Simple Pendulum Lab Report PDF

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PHYS 1007 Lab Report Simple Pendulum Etienne Rollin...

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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

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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 5.1 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 5.1 Simple Pendulum Diagram [from Lab Manual]

(1) The uncertainty of the amplitude of oscillation is given by,

(2)

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If there’s a small angle of oscillation, then the pendulum is in simple harmonic motion and its angular displacement is now given by, (3) Where t is the time of oscillation, and T is the period.

The period of the pendulum can be approximately determined for oscillations well below 60˚ by, (4)

L is the length of the string, and the gravitational acceleration is g. When there is an oscillation that is much larger than 60˚, the period can be found by the infinite series,

(5)

In the conservation of energy analysis, the vertical displacement from equilibrium position of the bob, (h) can be given by, (6)

In this system, there are no external forces doing any work and therefore the total mechanical energy is conserved. Then this relationship is shown by, (7) When the vertical displacement is at a maximum, then the potential energy of the

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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)

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Apparatus Materials used in simple pendulum apparatus: • String • Cylinder brass bob • Cylinder Aluminum bob • Metre Stick - Range: 100 cm - Resolution: ±0.1 mm • Scale - Range: 0-400g. - Resolution: ± 0.01 g. • Vernier Calliper - Range: 0-150 mm. - Resolution: 0.01 mm. • Vernier Photogate • LabQuest Mini Interface • Logger Pro

Figure 5.2 Simple Pendulum Apparatus

Observations Table 5.1 Masses and Dimensions of Aluminum and Brass bob for Simple Pendulum Mass (g)

Diameter (mm)

Length (mm)

47.97 ± 0.01

19.07 ± 0.01

19.02 ± 0.01

47.97 ± 0.01

19.10 ± 0.01

19.05 ± 0.01

47.96 ± 0.01

19.08 ± 0.01

19.03 ± 0.01

Average

47.97 ± 0.03

19.08 ± 0.03

19.03 ± 0.03

Aluminum Bob

14.18 ± 0.01

18.67 ± 0.01

19.08 ± 0.01

14.18 ± 0.01

18.67 ± 0.01

19.09 ± 0.01

14.18 ± 0.01

18.69 ± 0.01

19.08 ± 0.01

14.18 ± 0.03

18.68 ± 003

19.08 ± 0.03

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 5.2 and in Figure 5.3 that as the length is increased, a linear relationship with the period can be observed.

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Table 5.2 Periods of Oscillation for the two bobs at varying lengths Bob

Length, L (cm)

Period, T (s) of approx. 10 oscillations

Aluminum

30.0 ± 0.1

1.105 ± 0.0003970

9

1.106 ± 0.0004131

9

Brass Bob

Number of Samples, N

Average

1.106 ± 0.0008101

40.0 ± 0.1

1.271 ± 0.0003099

9

1.271 ± 0.0003745

9

Average

1.271 ± 0.0006844

50.0 ± 0.1

1.418 ± 0.0003096

9

1.419 ± 0.0001404

9

Average

1.419 ± 0.0002250

60.0 ± 0.1

1.556 ± 8.001E-5

7

1.556 ± 0.0002871

9

Average

1.556 ± 3.67E-4

70.0 ± 0.1

1.679 ± 0.0001720

9

1.680 ± 0.0004174

9

Average

1.680 ± 0.0005894

50.0 ± 0.1

1.421 ± 0.0001264

9

1.420 ± 0.0001317

9

Average Interpolated Value

1.421 ± 0.0002581 1.421

As shown in Figure 5.3 and Table 5.2, the interpolated value for the period of oscillation for a 50 cm pendulum length of the brass bob was consistent with the average measured value. Therefore the mass has no effect on the period of oscillation of the pendulum. Part C Table 5.5 Measurement of Length, Distance, and Velocity for the Brass Bob

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Trial

Brass Bob

Length (cm)

1

0.03430 -

50

30 1.354 ± 0.009035

0.04397 -

0.05225 -

50

32.5 1.473 ± 0.01009

0.05204 0.05218

5.6486

50

35 1.594 ± 0.007381

0.06094 -

50

35 1.640 ± 0.006879

0.06451 -

Average

1.617 ± 0.007130

0.06271

6.7260

50

37.5 1.739 ± 0.01257

0.07253 -

50

37.5 1.644 ± 0.01310

0.06483 -

Average

1.692 ± 0.012835 5

4.7059

32.5 1.476 ± 0.007026

1.475 ± 0.017116

4

0.03899

50

Average 3

Potential Energy, U (J)

30 1.196 ± 0.004920

1.275 ± 0.013955 2

Kinetic Energy, K (J)

50

Average

Average

Distance, x (cm) Velocity, V (m/ s)

0.06867

7.9661

50

40 1.792 ± 0.01272

0.07702 -

50

40 1.752 ± 0.01938

0.07362 -

1.772 ± 0.01605

0.07531

9.4117

The data for the angle of the amplitude of oscillation, 𝜃0, the distance, x, the period, velocity is contained in Table 5.4 on page 14 In Figure 5.4 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 5.3 Demonstrates the linear relationship of the length of the pendulum with the period of oscillation of the pendulum on page 13 Figure 5.4 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)

47.97g + 47.97g + 47.96g 3 x! ¯ = 47.97 g The same calculation was done for each of the measured quantities (mass, diameter, length, and period) x! ¯ = !

2. Standard Deviation (11)

𝜎=√(

1 (47.97 g-47.97 g)2 + (47.97 g-47.97 g)2 + (47.96 g-47.97 g)2) 3−1

𝜎 = 0.71 g The same calculation was done for each of the measured quantities (mass, diameter, length, and period) 3. Standard deviation of the mean

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𝜎mean =

0.71g 3

𝜎mean = 0.41 g The same calculation was done for each of the measured quantities (mass, diameter, length, and period) 4. Amplitude from the horizontal displacement of the bob (1)

The same calculation was done for each of the horizontal displacements

5. Uncertainty on amplitude of oscillation (2)

The same calculation was done for each of the horizontal displacements and the angle that was calculated from them

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6. Consistency test for a, b, and c values

(11)

t= !

1 − 0.2329

(0.0012 + 02) t = 0.77 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 √(0.3 m/ 9.81 m/s2) = 1.099 s This calculation was done for each length 8. Vertical Displacement (6) h = 0.50 m- √((0.50 m)2 - (0.30 m)2) h = 0.14 m The same calculation was done for each value of x

6. Kinetic energy

1 K = ! ! mv 2 2

(9)

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1 K = ! ! (0.04797kg)(1.617m /s)2 2 K = 0.06271 J The same calculation was done for each separate velocity

7. Gravitational Potential Energy (8)

U = mgh U = (0.04797 kg)(9.81 m/s2)(0.14 m) U = 0.0659 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 1! 1 of 14

pendulum. Even though from this experiment it was found that the mass didn’t affect the period of the pendulum, on something like a swing the periods are different for a child and an adult. This could be because of the different force of gravity from the child and the adult, which depends on the mass of the object. The oscillation of the amplitude was shown to decrease with time due to friction, which is why only the first three data points were included. After some more time passes, the system will eventually stop moving. This is when then the kinetic energy is now at a minimum (zero) and the potential energy is now back at a maximum. An experiment to find the energy dissipation rate which is the change in energy over a period of time could be having the pendulum in an enclosed case, and other different pendulums are used to find the sources of dissipation of energy (Atkinson, 1938). The oscillations started to wander out of the perpendicular path of the photogate, this could cause the periods of oscillation to be shorter, as it looses energy faster from travelling in that direction as it now has movement in both the x and y direction of the path. To be able to mesure the gravitational acceleration one could find a graph of T2 and L which is proportional to g, when using the small angle approximation model, (Nethercott, 2013).

References E C Atkinson. 1938. Proc. Phys. Soc. (50): 721 Nethercott, Quintin T. “Determining the Acceleration Due to Gravity with a Simple Pendulum.” University of Utah, 2013.

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