The Simple Pendulum and Hooke\'s Law PDF

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professor Olugbenga Adeyemi Olunloyo ...

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Physics 221 Section 009 Olugbenga Adeyemi Olunloyo Experiment Performed: 14 November 2017 Report Handed In: 28 November 2017 The Simple Pendulum and Hooke’s Law Introduction The purposes of the Simple Pendulum experiment are to study the motion of a simple pendulum, to study simple harmonic motion, to learn the definitions of period, frequency, and amplitude, to learn the relationships between the period, frequency, amplitude, and length of a simple pendulum, and to determine the acceleration due to gravity using the theory, results, and analysis of this experiment. A simple pendulum is described as a point mass suspended by a massless string from some point about which it can swing back and forth. The time for one complete oscillation is called a period and is defined as T = 2π   (m/k)1/2  , where m is the mass and k is the spring constant. The objectives of the Hooke’s Law experiment are to study simple harmonic motion, to learn the requirements for simple harmonic motion, to learn Hooke’s Law, to verify Hooke’s Law for a simple spring, to measure the force constant of a spiral spring, to learn the definitions of period and frequency and the relationships between them, to learn the definition of amplitude, to learn the relationship between the period, mass, and force constant of a vibrating spring with different masses, and to compare the measured periods of vibration with those calculated from theory. A simple harmonic motion is defined as a body that oscillates back and 1/2 forth. The time for one complete oscillation is called a period and is defined as T = 2π(m/k)   , where m is the mass and k is the spring constant. Procedure The apparatus shown in Figure 2 consists of a support stand with a string clamp, a small spherical ball with a 125 cm length of light string, a meter stick, a vernier caliper, and a timer. A computer with Microsoft Excel is also needed. An Excel spreadsheet is setup to be used based on Figure 3 in the lab manual. The pendulum was displaced and allowed the swing back and forth for 50 oscillations, time was then recorded into the spreadsheet. The period was then calculated and a scatter plot graph was made for period vs. length, period squared vs. length, and period vs. amplitude. The percent difference is then calculated for comparing our results to the accepted value and our results to the theoretical value. The equipment needed for the Hooke’s Law experiment is shown in Figure 1 and consists of a helical spring, a support stand with a mirror scale attached, a mass holder with 50 gram masses and a timer. A computer with Microsoft Excel is used as well. An Excel spreadsheet is setup to look like Figure 2 in the lab manual. Mass was added to a spring in increments of 50 grams and the position of the bottom of the spring was recorded. Displacement is then calculated and a Force vs. Displacement graph is made. Percent

difference is then calculated for the k constant, the mass of the spring, and the k constant from part one to part two. Data See attached. Analysis For both of these experiments, all calculations were done using Microsoft Excel. For the simple pendulum experiment, the radius of the spherical ball was first calculated from the measured diameter, as follows: R = d/2 = 0.01322m/ 2 = 0.00661m The length of the pendulum was calculated by adding the radius of the ball to the string length (ls) , as follows: l=ls + r = 0.39m + 0.00661m = 0.39661m The period (T) for each pendulum length was calculated by dividing the time for 50 oscillations (t50) by 50, as shown: T = t50  /50 = 65.01s/50 = 1.3002s The period of the pendulum was then graphed as a function of its length, with a power function trendline. This resulted in an R2 value of 0.9817, which indicates that the data are an accurate representation of a power function. This supports the equation T = 2 π (l/g)½ , where l is pendulum length and g is acceleration due to gravity, which indicates that the period should be directly proportional to the square root of the pendulum length. The period values were then squared, resulting in T2 , and graphed versus the pendulum length with a linear trendline. The expectation was that the relationship should be linear, as T2 = 4 π 2 (l/g). The R2 value of 0.98004 is extremely close to one, indicating the predicted linear relationship. The slope of this graph (a) was recorded and set equal to 4 π 2/g, and then used to solve for g, as follows: a = 4 π 2/g → 3.7436 = 4 π 2/g → g = 4 π 2/ 3.7436 = 10.53 m/s2 This was then compared to the accepted value for the acceleration due to gravity of 9.81m/s2 using the percent difference formula, as follows: Percent difference = [(experimental result - accepted value)/accepted value] x 100% = [(10.53m/s2 -9.81m/s2 )/9.81 m/s2 ] x 100% = 7.34%

This low percent difference value indicates that the experimental determination of the acceleration due to gravity and therefore the measured period values are accurate, and therefore support the equation T = 2 π (l/g)½ . The period of a pendulum of length 0.78661m was calculated as follows: T = 2 π (l/g)½ = 2 π (0.78661m/ 9.81m/s2 )½ = 1.77 s This was then compared to the measured period of 1.7862s at this pendulum length using percent difference, as follows: Percent difference = [(experimental result - accepted value)/accepted value] x 100% = [(1.7862s-1.77s)/1.77s ] x 100% = 0.92% Again, the small percent difference value indicates that the experimentally determined value for the period was extremely close to that calculated by the equation T = 2 π (l/g)½ . This supports both the validity of the equation and the experimental design. The period of the simple pendulum was then determined at various amplitudes, and the period was then graphed versus the amplitude. The period of the pendulum of length 0.6m was also calculated and included in the graph at an amplitude of zero. This calculation was done as follows: T = 2 π (l/g)½ = 2 π (0.6 m/ 9.81m/s2 )½ = 1.55s Based on the graph of period versus amplitude in the attached data section, amplitudes between 5 degrees and 25 degrees all resulted in amplitudes similar to that calculated for zero amplitude. However, beginning at an amplitude of 30 degrees, the experimentally determined period began to be greater than that calculated by the equation. The difference between these values got larger as the amplitude increased. Therefore, it can be determined that the equation T = 2 π (l/g)½ is only valid at very small amplitudes, and it begins to become invalid around an amplitude of 25 degrees. For the Hooke’s Law experiment, after the position and masses were recorded, the applied force, Fa, was calculated as follows: Fa = mg = 0.05 kg x 9.81m/s2 = 0.4905 N The displacement was calculated by subtracting the position of the spring with no mass attached (s0) from the position measured with mass attached (s), as follows: X = s - s0 = 0.065m - 0.002m = 0.063m The applied force was then graphed as a function of displacement. As predicted by Hooke’s Law, this resulted in a linear relationship, with an R2 value of 0.98137. This value is extremely close to one, and thus indicates a strong linear correlation. The slope of this graph was recorded as the spring constant, k.

In the second part of this experiment, the period (T) was calculated by dividing the total time by 50, as shown in the Simple Pendulum experiment calculations. T2 was also calculated in the same manner. T2 was then graphed as a function of mass. The result was a linear function with an R2 value of 0.99939, indicating a very strong linear correlation. This supports the equation T = 2 π [(m + ms/3)/k]½ , where m is the mass added to the spring and ms is the mass of the spring. The spring constant k was then calculated as follows, where a is the slope of the mass and T2 graph: k = 4 π 2/a = 4 π 2/4.3945 = 8.977 This was then compared to the spring constant found in the first part of the experiment using percent difference: Percent difference = [(k1 -k2 )/k1] x 100% = 3496% difference This is an extremely large percent difference value and can likely be attributed to errors in position measurement in the first portion of the experiment. The further the spring was displaced from equilibrium, the more it bounced and the more difficult it became to measure its position, resulting in random error.. It is therefore likely that the position measurements did not increase as much as they should have as the mass increased. This resulted in a less steep slope on the graph and thus a lower k value. Since time and oscillation number are easier to measure more accurately, it is likely that the second k value is closer to the true spring constant. The mass value at which the Mass on a Spring graph extrapolated to zero was determined by dividing the intercept (b) by the slope (a), as follows: b/a = 0.4242/4.3945 = 0.09653 This value served as the calculated ⅓ mass of the spring (ms) value, and was then compared to the 1/3ms value that was obtained by a balance using percent difference, as follows: Percent difference = [(experimental result - accepted value)/accepted value] x 100% = [(0.09853kg-0.0677kg)/0.0677kg] x 100% = 42.58% This percent difference value initially appears slightly higher than expected. However, the numbers involved in the calculation are rather small, with the first significant digit being in the hundredths place in each instance. In actuality, there is around a 0.03 difference between the measured mass and that given by the balance. Since the numbers are so small, this slight variation results in a higher calculated percent difference. The period for a 250 gram mass was then calculated, as follows: T = 2 π [(m + ms/3)/k]½ = 2 π [(0.25kg + 0.0677kg)/0.1222]½ = 10.13s

This was compared to the measured period value using percent difference: Percent difference = [(calculated value - measured value)/measured value] x 100% = [(10.13s1.238s)/1.238s] x 100% = 718.26% This large percent difference is attributable to the same error discussed with the percent difference of the two spring constant values. Since the period was calculated using the first k value, which is likely the more inaccurate one as already described, then this resulted in a less accurate calculated period. If the second, larger k value had been used, thus would have resulted in a smaller calculated period, thus making it more similar to the measured period and resulting in a smaller percent difference. Therefore, though these percent difference values do not appear to support the equation T = 2π [(m + ms /3)/k]½, this can actually be attributed to random experimental error in measuring displacement. Conclusions For the Simple Pendulum experiment, the prediction that T2 and the pendulum length would be linearly correlated was supported by the R2 value of 0.98004, which is extremely close to one. This was further supported by using the graph’s data to calculate acceleration due to gravity, which resulted in a value with only a 7.34% difference from the accepted value of 9.81m/s2 . The equation T = 2π (l/g)½ was further supported by it being used to calculate the period of a pendulum of length 0.78661mand comparing it to the measured value, which resulted in only a 0.92% difference. By graphing period versus amplitude, the equation T = 2 π (l/g)½ was established to be valid only at small amplitudes, becoming increasingly more inaccurate starting at an amplitude of 25 degrees. For the Hooke’s Law experiment, force and displacement were determined to have a linear relationship with an R2 value of 0.98137, thereby supporting Hooke’s Law. T2 and mass also showed a linear relationship with an R2 value of 0.99939, indicating a very strong linear correlation, as predicted. This supports the equation T = 2 π [(m + ms/3)/k]½ . The spring constant k was also calculated and compared to that found graphically in the first part of the experiment. The large percent difference value of 3496% indicated that one of values was inaccurate, likely that determined graphically since it was based on the measurement of displacement rather than time. In order to reduce errors in the future, it would be useful to focus on measurements of time, which are simpler to make accurately, rather than measurements of displacement, which are difficult to make accurately with a spring. This is further emphasized by the large percent difference between the measured and calculated period for a 250g mass, which was based on the k value from the graph. Overall, the Simple Pendulum experiment was extremely successful, and the Hooke’s Law experiment was successful for the second part which involved the measurement of time. The two labs together served to demonstrate simple harmonic motion, support the equations of period,

frequency, amplitude, and length, and to demonstrate Hooke’s Law. The results of the lab predominantly supported these equations and theories as expected....