Recently in my physics class, I got into deep thinking when my teacher derived the hook's law. The law is governed by the equation F=k (x). The equation was driven using the elasticity of spring, and in the explanation, the teacher mentioned that spring could only extend to a given limit beyond which it undergoes permanent deformation. I got challenged and was ready to explore more about Hook's law and its applications with the main aim being to understand the spring constant.
Springs exist in various forms and materials, and they are used in almost all the moderately complex mechanical systems, and this shows how important they are in human life. For instance, I was triggered to think about the racing car engines and the shock absorbers systems which all use the springs. Based on the derivation developed by the teacher, it was evident that spring can only regain its original position when a given maximum amount of force is applied (Yao, et al. 23). The elasticity is a fundamental property of springs. When a force is applied to a material, it stretches or compresses in response to the force and the exchange of change is proportional to the force applied. The strain is a measure of the change in the length of spring to its original length along the stress direction, and this is expressed as =DL/L0. Different materials respond differently to the force applied, and the chemical bonds within the material determine this (Yao, et al. 23). The spring constant is the constant of proportionality that determines the magnitude of the change of spring regarding shear and compression when a force is applied.
If the weight (Weight =mass x gravity, and g=9.98m/s2), I suspended at one end of the spring, it stretches by distance x, and then a proportional force acting on the opposite direction is generated by the spring so that it regains its original position. In case the weight applied is too high, the spring is likely to undergo permanent elongation or deformation; however, when the load is manageable, the force will reinstate the spring to its initial measurement. The force (F) is referred as the elastic force, and it is proportional to the amount of the elastic restoring force. According to the study conducted by Hooke during the 17th Century, the spring constant (k) refers to the stiffness coefficient of a material (Yao, et al. 23).
The plot of Force as a function of the change in the length of the spring has a linear segment, and this confirms that the spring's constant obeys the Hooke's Law and thus it becomes easy to find the spring constant (k). The other approach I considered in this experiment was the simple harmonic motion which refers to the linear motion and the Hooke's law restoring force (Yao, et al. 23). Borrowing from Newton's second law which presented F= -kx =ma
The negative sign appears in the equation above because the acceleration of the object in simple harmonic motion is in the opposite direction to the force inducing it. From the definition:
a = (d2x/dt2) = -k/m x
The simplification of the differential equation derived above is provided by (Bueche p.231), and the final expression of T is as follow:
T=2pm/k and K is the spring constant while m is the mass of the object under motion. The experiment can also be conducted using a different method, and the purpose of this study was to determine whether the spring constant conforms to the Hooke's Law.
All materials undergo deformation when subjected to stress. The elastic materials have an internal force that restores their sizes and shapes when stress is eliminated. In case the deformation is indirectly proportional to the stress applied, and the deformation caused by the resulting from applied pressure disappears upon the elimination of the force, then the material is referred as perfect elastic (Yao, et al. 23). Robert Hooke discovered the Hooke's law in the 17th century, and this was after observing that stress versus strain resulted to curve with a linear region for most of the materials. The law states that the force needed to stretch an elastic material is directly proportional to the change in length recorded. The law is expressed as F =-kx where F is the force applied while the X is the magnitude of change in the length of the spring. Hooke's law was the initial classical illustration of the elasticity.
The law applies in various areas including the inflation of the balloons and pulling off the rubber bands during the measuring of the wind force among others. In addition, the creation of the balance wheel promoted the creation of the mechanical clocks and spring scales (Yao, et al. 23).
The experiment aims at investigating how springs behave when stretched under the influence of the external forces.
The spring used in the experiment
Changing mass and the weight force was the independent variables
Dependent variable was the extension of the spring
The control variables included the dimensions of the springs regarding mass and radius as well as the height of the retort stand.
The initial mass was determined by weighing the support table and then attaching it to the mass to spring. The position of the end of the spring after being attached to the table was determined and recorded as the initial position. The mass attached to the spring was increased up to 120 grams, and the corresponding measure of the spring position examined. The calculation of the force applied was based on Newton's gravitational force using the formula: F=Ma. Also, F=spring constant x change in the length of the spring. The value of k is unknown in the experiment; therefore, the data collected from the measurements were used to calculate F. Graph of F vs. change on the length of the spring constant was constructed and used to verify the Hooke's law (Yao, et al. 23).
The masses were added to the spring as randomly indicated in the results table to observe the correlation between the load and the elongation of the spring (Kaanta, Leila, Gabriele, and Arja 25). The SHM part involved the use of a single mass of 4 kilograms which was hung from the spring and the spring to execute a given number (N) of the oscillations measured using a digital watch. The procedure was repeated for three times and the data recorded accordingly.
Results and Data
Load (Kg) L0 (cm) L1 L(L1-L0) Change in L(cm) Change in L (m)
0 30.9 71.7 40.8 - 0 31.1 71.9 40.8 40.8 0
1 31 72.1 41.1 41.1 0.03
2 31.1 75.3 44.2 44.2 2 31.3 75.6 44.3 44.3 0.035
3 31.1 80.1 49 49 0.065
3 31.3 82.1 50.8 50.8 0.082
4 31.5 93.6 62.1 62.1 0.123
5 31.8 98.2 66.4 66.4 0.135
The calculation of the spring constant from the load extensions was done as follow:
Oscillations Time T
50 43.60 0.86
25 22.14 0.85
40 34.91 0.85
Considering that the T = 0.85 0.02 seconds
The timing oscillator - load = 4kg
The spring constant was calculated using the resonance frequency method
The approach considers that there are uncertainties which must be calculated; the standard deviations of the collected data for the value of T. Also the standard deviation for the slope of the graph was determined using the computer program and the line of best fit constructed which was used in determining the spring constant.
The graphical representation of the data is as indicated below.
The figure above indicates the graph of the load applied on the spring as a function of the resulting change in the length.
Discussion and analysis
The above diagrams indicate that the load greater than 1.1 as evident in the intercept of the line of fitness, there is a linear relationship between mass and the extension of the spring. However, since the fit does not curve the y-axis at 0, I assumed that the spring required some load for it to overcome. Having this assumption, I ignored loads of weight between 0 and 1 kg. The plotline was used and ignoring loads of less than 2kg; it was found that the gradient of the line was 22.2. From the spring constant equation, there was a need to multiply the quantity by the g to calculate the value of the spring constant; therefore, k=217.41.8N/m. In the determination of the of k dynamically the value of one load (4kg) was used and considering that the mass of the spring was 7.0 grams, the amount of mass used in the calculation of the value of K from equation 2 was 4.023. The average time was 0.85; therefore, substituting the values in the formula K= 4p2m/t2, the value of the spring constant was 217.7N/m.
The values of spring constant obtained from the experiments were in agreement with the limits of accuracy, and thus they obey the Hooke's law. In case the dynamic experiment was extended through the use of a range of masses and graph of T2 plotted against m, the gradient of the line would be linear (Greene, Nathaniel, and Stephen139).
T2 = (2p)2/k (m+m1)
The m represents the effective mass of the spring. The use of the above formula would give a more reliable finding compared to the estimated 1/3of the spring-mass applied in this experiment.
The static method of determining the spring constant involves the addition of weight on the spring and then measuring the extension. The spring is fixed in one end as evident in the experiment I carried out. Basically, the value of the spring constant is determined using the formula F= ke (Greene, Nathaniel, and Stephen138). On the other hand, the dynamic method incorporates the concept of oscillation and the angular velocity in determining the spring constant.
The Hooke's law derives the relationship between the mass applied on a spring and the magnitude of displacement following the application of the mass. In my imagination, I thought that there could be other forces acting on an object suspended on the spring; however, I was limited to use the model derived by Hooke which is governed by an equation: F=km (Gross, Dietmar, et al. 22). To confirm the relationship, two approaches were used in the study, and they included the dynamic and static method; I did this intentionally so that I could compare the findings from the two experiments and determine whether they corroborate with each other and the Hooke's Law. The findings indicate that both approaches gave almost the same results and there was a direct relationship between the mass applied and the length of extension of the spring (Greene, Nathaniel, and Stephen138). One of the approaches used the direct extension and mass applied while the other approached employed the oscillations in determining the elasticity of the material. There were some limitations in the experiment; however, the findings indicated that the experiment was conducted with a lot of precision considering that there was a sight standard deviation from the values obtained in either experimental approach (Yao, et al. 50).
Improvement of the experiment
Some of the limitations I encountered in the experiment included lack of a proper way of ensuring that the spring attached to the stand or table remain linear all the time and especially in the dynamic method of determining the spring constant, the number of oscillations could be missed out. The most probable ways of improving the experiment include using video cameras with high resolutions that can help in counting all the oscillations. The findings indicate that there is a close relationship between the spring constant of a spring measured using either the dynamic or static method (Yao, et al. 49). I consulted with the teacher and agreed that I could make the changes and ensure that I get more accurate answers and values.
Also, my experiment and data collection did not consider the errors that may be associated with calibration and inaccurate balance reading. Also, the other possible limit...
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