Investigation of Elasticity - Lab Report Example

Investigation of Elasti by s s Table of Contents 1Introduction 3 1Aim 3 2Objectives3 1.3Hypotheses 3 1.3.1Hypothesis 1 3 1.3.2Hypothesis 2 3 2Theory 4 3Apparatus 6 4Experimental Procedure 6 5Analysis 6 6Results 8 6.1Table 1: Measurements of extension of spring 1 9 6.2Graph 1: Extension versus Mass for spring 1 9 6.3Table 2: Measurements of extension of spring 2 10 6.4Graph 2: Extension versus Mass for spring 2 11 6.5Table 3: Measurements of extension spring 3 12 6.6Graph 3: Extension versus Mass for spring 3 12 7Discussion 13 7.1Error analysis 15 8Conclusion 16 1 Introduction 1.1 Aim The purpose of this experiment was to determine the constant of elasticity of three different springs 1.2 Objectives The objectives of the experiment were: To calculate the constant of elasticity of three different springs To determine the elastic potential energy of the same springs 1.3 Hypotheses 1.3.1 Hypothesis 1 Theoretically, when the thickness of the spring increases, the constant of elasticity will also increase. This is as a result of increase in the second moment of area. The second moment of area depends on the density of material within the cross section of an object (Furness, Charles & Crane, 1997, p. 90) 1.3.2 Hypothesis 2 As the mass hanged on the spring is added, the elastic potential energy of the spring also increases. Elastic potential energy is the energy that is stored when spring is deformed especially after stretching (Hyperphysics, 2015). Therefore, theoretically, the elastic potential energy should be directly proportional to the mass loaded on the spring. 2 Theory Elasticity is the property of a solid material that defines the ease with which it returns to its original shape and size after the force deforming it has been removed (Physics.info, 2015). The deformation may be in the form of stretching, compression or bending. An object that is easy to deform is said to be more elastic, that is it returns to its original shape with ease. A material can be said to be perfectly elastic if it satisfies the following conditions as described by (Kazimi, 2001, p. 17). The deformation and recovery are instantaneous when loading and unloading. There is no permanent set left after unloading, that is, the recovery is complete and immediate. The load-deformation curve maintains the shape when loading or unloading. A rubber band will stretch and snap back to almost its original length when released. It is, otherwise, not as elastic as a guitar wire. Guitar wires are harder to stretch, but in reality, they are more elastic than rubber bands due to the high precision of returning to the original size. The wires can handle several strokes without much stretching, which might result in loss of tune. Theory of elasticity is a branch of continuum Mechanics dealing with deformable solid bodies having the physical property called elasticity (Wan, 1982, p. iii). The concept of elasticity can be explained by studying springs. According to Bethell and Coppock (1999, p. 7), springs are useful because they stretch evenly when subjected to stress). Springs are elastic bodies, whose purpose is to disfigure when loaded and to recover their initial shape when the mass is removed (Khurmi & Gupta, 2005, p. 841). They are used in various applications including watches to store energy, in automobile suspensions to absorb energy, retractable ball-point pens, playground toys, and in balances and engine indicators to measure forces. Their utilization depends with their type, which is either helical, leaf, or conical etcetera (Gubeljak, Vejborny and Predan, n.d.). Springs have a predictable behaviour when a stretching or compression force is applied. They exert a restoring force that brings them back to their original size after stretching. This behaviour is defined by the Hookes Law, which states that the extension on the spring varies directly with the application of force (Beer, Johnstone & Dewolf, 2004, p. 56). The Law is represented by the following equation (Horibe, n.d). Whereby F is the applied force k is the spring constant, which varies with the type of spring. It is the force per unit length. The value of k is large for springs with high modulus of elasticity and small for loose springs (Pickover, 2008, p. 74). Hookes Law is valid for any material provided the proportional limit is not exceeded (Roesler, Harders, and Baeker, 2007, p. 39). There is no longer a linear relationship between the applied force and the spring extension beyond this point. When the elastic limit is exceed, the spring deforms completely and will not return to its original size and shape after the applied force is removed. The proportional limit varies directly with the elastic limit. It is, therefore, the objective of this experiment to examine various springs that are classified as elastic and also to determine their constants of elasticity. 3 Apparatus The apparatus used in this experiment were: 1. Ruler 2. Mass hanger 3. 3 steel springs 4. Retort stand and 5. Loading weights 4 Experimental Procedure i. One end of the first spring was fixed to the retort stand and the other end to the mass hanger. ii. The length of the spring was measured as relaxed length l1 with no mass attached. iii. A suitable range of masses was determined as 10g, with which to measure the extension of the spring. A mass of 10 g was added and the length recorded as length l2. iv. Successive masses were added at a range of 10g up to a load of 100g and the individual lengths recorded in table 1. The extensions were also calculated and recorded on the same table. v. The data in the table was used to plot the graph of extension (mm) against the mass (g). vi. The first spring was then removed from the retort stand and the second one fixed to the screw of the clamp instead of clamping it on the clamp stand since it was small. vii. The same procedure was repeated for the second and third springs, and their results recorded in tables 2 and 3 respectively. 5 Analysis When a mass M is hanged on a spring, it applies a force F, which is equal to its weight Mg, whereby g is the gravitational force equal to (Physicsclassroom, 2015). The spring extends by an amount equal to the change in length so that the weight Mg or downward force is balanced by the springs upward restoring force Fs. As shown in the figure below, the mass is at rest in its equilibrium position, that is Fs = - Mg. Since the extension has a linear proportion with the load, then, therefore, the restoring force is directly proportional to extension. This can be written as, but F=Mg, therefore , , k is the constant of proportionality Therefore the gradient of the graph vs enables the ratio to be determined. Theoretically, this graph should form a straight line from the origin to a point whereby the elastic limit is achieved (Avison, 1989, p. 164). A permanent stretching develops beyond this point and the graph seems to curve as shown in figure 2. Figure 1: Spring at equilibrium (http://www.physics.dcu.ie/~jpm/PS128/P4-experiment.pdf) Figure 2: Steel spring stretched beyond its elastic limit (Avison, 1989, p. 164) 6 Results Mass M (g) 10 20 30 40 50 60 70 80 90 100 (mm) 15 16 17 20 22 23 25 26 27 30 (mm) 14 14 14 14 14 14 14 14 14 14 (mm) 1 2 3 6 8 9 11 12 13 16 EPE (J) )= 33.9 135.4 304.7 1218.6 2166.4 2741.9 4095.9 4874.4 5720.7 8665.6 EPE is the elastic potential energy 6.1 Table 1: Measurements of extension of spring 1 6.2 Graph 1: Extension versus Mass for spring 1 The two points chosen were (90, 13) and (35, 5). These points were selected because they were closest to the Trendline. Therefore, the gradient Since then So we have , The SI units were left out since it was a constant. Mass M (g) 10 20 30 40 50 60 70 80 90 100 (mm) 20 20 23 26 30 34 38 42 46 50 (mm) 20 20 20 20 20 20 20 20 20 20 (mm) 0 0 3 6 10 14 18 22 26 30 EPE (J)= 0 0 117.9 471.6 1310 2567.6 4244.4 6340.4 8855.6 11790 6.3 Table 2: Measurements of extension of spring 2 6.4 Graph 2: Extension versus Mass for spring 2 The two points chosen were (90, 25) and (50, 10). These points were selected because they were closest to the Trendline. Therefore, the gradient Since then So we have , The SI units were left out since it was a constant Mass M (g) 10 20 30 40 50 60 70 80 90 100 (mm) 20 20 21 21 22 24 27 31 35 38 (mm) 20 20 20 20 20 20 20 20 20 20 (mm) 0 0 1 1 2 4 7 11 15 18 EPE (J)= 0 0 29.4 29.4 117.4 469.6 1458.2 3551.4 6603.8 9509.4 6.5 Table 3: Measurements of extension spring 3 6.6 Graph 3: Extension versus Mass for spring 3 The two points chosen were (55, 6) and (100, 13.5). These points were selected because they were closest to the Trendline. Therefore, the gradient Since then So we have , The SI units were left out since it was a constant 7 Discussion The results of the experiment showed that the mass loaded on each of the springs varied with the extension but not proportionately. However, the second and third springs did not register any extension in their length for masses that were less than 30g. This was because the exerted force was less than the restoring force and, therefore, more weights needed to be added so as to overcome the resistance and record an extension. The obtained values of the extension and loaded masses were plotted in scatter graph. A trendline was used also plotted. The Trendline was used to get points, which lay along its path so as to calculate the slope of the graph. The gradient of each graph was calculated by using the formula of . Other than utilizing the formula, the slope could also be obtained by inserting the line of best fit. The values obtained were then used to calculate the spring constant k for each of the springs. The K (Spring constant) values for springs 1, 2, and 3 were 67.7, 26.2 and 58.7 respectively. The first spring had the highest proportionality constant. This meant that it had the highest elasticity and, therefore, the highest restoring force followed by the third spring and finally the second one. The elastic potential energy also varied with an increase in the loaded mass, though not uniformly. The values were calculated using the formula EPE = (Cutnell & Johnson, 2014, p. 260). For example, the EPE value of the first spring when a mass of 10g was loaded is given as follows k, which is the springs constant is 67.7 and , which is the change in length is 1mm Other calculations for the Elastic potential energy Spring 1 Mass of 20 g Mass of 30 g Mass of 40 g Mass of 50 g Mass of 60 g Mass of 70 g Mass of 80 g Mass of 90 g Mass of 100 g All the other values for springs 2 and 3 were calculated using the same formula and the values of k as 26.2 and 58.7 respectively. 7.1 Error analysis The springs used in the experiment were not new, which resulted in deviations from the expected values of the spring constants. Springs 1 and 2 had lost their elasticity, thereby, recording lower values k. The expected values of the slope for springs 1, 2 and 3 were 0.165, 0.355 and 0.204 respectively. However, the values obtained from the calculations were 0.145, 0.375 and 0.167. The values of k resulting from these gradients were 67.7, 26.2 and 58.7 respectively. The expected spring constants were Therefore, the percentage error in the experiment for each of the springs is Spring 1 Spring 2 Spring 3 8 Conclusion The purpose of the experiment was met, which was to determine the spring constant of 3 different springs. These k values showed that spring 1 was the stiffest of all the springs, that is, it had the highest elastic modulus. Also, from the graphs, it was evident that as the mass was loaded, the change in length also increased. The mass was directly proportional to the extension and also the extension varied proportionately with the elastic potential energy, which can be represented by the following equation. The experiment was conducted with the highest level of accuracy, but there were a few errors. They were systematic errors and environmental factors, which resulted due to vibration oscillations of the springs after loading of weights. The variation between the experimental and the expected values resulted since the springs had worn out and, therefore, they could not sustain their restoring forces. The values were also not uniform because the experiment involved using the naked eye to read measurements from the ruler. Theoretically, the springs should have a uniform extension since the weights were loaded at a range of 10g. However, this could not be realized due to the systematic errors, which meant that the springs extended by a larger and smaller ranges than the expected. In order to ensure accurate results, the experiment should be conducted with the highest amount of professionalism. The masses should be loaded gently without causing vibration oscillations to the springs. Better results can also be ensured by using brand new springs to help solve the problem of elasticity. Brand new springs can sustain their restoring forces that are directed upwards. The measurements should also be performed using a digital ruler, which would have more accurate than the measurements performed by a naked eye. The experiment should also be conducted in an isolated and quiet area where there is less interference whatsoever. References ANON, (2015). 1st ed. [Online] Available at: https://www.engineersaustralia.org.au/sites/default/files/shado/Learned%20Groups/Natio nal%20Committees%20and%20Panels/Engineering%20Design/Part%204.pdf AVISON, J. (1989). The world of physics. Walton-on-Thames, Nelson. BEER, F. P., JOHNSTON, E. R., & DEWOLF, J. T. (2004). Mechanics of materials: Third edition in SI units. Boston, Mcgraw-Hill. BETHELL, G., & COPPOCK, D. (1999). Physics first. Oxford, Oxford University Press. CRANE, F. A. A., CHARLES, J. A., & FURNESS, J. (1997). Selection and Use of Engineering Materials. Burlington, Elsevier. CUTNELL, J. D., JOHNSON, K. W., YOUNG, D., & STADLER, S. (2015). Physics. Louisiana, Wiley and Sons. 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