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Simple Pendulum Lab Report
Elementary University Physics I (Phys 1007)
Carleton University
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Carleton University Laboratory Report Course #: PHYS 1007 Experiment #: 5 Simple Pendulum Experiment Charlotte Bakker 101102402 Date Performed: November 23, 2018 Date Submitted: November 30, 2018 Lab Period: Friday, PM, L7 Partner: Philippe Beaulieu Station #: 25 TA: Mohamed Page 1! of 14 Abstract The goal of the experiment was to identify the relationship between the period of a pendulum and the length of the string, mass of the bob, and the amplitude of the oscillations, and to then confirm the result with a conservation of energy analysis of the pendulum. The final results from this is that there is no relationship between the mass and the period of the pendulum; there is a linearly proportional relationship between the length of the string and the period of the pendulum; and there is also a proportional relationship between the period and the amplitude of oscillations. The consistency test for the value of a was consistent with the accepted values, but the values for b and c were inconsistent. Theory As shown in Figure 5 the simple pendulum has a bob suspended from a string across a pivot point where it is able to swing freely. The bob is affected by just the force of gravity (Fg = mg ) when air resistance is ignored and the tension of the string on the bob (FT). The horizontal displacement (x), is when the pendulum is released and it then will oscillate about equilibrium position because the gravity force acting on it that restores it. The time it takes for the pendulum to complete one full cycle back and forth is the period of oscillation (T). To find the amplitude of oscillation (𝜃0), trigonometry can be used by relating the length (L) and the horizontal displacement (x) of the bob Figure 5 Simple Pendulum Diagram [from Lab Manual] (1) The uncertainty of the amplitude of oscillation is given by, (2) Page 2! of 14 1 pendulum (U) = mgh (8) and the kinetic energy (K) = ! !mv 2 (9) is then at its minimum. 2 When the pendulum is vertical then the potential energy is all converted into kinetic. So at the equilibrium position (rest) the kinetic energy is at a maximum and constant velocity, and the potential energy must then be at a minimum. To calculate the average for all the trials, the following is used, where x! ¯ is the average (10) The standard deviation is given by, (11) The standard deviation of the mean is found by, (12) For possible discrepancies in the data, a consistency test is performed (13) Page 4! of 14 Apparatus Materials used in simple pendulum apparatus: • String • Cylinder brass bob • Cylinder Aluminum bob • Metre Stick - Range: 100 cm - Resolution: ±0 mm • Scale - Range: 0-400g. - Resolution: ± 0 g. • Vernier Calliper - Range: 0-150 mm. - Resolution: 0 mm. • Vernier Photogate • LabQuest Mini Interface • Logger Pro Figure 5 Simple Pendulum Apparatus Observations Table 5 Masses and Dimensions of Aluminum and Brass bob for Simple Pendulum Mass (g) Diameter (mm) Length (mm) 47 ± 0 19 ± 0 19 ± 0 47 ± 0 19 ± 0 19 ± 0 47 ± 0 19 ± 0 19 ± 0 Average 47 ± 0 19 ± 0 19 ± 0 Aluminum Bob 14 ± 0 18 ± 0 19 ± 0 14 ± 0 18 ± 0 19 ± 0 14 ± 0 18 ± 0 19 ± 0 14 ± 0 18 ± 003 19 ± 0 Brass Bob Average Part A The effect of the pendulums length on the period is that it increases the period of oscillation. It is evident in Table 5 and in Figure 5 that as the length is increased, a linear relationship with the period can be observed. Page 5! of 14 Trial Brass Bob Length (cm) 1 0 - 50 30 1 ± 0 0 - 0 - 50 32 1 ± 0 0 0 5 50 35 1 ± 0 0 - 50 35 1 ± 0 0 - Average 1 ± 0 0 6 50 37 1 ± 0 0 - 50 37 1 ± 0 0 - Average 1 ± 0 5 4 32 1 ± 0 1 ± 0 4 0 50 Average 3 Potential Energy, U (J) 30 1 ± 0 1 ± 0 2 Kinetic Energy, K (J) 50 Average Average Distance, x (cm) Velocity, V (m/ s) 0 7 50 40 1 ± 0 0 - 50 40 1 ± 0 0 - 1 ± 0 0 9 The data for the angle of the amplitude of oscillation, 𝜃0, the distance, x, the period, velocity is contained in Table 5 on page 14 In Figure 5 the relationship between the period and the amplitude of oscillation angle is demonstrated, From this relation, it is found that as the angle is increased, so is the period of oscillation Graphs Page 7! of 14 Figure 5 Demonstrates the linear relationship of the length of the pendulum with the period of oscillation of the pendulum on page 13 Figure 5 demonstrates the relationship between The period of the pendulum and the angle of amplitude. The relationship can be viewed on page 14 Sample Calculations 1. Average measurements (10) 47 + 47 + 47 3 x! ¯ = 47 g The same calculation was done for each of the measured quantities (mass, diameter, length, and period) x! ¯ = ! 2. Standard Deviation (11) 𝜎=√( 1 (47 g-47 g)2 + (47 g-47 g)2 + (47 g-47 g)2) 3−1 𝜎 = 0 g The same calculation was done for each of the measured quantities (mass, diameter, length, and period) 3. Standard deviation of the mean Page 8! of 14 6. Consistency test for a, b, and c values (11) t= ! 1 − 0 (0 + 02) t = 0 for the value of a The same consistency test was done for values of b (t >>2) and c (t >>2) as well. 7. The period of oscillation (4) T = 2π √(0 m/ 9 m/s2) = 1 s This calculation was done for each length 8. Vertical Displacement (6) h = 0 m- √((0 m)2 - (0 m)2) h = 0 m The same calculation was done for each value of x 6. Kinetic energy 1 2 K = ! mv ! 2 (9) Page 10 ! of 14 1 /s)2 K = ! (0)(1 ! 2 K = 0 J The same calculation was done for each separate velocity 7. Gravitational Potential Energy (8) U = mgh U = (0 kg)(9 m/s2)(0 m) U = 0 J The same calculation was done for each different vertical displacement 8. Conservation of Energy Analysis From the calculations above it is shown that ∆K ≠ ∆U and the sum of the two is not equal to zero. Discussion The goal of the experiment was to illustrate the relationships of the period of oscillation with many variables, such as the length of the string, the mass of the bob used, and the amplitude of oscillations. A conservation of energy analysis was also performed for the pendulum. From this experiment, it was found that the mass doesn’t not have a direct effect on the period of oscillation of the pendulum. When the length of the pendulum was changed, the period of oscillation also changed, indicated a proportional relationship between the two. There is also a proportional relationship between the amplitude of oscillation and the period of oscillation. From the consistency test for each of the values of a, b, and c, it yielded that a is consistent, but b and c are not. From the conservation of energy analysis is was found that not all of the energy was conserved. If there was access to a force sensor then we could measure the force that the pendulum was dropped with to more accurately measure the kinetic energy of the Page 11 ! of 14
Simple Pendulum Lab Report
Course: Elementary University Physics I (Phys 1007)
University: Carleton University
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