Study of Laws of Simple Pendulum Motion to Teach About Simple Harmonic Motion

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Abstract

Simple harmonic motion is the motion of a body travels back and forth over a fixed path, returning to each position and velocity after a certain interval of time. A simple pendulum definitely can be used to demonstrate simple harmonic motion. The objectives of this assignment are to investigate the relationship between length of the simple pendulum and the period as well as determine the frequency of a simple pendulum and build a pendulum wave. A simple pendulum can be defined to have an object that consist of a small mass which is hung from a string at a fixed pivot point. A prototype was done by the student to prove and support the Physics concept. A combination of 15 pendulums which made up of similar nuts and strings with different length was able to produce a pendulum wave. Besides that, the frequency of each pendulum was calculated after the period was taken. The displacement and the velocity of each simple pendulum can be calculated as well since having the frequency. Last but not least, the details and the analysis of data were discussed. To conclude, the objectives were managed to achieve and the student had learned more about simple harmonic motion.

|Keywords|: Simple harmonic motion; pendulum; length; period; frequency; and mass.

Introduction

The story began with perhaps, apocryphal, observation of the revolving chandeliers at Pisa cathedral by Galileo. By using his heart rate as a clock, Galileo undoubtedly made the quantitative discovery that the time or length of a swing for a given pendulum is independent of the amplitude of the displacement of the pendulum (Baker & Blackburn, 2005). A pendulum can be defined as mass, or also known as bob. This bob connected to a rod or rope, and when it swings without friction, it will experience one phenomenon that is call as simple harmonic motion. When the mass of the pendulum is hanging downward, it can be said that the pendulum is at the equilibrium position. The period, the time interval, T required to complete one oscillation of a simple pendulum can be defined as acceleration of the gravity, g, and the length of the pendulum, L (N.A).

In this project, a pendulum wave was created by using the laws of simple pendulum motion to teach about simple harmonic motion. Pendulum wave is known as a device where many pendulums of different length that increasing from the first pendulum until the last pendulum start swinging at the same time. The difference in the length of the pendulum will results in different periods. As the pendulums move in and out of motion, they create a sequence of cycling visual wave pattern (Crockett, 2010). The lengths of the pendulum were adjusted such that if the longest pendulum performs L oscillations in a time interval, T, then each consecutively shorter pendulum must execute an additional oscillation in that same time interval. This creates an array of the pendulum in which the phase change per unit time is the same for any two adjacent pendulums.

Thus, if all the pendulums started in motion in phase at some time, t= 0, after the time interval T they will again all be in phase. As the pendulum goes through their gyrations the system passes through a sequence of traveling waves, various standing waves, periods of apparent random motion, and a sequence of traveling waves moving in the direction opposite to the original traveling waves, with all pendulums ultimately coming back in-phase (Berg, 1990). The objectives of this project are to determine the relationship between the length of the pendulum and the period, to determine the frequency of simple pendulum, and to build a pendulum wave. Based on the previous study of simple harmonic motion, a simple pendulum that oscillates will have its own frequency f, which is the number of vibrations the system makes per unit of time. The frequency of the pendulum can be estimated from the calculated period.

Objectives

  1. To build a pendulum wave.
  2. To determine the relationship between the length of the simple pendulum and the period.
  3. To determine the frequency of a simple pendulum.

Problem statement

The students were confused about simple harmonic motion and was having difficulties in understanding the relationship between simple harmonic motion and pendulum. Therefore, the prototype of pendulum wave can help the students to understand this physics concept better by understanding the relationship between length of simple pendulum and its period.

Literature review

The concept of a simple pendulum is a particle suspended by a weightless string. In action, it consists of a small object, normally a sphere, suspended by a string where its mass is negligible relative to that of the sphere, and where the length is much greater than the sphere's radius. Under these conditions, the mass of the system can therefore be regarded as being concentrated in a poi

Consider the simple pendulum diagram shown in Figure 1. The displacement, arc length x, for small angle q that the string makes with the vertical is given by:

The vibration duration is the time it takes to go through one loop (i.e. the time for the pendulum to travel back to the same point from some point on its path with motion in the same direction) and is related to w (the angular velocity) by the T=2p / w relationship.

Frequency is the number of occurrences of a repeating event per unit of time (Davies, 1997). It can be determined by using formula:

Materials and apparatus

  • Wood
  • String
  • Nuts
  • Nails
  • Stopwatch
  • Thumb tack

Methodology

To make a pendulum wave, the main thing that needed is its model. To build the model, the main material used is a wood. In this project, five woods were prepared. There are two types of these woods, four of them are in about 16.5 inch which is used to make a stand while one of them is in about 30 inch which is used place the end string of the pendulums. Firstly, every two 16.5 inch woods of the four woods are grafted to make a ‘V’ shape and affixed to the base to provide stability, making a “bridge” connecting the stand. After the body of the model prepared, a strings were cut and prepared on a specific length which is 89mm, 94mm, 99mm, 105mm, 110mm, 117mm, 124mm, 131mm, 140mm, 149mm, 159mm, 170mm, 183mm, 196mm, 211mm.

Then, each end of the strings was attached with a same weight which is the steel nut with equal mass, size and shape. After the strings were attached to the steel nuts, each piece of string was tied located every 3.75 cm along the ruler roughly. Lastly, the prototype was constructed and all the fifteen pendulums firstly must be brought to a position of about 20° from their vertical position using a ruler, large book or another plank wood. To test the model whether it works, the pendulums were released at the same time so that they swing perpendicular from where were they hanging.

Result

The period of every simple pendulum can be calculated using equation:

Where L is length of the string, m that attached to the mass or bob and g is the acceleration of the gravity

  • Table 1: Relationship Between Length of Simple Pendulum and the Period

Pendulum

Length, m

Period, T (S)

Period, T (S)

1. 0.211 / 1.42

2. 0.196 / 1.26

3. 0.183 / 1.03

4. 0.170 / 0.98

5. 0.159 / 0.90

6. 0.149 / 0.86

7. 0.140 / 0.83

8. 0.131 / 0.79

9. 0.124 / 0.75

10. 0.117 / 0.72

11. 0.110 / 0.69

12. 0.105 / 0.67

13. 0.099 / 0.63

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14. 0.094 / 0.61

15. 0.089 / 0.59

Based on the result obtained, it shows that the greater the length of the pendulum the longer the period calculated. Thus, the relationship between length of the pendulum and period can be determined where the length of the pendulum is directly proportional to the period.

By obtaining the period of every pendulum, the frequency of the pendulums can be calculated using equation:

Where g is acceleration of the gravity, and L is length of the pendulum, m

  • Table 2: Length of the Pendulum, the Period and Frequency of the Pendulum

Pendulum

Length, m

Period (T)

Frequency, Hz

1. 0.211 /. 0.921

2. 0.196 / 0.888

3. 0.183 / 0.858

4. 0.170 / 0.827

5. 0.159 / 0.800

6. 0.149 / 0.774

7. 0.140 / 0.750

8. 0.131 / 0.726

9. 0.124 / 0.706

10. 0.117 / 0.686

11. 0.110 / 0.665

12. 0.105 / 0.650

13. 0.099 / 0.631

14. 0.094 / 0.615

15. 0.089 / 0.598

Table 2 shows the decrease of the period of the pendulum result in increasing of frequency of the pendulums. It also can be said that the shorter the length of the pendulum the greater the frequency calculated. Therefore, it can be deduced that the frequency of the pendulum is inversely proportional to the period and length of the pendulum.

Discussion

Simple harmonic motion is the motion in which a body moves back and forth along a fixed path and returns for each position and speed after a definite period of time (N.A, 2019). One of the ways to demonstrate simple harmonic motion is by using simple pendulum. A simple pendulum consists of a mass hanging from a string and fixed at a pivot point (Suzuki et al, 2008). The mass hanging is release from an initial angle and will swing back and forth. The simple pendulum was further studied by producing a pendulum wave which is the combination of 15 simple pendulums with different length as well as by determining the relationship between the length of the pendulum and period and the frequency for each of the simple pendulum.

It was started by producing a pendulum wave which are a combination of 15 pendulums that made up of nuts and strings of the same weight. The mass of nuts is remain constant to ensure the period is not affected by the mass of pendulum. Besides that, the separation of two strings attached to the nuts must be equal which is 1.5 inches. This is to ensure that the distance between the nuts is equal so that the pattern of oscillated pendulums produced will not be affected. Then, the pendulum was hung from the shortest to the longest length of pendulum. However, the length of pendulum is increasing but not linearly when it was arranged. This is because the pattern produced would be different when the length of pendulum is increasing linearly as the period produced would be different.

After the pendulum wave has been made, the relationship between the length of pendulum and period can be determined. To determine the relationship between the length of pendulum and the period, all of the pendulums was swung and the period which is the time taken for one complete oscillation was determined. Then, the period of each pendulum were taken by using stopwatch. Based on the results obtained, it shows that the longer the length of pendulum, the longer the time taken for one complete oscillation. Thus, the length of pendulum is directly proportional to the period. This is because the period is not affected by the mass hanging to the pendulum but affected by its length (N.A, 2012). Besides that, the period can be determined by calculation and was calculated by using equation (2). Theoretically, the period was also directly proportional to the length of pendulum. Therefore, when different length of the pendulum is swing at the same time, it produces a wave-like pattern due to different period is produce.

Figure 1: The wave-like pattern produced by oscillating different length of pendulum.

To determine the frequency for each of the pendulum, the value of time taken for one complete oscillation for each of the pendulum was inserted in the equation (3). Based on the results obtained, it shows that the longer the period or the time taken for one complete oscillation, the lower the frequency. Thus, the frequency of the pendulum is inversely proportional to its period. This is because the definition of a frequency is the rate at which something occurs over time. Therefore, by reducing the time taken, it increases the frequency. Similarly, by increasing the time taken, it reduces the frequency (Bakshi, 2008). On top of that, the frequency is also inversely proportional to the length of pendulum. This is because as the length of pendulum increase, the period also increase. Thus, the frequency decrease.

Lastly, there are some precautionary steps that needs to be follow in order to fulfil the objectives. Firstly, the mass of nuts need to be equal so that the period will not be affected. Next, the separation between two strings attached to the nuts must be same to ensure the distance between the nuts is equal. This is because the pattern produced will be different when the separation between nuts is not same. Moreover, the length of pendulum were set and arranged increasingly but not linearly. This is because the pattern produced by the swinging pendulums will be affected when the it is increasing linearly. On top of that, the thumb tacks that hold the strings together need to be pressed tightly to ensure the strings will not swing randomly and disrupt the period.

Conclusion

In conclusion, the assignment was done by the students to describe the relationship between the length of the simple pendulum and the period in order to achieve the first objective. Based on the results that obtained, it can be clearly seen that when the length increases, the time taken for one complete oscillation increases. Hence, the results had shown that the length is directly proportional to the period. Next, for the second objective, the calculation was done by the students to determine the frequency for each pendulum. However, the frequency of each pendulum is inversely proportional to its period as the longer the period, the lower the frequency. Last but not least, the pendulum wave was able to be built. This is due to the fact that different period is produced when the pendulums with different length are swung at the same time, producing wave-like pattern. In a nutshell, the assignment carried out by the student was able to be completed precisely and the three objectives were achieved. Therefore, students have to be cautious and focus while taking any type of measurement where student can double check all the measurements for accuracy and make sure the position of the eyes is perpendicular to the scale of the measurement tool to reduce error.

References

  1. Baker, G.L. & Blackburn, J.A. (2005). The Pendulum- a case study in physics. Page (8 & 9). United States: Oxford University Press Inc., New York.
  2. Bakshi, K.A. (2008). Electronic Measurement Systems. US: Technical Publications. pp. 4- 14. ISBN 978- 81- 8431- 206- 5.
  3. Berg, R.E. (1990). Pendulum Waves: A demonstration of wave motion using pendula. Page 186. United States: American Association of Physics Teachers.
  4. Crockett, C. (2010). Science Project- Pendulum Waves. Retrieved from https://www.education.com/science-fair/article/pendulum-waves/
  5. High Definition Books (N.D). GK: General Knowledge- Motion, Waves and Echo. Page 403. High Definition Books.
  6. N.A., (N.D.). Experiment 2- The Simple Pendulum. Retrieved from http://www.aui.ma/personal/PHYSICS/1400/experiments/experiment_2.htm
  7. OpenStax College Physics, & Rice University (2012). The Simple Pendulum. Retrieved from https://opentextbc.ca/physicstestbook2/chapter/the-simple-pendulum/
  8. Suzuki, M.S. & Suzuki, I.S. (2008). Physics of simple pendulum. Department of Physics, State University of New York at Binghamton.
  9. The Editors of Encyclopaedia Britannica (2019). Simple harmonic motion. Retrieved from https://www.britannica.com/science/simple-hramonic-motion
  10. Davies, A. (1997). Handbook of Condition Monitoring: Techniques and Methodology. New York: Springer. ISBN 978-0-412-61320-3.
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Study of Laws of Simple Pendulum Motion to Teach About Simple Harmonic Motion. (2022, September 27). Edubirdie. Retrieved April 18, 2024, from https://edubirdie.com/examples/study-of-laws-of-simple-pendulum-motion-to-teach-about-simple-harmonic-motion/
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