Simple Pendulum (g)

Fawaz Alazmi

28552

Engineering Principles

Lab Report

Aim

The purpose to find out the gravitational acceleration due to gravity. In addition, the experiment was managed to compare the theory and experiment part of acceleration due to gravity.

Introduction

A simple pendulum is pretended to be a point mass which is suspended freely by a string that is also massless (“Measurement of g: Use of a simple pendulum”). The string allows the pendulum to oscillate from the same point where the string is attached. The motion assumed by the pendulum is periodical as it swings back and forward (Nethercot & Walton, 2013). According to experiment, the time of oscillation of the pendulum is changed only by the length of the pendulum as long as it oscillates at small distances (Mohazzabi & Shankar, 2017). The size of the angle resulting from the pendulum swing and the (m) of the pendulum have no influence on the period of the pendulum. Furthermore, the (g) also affects the period of the pendulum. The equation of a pendulum that defines the period is expressed as given in equation 1 below (Parks, 2000).

(1)

Where;

T= period (s)

l= of pendulum (m)

g= gravitational acceleration (m/s^2)

Equation 1 is obtained from the beginning where the pendulum is placed to small arc (displaced at small angles) (Aggarwal, Verma & Arun, 2014). It is important to note from the formula that the period depends on two parameters (acceleration due to gravity and the length).

Apparatus

• The meter stick

• String

• Stopwatch

• Support stand

• Clamp

• Spherical ball

The apparatus was arranged as shown below

Figure 1: The simple pendulum

Method

The pendulum was setup as shown in figure 1 above. The string was to suspend the ball. The string was held in place by two wooden pieces to ensure that the pivoting point remained constant. The length of the pendulum was measured the first length was 0.200 m then it was displaced at an angle of 100. The time the pendulum took to finish ten motions timed using a stopwatch. The time for 10 oscillations to complete was measured and repeated for the same length and the two times were averaged. The length of the pendulum was changed and the timing of the oscillations was repeated for six different lengths. For each of the lengths, 10 oscillations were timed twice then by averaging and the outcomes recorded in table 1 underneath. The period was isolated by 10 to decide the time of a solitary wavering for every length.

Data and Results

Table 1: The table shows the time of oscillations of the pendulum at different lengths.

Length, l (m)

Time for 10 oscillations in seconds

Time period T (s)

4pi^2l (m)

T^2 (s^2)

1

2

Avg

0.200

9.21

9.18

9.195

0.92

7.90

0.85

0.400

12.90

12.78

12.84

1.28

15.79

1.64

0.600

15.52

15.38

15.45

1.55

23.69

2.40

0.800

18.08

18.06

18.07

1.81

31.58

3.28

1.00

19.97

19.60

19.79

1.98

39.48

3.92

1.200

21.8

21.50

21.65

2.17

47.37

4.71

Analysis

Figure 2: A plot of 4pi^2l vs T^2: The graphs shows the difference of the 4pi^2l against the square of the period of the oscillations of the pendulum.

The gradient of the graph= 10.215 m/s^2

Accordingly, the experimental value is equal to the slope of the graph which is 10.215 m/s^2.

From research, value of gravitational acceleration, g= 9.81 m/s^2.

When comparing the g=9.81 m/s^2 to experimental value for acceleration due to gravity the error is calculated as;

% error = ((experimental value – theoretical value)/ theoretical value) * 100

wherefore, percent error in g= ((10.215 m/s^2 – 9.81 m/s^2)/ 9.81 m/s^2)* 100 = 4.13%

The error between the experiment and the measured value was quite small (less than 5%), so, the experiment was relatively accurate.

Discussion

The slope of the graph of 4pi^2l against the square of the period was 10.215 m/s^2. Therefore, from equation 1, the slope represents the acceleration due to gravity, so, the measured g= 10.215 m/s^2. The percent difference between the two values (theoretical and experimental) was 4.13%. The percent was positive as the observed g was greater than the theoretical g. In spite of the experimental error noted by the percent difference, the accuracy of the results were high as the error was less than 5%. It was established that the period increased with an increase in the pendulum length. Furthermore, the square of the period varied proportional to the pendulum length. The small error encountered during experiment could be apply to the sources of error that could be minimized. However, the error are can be minimized further with proper consideration of some improvements as discussed below.

Experiment Improvements

The effect of air resistance could have increased the time of the oscillations. Air resistance can effect of dragging the spherical ball, therefore, reducing its energy due to friction. The time recorded for the oscillations would be lower than the actual times of swings, so, reducing the periodic time. The gravitational acceleration changes when the period changes, therefore, the drag had effect on the value of g. We could’ve have recorded longer time of oscillations due to longer reaction times when they were timing the oscillations.

The accuracy of the results can be improved by ensuring that the reaction time is as minimal as possible. The timer requires being aware to ensure that he starts the timer at the same time with the release of the pendulum to reduce the timing delay. Furthermore, the use of spherical ball with smaller diameters and less mass can minimize the effect of drag and friction leading to better results. The angle displacement of the pendulum should be small (<100) for the equation relating the square of the period to the length and the acceleration due to gravity to hold.

Conclusion

In summary, the experiment measured gravitational acceleration, g, using the oscillations of a simple pendulum. The observed and the measured value of acceleration due to gravity were compared to ascertain the accuracy of the results. The measured value of g was 10.215 m/s^2, it was greater than the observed value which is 9.81 m/s^2. The percent difference between the theoretical value and the measured value was 4.13%. Markedly, the error was almost minor, this could be understood to mean that the results were relatively accurate. The period increased as the length of the pendulum was increased. The acceleration due to gravity obtained in the experiment was close to the observed value, therefore, the experiment was successful.

### References

Aggarwal, N., Verma, N., & Arun, P. (2014). Simple pendulum revisited. European Journal of Physics.

Elbori, A., & Abdalsmd, L. (2017). Simulation of simple pendulum. International Journal of Engineering Science Invention. 6(4), 33-38. Retrieved from http://www.ijesi.org/papers/Vol(6)4/F06043338.pdf.

Measurement of g: Use of a simple pendulum. Retrieved from http://www.kbcc.cuny.edu/academicdepartments/physci/science25/Documents/Exp2.pdf.

Mohazzabi, P., & Shankar, S. P. (2017). Damping of a simple pendulum due to drag on its string. Journal of Applied Mathematics and Physics. 5, 122-130. https://file.scirp.org/pdf/JAMP_2017012515591136.pdf.

Nethercott, Q. T., & Walton, M. E. (2013). Determining the acceleration due to gravity with a simple pendulum. Retrieved from http://www.physics.utah.edu/~ewalton/lab_report.pdf.

Parks, J. E. (2000). The simple pendulum. Retrieved from http://www.phys.utk.edu/labs/simplependulum.pdf.

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