus mass graph. Although the addition of mass results to a consequent increase in the extension, springs have a point beyond which additional masses do not result to an increase in the extension. This point is referred to as the limit of proportionality (Sang, 2015). Beyond the limit of proportionality, the springs undergoes permanent deformation and consequently loses its elasticity (Sang, 2015).
In a spring, once the spring is stretched or compressed, it exerts a force which either pulls back or pushes away the force (Halliday, Resnick, and Walker, 2014). The force that either pushes or pulls back the spring is referred to as the restoring force (Halliday, Resnick, and Walker, 2014). In case of a vertical spring, exerting a force by attaching a block of a given mass and pulling the block slightly causes the spring to oscillate back and forth about the equilibrium point. The equilibrium point is the reach of the spring when there is no force exerted. The oscillation of a mass on a vertical spring is affected by the force of gravity (Giordano, 2013). The force of gravity causes the equilibrium point of the spring to shift downwards. As such, it affects the period of the oscillation. The period of oscillation is the time taken for the mass on the spring to complete one cycle about the equilibrium point (from equilibrium-maximum-equilibrium-minimum-equilibrium) (Newman, 2008). As observed in Section 2, the period of oscillation is affected by the mass as well as the gravity of oscillation. However, this only applies to vertical springs. Horizontal springs do not exhibit the same extension as the vertical spring.
1.4 Experiment Introduction:
In the current experiment, a mass is attached to a vertical spring and the corresponding extension registered. For a linearly elastic spring, the addition of mass should be proportional to the extension recorded. As such, the extension versus mass graph should be a straight line unless the limit of proportionality is exceed. The gradient of the straight would be the spring constant as given by the formula below (Serway, Faughn, and Vuille, 2010).
Where F is the force of the applied mass is the displacement of the spring, and k is the spring constant which is normally given in newtons per meter. The equation has a negative sign which represents the restoring force which is opposite to the applied force (Serway, Faughn, and Vuille, 2010). As such, the resulting force is always negative to the applied force as it goes towards the equilibrium where the displacement is zero, x=0. For a spring which is considered to be elastic, Hooke’s Law can be assessed and verified by checking whether the curve formed by the extension and mass graph is straight.
The force of gravity affects the extension of a spring with a mass attached in the vertical position. The equilibrium point slightly shifts downwards compared to the horizontal spring. As such, the acceleration due to gravity can easily be obtained from an expression associating the period of oscillation of the spring, the mass and the extension. The period of oscillation, T, is easily obtained from measuring the time taken for the spring to complete one cycle about the equilibrium point. The graph of the square of the period and the mass gives a straight one whose gradient, alongside the gradient of the extension versus mass curve, can be used to determine the acceleration due to gravity, g, using the formula given below.
Normally, the acceleration due to gravity is 9.80665m/s2 (Tate, 1968). The acceleration due to gravity obtained in this experiment can, therefore, be determine and verify the acceleration due to gravity. Furthermore, the comparison of the experimental and known gravitational acceleration is important in the assessment of errors in the entire experiment.
1.5. Hypothesis
It is hereby hypothesised that the extension graph (extension versus mass graph) will have a straight line which will confirm that, indeed, the spring obeys Hooke’s Law. It also hypothesised that the period graph (square of period versus mass) will have a straight line. It is expected that the acceleration due to gravity will be obtained to be approximately .
2. Theory and Equations
According to Serway, Faughn, and Vuille (2010), Hooke’s Law is expressed by the equation shown below.
The equation can be explained by considering a spring attached to a wall with its length at the equilibrium position and a mass exerted on the spring.
In Figure 1 (a), the spring is at the equilibrium position. At this position, it is neither stretched neither compressed. A block of mass m is then brought next to the spring, and presses the spring against the wall to its minimum length as shown in Figure 1 (b). Assuming that the compressed length is x and that the surface is frictionless, then the stored energy in the spring becomes;
In Figure 1 (c), the block is released and the potential energy stored in the spring is transferred into kinetic energy, which then pushes the block in the opposite direction at a velocity v. As such, the resultant force exerted by the spring is negative to the applied force while compressing it. Hooke’s Law, therefore, becomes;
In a vertical spring such, once a mass is attached to the spring as shown in Figure 2, the spring will stretch to a new height. The additional length is said to be x1. In case the mass is replaced with a higher mass, more force will be exerted to the spring. The spring, thereby, stretches to new length with the additional length (from the original equilibrium position) is x2.
Using the forces of the two masses, F1 and F2 and the extension of each mass, x1 and x2, a graph of F-x can be plotted as shown in Figure 3 below.
The gradient m gives the spring constant of the spring.
Now let us considered the horizontal spring shown in Figure 1.
According to Giordano (2013), the frequency of the oscillation of the spring is given by;
However, period is related to the frequency by;
As such, the period of the spring is given by;
However, as shown earlier,
Assuming that the spring is vertical, F becomes mg
Hence;
Inserting equation (viii) in (xi);
The ration of the gradients of the extension graph and the period graph will be expressed by;
As such, the acceleration due to gravity is given as;
The acceleration due to gravity can also be obtained by;
Therefore;
3. Materials and Methods
3.1 Tools and Materials:
The following tools and materials were utilised in the experiment:
- A retort stand
- A linearly elastic spring
- A stop watch
- 150g masss
- 410g masses
- Ruler
- Tape measure
- Laboratory coat
- Laboratory goggles
- Calculator
- Sheets of paper
- Graph papers
- Pens and penscils
3.2 Safety Precautions:
The following precautions were taken before starting the experiment:
1. Due to the elastic nature of the spring, all the students were required to wear safety clothing, which include lab coats, goggles and closed shoes to prevent injuries in the body in cases of dropping masses or springs bouncing off the stand after stretching.
2. The lab technician checked the retort stands to make sure that they were firm and authorised the commencement of the experiment.
3. To prevent harm to the springs, students were required to make small stretches and short oscillations such that the extension of the spring did not exceed the length of the spring at its equilibrium position.
3.3 Methodology:
The tools were prepared according to the set—up shown in Figure 5. The spring was tightly clamped to the retort stand fitted with a cramp. A mass hanger with a pointer was then
connected to the hook of the experiment as shown in Figure 5.
A ruler was then connected next to the spring such that the pointer pointed at the calibrated end of the ruler. The reading of the pointer at 0 mass connected was read and recorded. A mass of 50g was then connected to the spring, the extension measured, and recorded, this was repeated for the masses of 100g, 150g, 200g, 250g, and 300g. All the measurements were recorded ready to plot the extension graph. The second segment involved making oscillation at a fixed mass while varying the extension of the mass below its new equilibrium point. A 100g mass was attached to the hook of the spring. The mass was pulled for 20mm below the equilibrium point. The time taken to complete 5 oscillations was measured using the stop watch. The period was calculated and recorded. At the same mass, this was repeated for 30mm, 40mm, 50mm, 60mm, and 70mm. The 100g mass was then replaced with a 150g mass and the process repeated for the lengths of 30mm, 35mm, 40mm, 45mm, 50mm, 55mm, and 60mm.
4. Results and Calculations
4.1 Experiment Results:
Table 1: Results of the extension against mass
Mass(kg)
Extension (Dx) (cm)
0
0.00
0.5
0.16
1
0.33
1.5
0.49
2
0.64
2.5
0.81
3
0.97
Table 2: Results of the Oscillation using mass 100g
Length (mm)
Time Period( 5 Swings) (sec)
Avg Time Period (sec)
Time Period (sec)
T2 (sec^2)
20
4.34
4.470
0.894
0.799
4.53
4.54
30
5.32
5.337
1.067
1.139
5.28
5.41
40
6.59
6.500
1.300
1.690
6.41
6.5
50
7.16
7.187
1.437
2.066
7.18
7.22
60
8.04
8.003
1.601
2.562
8
7.97
70
8.22
8.207
1.641
2.694
8.22
8.18
Table 3: Results of the Oscillation for 150g mass
Length (mm)
Time Period( 5 Swings) (sec)
Avg Time Period (sec)
Time Period (sec)
T2 (sec^2)
30
5.75
5.690
1.138
1.295
5.63
5.69
35
6.04
5.950
1.190
1.416
5.74
6.07
40
6.44
6.430
1.286
1.654
6.41
6.44
45
6.84
6.810
1.362
1.855
6.78
6.81
50
7.19
7.180
1.436
2.062
7.16
7.19
55
7.57
7.490
1.498
2.244
7.43
7.47
60
7.81
7.873
1.575
2.480
7.9
7.91
4.2 Variables:
Extension experiment
Independent variable – Mass (kg)
Dependent variable – Spring extension (m)
Controlled variable – Spring constant
Period experiment
Independent variable – Extension length (m)
Dependent variable – Oscillation time (sec)
Controlled variable – Mass
4.3 Graphs and Calculations:
Graphs
The extension graph was obtained using the table below.
Mass(X)
Dx(Y)
0
0.00
0.5
0.16
1
0.33
1.5
0.49
2
0.64
2.5
0.81
3
0.97
The graph obtained was:
Figure 6: Extension Graph
The next graph is a tension graph which was obtained from the length and T2 reading of Table 2.
Figure 7: Period Graph (100g)
The next graph is a tension graph (150g) which was obtained from the length and T2
reading of Table 3.
Figure 8: Period Graph (150g)
Calculations
The extension graph forms a straight line with the equation being . The gradient is 0.3229
The spring constant is given as;
The period graph (100g) forms a straight line with the equation . The gradient is 0.2437
The ratio of for the 100g period graph was 0.2437.
The acceleration due to gravity was found to be;
The period graph (150g) forms a straight line with the equation . The gradient is 0.2484
The ratio of for the 100g period graph was 0.2484.
The acceleration due to gravity was found to be;
The average acceleration due to gravity is;
According to Etkina, Gentile, and Van Heuvelen (2014), the theoretical acceleration due to gravity is 9.81.
5. Uncertainties and Errors
5.1 Error Analysis:
There are various errors in the Hooke’s Law experiment. For instance, the vibrations of the spring were not account for in the experiment. In addition, the continuous application of masses to the springs probably caused the equilibrium position of the spring to shift downward interfering with the extension measurements. Reading of the extension on the ruler was done using naked eyes, which incurs errors due to parallax. On the period experiment, there is no mechanism to determine that the cycles of the oscillations are equal. It is assumed that the oscillation at each mass are equal. However, this is not true due to the damping effect of the spring. However, these errors were catered for through the modification of data collection and inclusion of the uncertainties in the measurements.
5.2 Random Errors:
Among the random errors incurred in the experiment include the resolution of the instruments such as the stopwatch and the ruler. These instruments have the minimum possible measurements that they can measure beyond which their precision becomes limited. The uncertainty in the stopwatch was ±0.01s while the uncertainty in the ruler was ±0.05m. Errors due to reading of the ruler due to parallax also constituted the random errors. Other random errors include the carelessness during the experiment such as stretching the spring for displacements exceeding its length in the period and oscillation experiment.
5.3 Systematic Errors:
Systematic errors in the experiment included the state of the springs. Since the springs utilised in the experiment were not brand new and may have had their equilibrium shifted in previous experiments. The ruler might also have had zero errors which would be replicated throughout all the extension measurements. This kind of errors can only be eliminated through frequent recalibration of the measuring instruments.
5.4 Error Calculations:
Figure 9: Period Graph (100g)
The line of best fit was drawn through the centroid point which is (1.83, 0.45).
The gradient for the Mmax was;
The gradient for the Mmin was;
The uncertainty is therefore;
The gradient and its related uncertainty is given as 0.244 ±0.076
Figure 10: Period graph (150g)
The line of best fit was drawn through the centroid point which is (1.86, 0.45).
The gradient for the Mmax was;
The gradient for the Mmin was;
The uncertainty is therefore;
The gradient and its related uncertainty is given as 0.248 ±0.095
5.5 Average Error Calculation:
The average error is calculated from the uncertainties recorded and the individual measurements used in the calculation of the acceleration due to gravity. The expression of the acceleration due to gravity was given as;
The average error can therefore be calculated as;
The average error for the acceleration due to gravity is calculated as:
And
5.6 Percentage Error:
The percentage error for the acceleration due to gravity is calculated as;
6. Discussion
In his refined research in 1968, Tate established the acceleration due to gravity to be 9.80665m/s2 (Tate, 1968). Experiments since then has since aimed at verifying and supporting this finding. In the current experiment, the acceleration due to gravity was found to be 9.714m/s2. With this value, a percent error of 0.979% was obtained. The percent error is very minimal compared to most research (Etkina, Gentile, and Van Heuvelen, 2014). The little percent improves the validity of the experiment, especially in the initial part involving the demonstration of Hooke’s Law in the spring. As seen in the background research, Hooke’s Law provides the basis for the behaviour of linearly elastic springs pertaining the stress and strain of the spring (Serway, Faughn, and Vuille, 2010; Nolan, 2015; Halliday, Resnick, and Walker, 2014). For a linearly elastic spring, the slope of the extension against mass graph is linear with the gradient being the spring constant (Wynne, 2008). In this experiment, the graph of the extension and mass exhibited a linear curve whose gradient was 1.2558. The spring utilised in this experiment is therefore a linearly elastic material or Hookean material as earlier observed. Being a Hookean material, an increase in the mass or force applied to the spring results to a corresponding increase in the deformation in the spring (Radi, 2013; Purrington, 2009). The masses utilised in this experiment were such that the spring would not undergo permanent deformation. Had the masses or the force applied to the spring exceeded the elastic limit of the spring, the spring would have gone beyond the limit of proportionality and, thereby, permanent deformation. Although, the graph of the extension and mass was linear, there were minor errors that caused the obtained values to differ to the literature values. Among them include systematic and random errors caused by the instruments and the experimenters. The errors observed in the experiment can be minimised through use of regularly calibrated instruments. In addition, digital instruments such as digital rulers would minimise errors due to the observation of the experimenter (Halliday, Resnick, and Walker, 2014). Hooke’s Law has wide application in most science and engineering fields. Most importantly, Hooke’s Law forms the fundamental principle behind the operation and design of the spring scale, balance wheel in a clock, and manometer (Etkina, Gentile, and Van Heuvelen, 2014). Hooke’s Law also finds application in the design of acoustic and seismology instruments which are used detection of earthquakes.
7. Conclusion
The objective of the current experiment was to investigate and verify Hooke’s Law as well as establishing the acceleration due to gravity using a spring and a set of masses. The experiment found the spring to obey Hooke’s Law since the extension increased proportionally with the mass. The spring constant was obtained to be 1.2558. The acceleration due to gravity was found to be 9.714m/s2. Compared to the theoretical value of g, a percent error of 0.979 percent was observed. There minimal errors accrued in the experiment which were both systematic and random, mostly due to the instruments. The errors can be improved by regularly calibrating the instruments, use of digital instruments, and conducting the experiment carefully.
8. References:
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