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Hooke's Law Lab Report PHY 113
General Physics (PHY 112)
Arizona State University
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PHY 113: Hooke’s Law/Springs Objective: The objective of this lab was to test Hooke’s Law by measuring the spring constants of different springs and spring systems and to investigate whether all elastic objects obey Hooke’s Law. The aforementioned was tested by physically attaining values for the spring constant (k), the slope of the graph from the force vs. displacement graph which is equivalent to the spring constant, force (N), and displacement/position of spring (m). Equipment: long rubber band, two nearly identical springs, support stand with a meter stick, DataStudio data collection software, LoggerPro graphical software, pulley, string, set of masses from 100 g up to 600 g, balance scale, Science Workshop interface with force sensor and rotational motion detector used as linear sensor Procedure: This experiment consisted of three parts. The first part was subdivided into 2 methods. The first method was the classical method in which masses were added to a weight hanger attached to a singular spring. Masses were added and for each additional mass, the position of the spring was measured with a meter stick. This process was repeated for a maximum of 600 g added to the weight hanger. The process was then conducted in reverse where individual masses were removed and position was manually measured and recorded with a meter stick. These measurements were used to calculate force and displacement through LoggerPro. The second subdivision of part 1 involved the use of a pulley and string to manually stretch the spring with human force from its initial position until the force meter read approximately 10 N. The graph produced by DataStudio was used to analyze the two different methods of part 1. Part 2 involved springs in series and in parallel. The applied force of approximately 10 N was repeated for a series of springs in parallel to each other and in series (one attached linearly to each other). Two separate graphs were produced. Part 3 investigated the application or lack thereof of Hooke’s law on a nonlinear springy rubber band. The rubber band was stretched until the force sensor read approximately 10N. Experimental Data: The following is a data table that displays the data obtained during the experiment in all 3 parts which was used (slopes of the graphs produced and their uncertainties) which were used to analyze their relevance to the applicability of Hooke’s Law. Part 1, Method 1: ks1_exp (26土 0) Part , Method 2: ks2_exp (25土 2-2) Part 2, Springs in Series: kser (12土 4-2) Part 2, Springs in Parallel: kpar (49土 0) It is important to note that although Tables 1 and 2 appear similar, Table 1 merely provides slope along with uncertainty whereas table 2 utilizes the slope in context of the experiment as k (the spring constant). This has to be done separately to thoroughly understand the relationship between slope and the spring constant. This is done through using Hooke’s Law: Fel = k|Δx| (the negative sign in front of k is omitted through the absolute value of displacement because it is acknowledged that the restoring force is opposite in direction of stretching of the spring. A simple rearrangement of the equation results in k=|Δx|/Fel. The experimental values for force and displacement calculated from the sensor and workshop interface can be plotted with displacement on the x-axis and force on the y-axis and the slope is indicative of k because it expresses the same relationship. Discussion and Conclusion: The purpose of this lab was to investigate the applicability of Hooke’s Law through measuring the spring constant of various spring systems and also with a nonlinear springy object (rubber band). This resulted in 4 experimentally obtained slopes through a classical method by adding and removing masses and 2 springs in series and parallel to each other - these slopes are, respectively, (26土 0), (25土 2-2), (12土 4-2), and (49土 0) N/m. These experimental slopes allowed for the theoretical spring constant of springs 1 and 2, which are respectively (25土 0) for spring 1 and (24土 0) N/m. These spring constants were calculated from the slopes of the experimentally produced graphs. Theoretically, Hooke’s Law is given by Fel = k|Δx| where F is elastic force, k is the spring constant, and |Δx| is displacement or difference between unstretched equilibrium position of spring and a specific stretched positionel and Fg are equivalent to each other because Fel and Fg = 0 since both forces act in opposite directions in the y-axis. Therefore, Fel= Fg = mg. This basic manipulation was utilized to calculate force in part 1, method 1 to determine slope and therefore k. The force sensor produced values for F and the Science Workshop interface recorded values for displacement (these values were manually measured with a meter stick in part 1 or the classical method). Since the values of force and displacement are experimentally produced and Hooke’s Law can be rearranged as, k = F/|Δx|. Therefore, the slope of a graph with displacement |Δx| on the x axis and F (N) on y-axis can be equated to the spring constant. As such, the slopes were used as the experimental spring constants. The percent discrepancies of 4% for springs in series and 0% for springs in parallel indicate that the experimental spring constants were accurate as both percent discrepancies are less than 10%. Since the results are accurate, it can be said that Hooke’s Law holds for springs, and also springs in series and parallel. The graphs further support this assertion. All 4 graphs for parts 1 and 2 combined (only parts 1 and 2 deal with Hookean springs) evidently portray a strong, positive, and directly proportional relationship between elastic force and Δx, as given by the positive slopes, which is expected since Hooke’s Law defines that Fel = k|Δx|. The slopes of the graphs for part 2 also reveal that the spring constant for springs in series is half that of a single spring (12 = kser is approximately half of 26 = ks1_exp). The slope of the second graph for part 2 highlights that when the springs are parallel, the spring constant is doubled compared to a single spring (49 = kpar is approximately twice the value of 25 = ks2_exp). The experimentally produced graph for part 3 of a nonlinear rubber band has a leaf like appearance. No clear relationship can be derived between force and displacement and a slope cannot be calculated because it constantly fluctuates (as outlined by the leaf like appearance). This indicates that a spring constant does not exist for the rubber band because a definitive slope cannot be calculated. Therefore, it can be concluded that Hooke’s Law cannot be applied to non-Hookean objects such as rubber bands because a spring constant does not exist/impossible to calculate because slope constantly changes. Hooke’s Law cannot be applied to to all elastic objects even though the rubber band is spring like and nonlinear. Placed in context, a spring constant is indicative of stiffness with stiffer, more difficult springs having a larger k. A rubber band does not have one which can be interpreted to mean that it is extremely easy to stretch and its elasticity is due to IMFs in the material and rearrangement of the polymers within the rubber band material rather than the elastic force like that found in a spring. Since the spring constant is halved when the springs are in series, the smaller k indicates that springs in series are easier to stretch compared to springs in parallel where k is doubled because a larger spring constant is indicative of increased stiffness/difficulty to stretch. The experimental results strongly support this aspect of Hooke’s Law in relation to springs as opposed to rubber bands. Even though the experiment was highly successful and the percent discrepancies are small, the existence of the percent discrepancies can be attributed to systematic error. As both experimental spring constants, (26土 0) and (25土 2-2) N/m were larger than the theoretical spring constants of (25土 0) and (24土 0) N/m, the elastic capacity of the spring might have been increased due to the condition of the springs. Both springs appeared unused, new and had a bright metallic luster which means that they probably were at their peak performance capability and have greater elastic force/restoring force compared to springs that have been repeatedly used in the past. As a result, the experimental spring constants are higher than the theoretical values since elastic force and k are directly proportional according to the equation Fel = k|Δx|. A larger restoring force corresponds to a larger spring constant since displacement and restoring force are directly proportional so an increase in one corresponds to an increase in the other which overall leads to a larger k value. This systematic error could have been mitigated by averaging the results of the students or by using a spring that has been used in the past which could have less rigidity and led to a more accurate experimental spring constant. Another possible systematic error lies within the force sensor of the apparatus. The force sensor read a slightly higher force than expected when a 1 kg weight was hung which means that all of the consequently recorded force values were probably slightly higher than the true values. As a result, both experimental k’s are higher than the theoretical values since force is directly proportional to the spring constant k which means that increasing force increases the value of k. This error could have been mitigated by being more cautious when calibrating the equipment and vigilant of the calibration values and set points before starting the experiment. Another minor source of error could have risen from frictional forces between the spring and air or possibly between the wheel of the pulley and the string could have acted in the same direction as the
Hooke's Law Lab Report PHY 113
Course: General Physics (PHY 112)
University: Arizona State University
