College of Science and Computer Science

Physical Sciences Department

Physics 2

Experiment #2

SIMPLE HARMONIC MOTION

(THE SIMPLE PENDULUM)

**1. Abstract**

In this experiment periodic motion involving a swinging pendulum was studied. The period is the time it takes for a vibrating object to complete its cycle. In the case of pendulum, it is the time for the pendulum to start at one extreme, travel to the opposite extreme, and then return to the original location. As the pendulum oscillated, we gathered its data. When analyzed, the period of the motion was found to be given by: 2 Therefore, the period, or time to complete a full oscillation, of a pendulum was found to be dependent on its length. To find out what is the experimental determination of our g we used the formula: T^{2 }=4^{2} (L/g)**. **The value that we solved for g is 983.88 m/s^{2}. With the assumption of small angles, the frequency and period of the pendulum are independent of the initial angular displacement amplitude. All simple pendulums should have the same period regardless of their initial angle and regardless of their mass.

**2. Actual Materials Used**

- 5 meter light string
- Iron stand with horizontal bar extension
- Stopwatch
- Protractor or angle indicator
- Small bob (spherical object)
- Vernier caliper
- Meter stick and ruler
- Triple beam balance

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** 3. ****Data and Results**

** ****Mass of the bob: m = **11.3 g

**Length of the string: l = **120 cm

**Diameter of the bob: d = **3.59 cm** r = **180 cm

**Angular displacement: ****= **10^{0}

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**Table 1: Length of pendulum, Time per vibration, period, % error, Frequency, Angular frequency **

Length of pendulum (cm) |
Average time for 50 vibrations |
Period, T (sec) |
Period^{2}, T^{2} (sec^{2}) |
Linear frequency, f (Hz) |
Angular frequency, w (rad/s) |
% error |
|||||||

EV |
TV |
EV |
TV |
EV |
TV |
EV |
TV |
T |
T^{2} |
f |
w |
||

120 + r | 64.42 | 2.15 | 2.22 | 4.62 | 4.93 | 0.47 | 0.45 | 2.92 | 2.83 | 3.15 | 6.29 | 4.44 | 3.18 |

100 + r | 59.88 | 2.00 | 2.03 | 4.00 | 4.12 | 0.50 | 0.49 | 3.14 | 3.08 | 1.48 | 2.91 | 2.04 | 1.95 |

80 + r | 51.85 | 1.73 | 1.82 | 3.00 | 3.11 | 0.58 | 0.55 | 3.63 | 3.46 | 4.95 | 9.37 | 5.45 | 4.91 |

60 + r | 46.51 | 1.55 | 1.58 | 2.40 | 2.50 | 0.65 | 0.63 | 4.05 | 3.96 | 1.90 | 4.00 | 3.17 | 2.29 |

40 + r | 38.40 | 1.28 | 1.30 | 1.64 | 1.69 | 0.78 | 0.77 | 4.91 | 4.84 | 1.54 | 2.95 | 1.30 | 1.45 |

20 + r | 29.38 | 0.98 | 0.94 | 0.96 | 0.88 | 1.02 | 1.06 | 6.41 | 6.66 | 4.26 | 9.09 | 3.77 | 3.75 |

80 + r
30 |
34.2 | 2.04 | 1.82 | 4.16 | 3.31 | 0.49 | 0.55 | 3.08 | 3.46 | 12.09 | 27.68 | 10.91 | 10.98 |

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**Graph 1**

**Based on the data recorded, plot the curve to show the relationships between the periods, T (ordinate) and length of the pendulum, L (abscissa).**

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**Discuss the significance of the shape of the graph:**

The importance of the relationship between the period and the length is to show how we can calculate the gravity. The period T increased steadily with respect to the length l (Graph 1). According to the graph for the EV, at 20 m long, the square of the cycle was 0.98 s. At the length of 120 m, it increased and reached a peak of 2.15 s. And for the TV, at 20 m long, the square of the cycle was 0.94 s. At the length of 120 m, it increased and reached a peak of 2.22 s. The Length of the string will affect the time period of a pendulum because it will mean that the pendulum travels a greater distance in its oscillation.

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**Graph 2**

**Based on the data recorded, plot the curve to show the relationships between the square of the period, T**^{2}(ordinate) and length of the pendulum, L (abscissa).

** **

**Discuss the significance of the shape of the graph:**

The importance of the relationship between the square of the period and the length is to show how we can calculate the gravity. The square of the period T^{2} increased steadily with respect to the length l (Graph 2). According to the graph for the EV, at 20 m long, the square of the cycle was 0.96 s. At the length of 120 m, it increased and reached a peak of 4.62 s. And for the TV, at 20 m long, the square of the cycle was 0.88 s. At the length of 120 m, it increased and reached a peak of 4.93 s. The Length of the string will affect the time period of a pendulum because it will mean that the pendulum travels a greater distance in its oscillation.

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** 1. ****Compute the value g, the acceleration due to gravity from the slope of the graph in question no. 2 use the equation below.**

**T ^{2 }=4**

^{2}**(L/g)**

**Write the computations:**

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**2. Compare the value of g obtained in question no. 3 with the standard value of g. include the computation of the % error.**

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The true value is more close to the standard value of gravity 975 m/s^{2} vs 981 m/s^{2}. Also the percentage error of the true value is close to zero 0.61% compare to the experimental value which is 6.12%. A percentage very close to zero means you are very close to your targeted value, which is good.

** 3. ****Compare the period when the angle **** is over 30 ^{0} to that of the period **

**10**

^{0}8 + r 10^{0 }= 1.73

8 + r 30^{0} = 2.04

Changing the starting angle of the pendulum (how far you pull it back to get it started) has only a very slight effect on the frequency. The smaller the angle, the shorter the period will be. For larger amplitudes or angles, the amplitude does affect the period of the pendulum, with a larger amplitude leading to a larger period. However, for small amplitudes (typically around a few degrees), the amplitude has no effect on the period of a pendulum.

**4. Interpretation of Data**

In the examination of a pendulum involves the study of its periodic motion. Objects that exhibit this type of motion follow sinusoidal paths and experience oscillations between their maximum values of position. Through this experiment we are able to do an accurate way to represent and study periodic motion that is through configuring an oscillating pendulum and analyze its motion in relation to different lengths of attached string. As can be seen in the table, alterations in length definitely have an effect upon the period of the pendulum. As the string is lengthened, the period of the pendulum is increased. There is a direct relationship between the period and the length. The table shows us that in a simple pendulum system, only the change of the length affects the period, but not the change of mass.

Overall, the observed data was found to produce values for the period that were related to the length of the string by the following equation: T^{2} =4π^{2} (L/g). When the length is changed, the pendulum will take more or less time to oscillate, depending on its length and acceleration due to gravity. Therefore, the period may be varied by changing either of these two factors the length or gravity. Since acceleration due to gravity is constant on Earth (g = 9.8 m/s^{2}), the only dependent factor is the length of the pendulum.

The sources of error in our experiment is random, systematic and human errors during the experiment and these were:

- The human error of reaction time when counting the number of cycles completed. Because the spring and pendulum were moving quite fast, it was hard to count the number of cycles that had completed.
- The systematic error of not using ideal equipment such as a frictionless and massless string for the pendulum.
- The systematic error of air resistance (friction) for the simple pendulum. Due to air resistance (friction), the pendulum can started to slow down and not return to the same height as more cycles were completed.
- The random error of not releasing the pendulum at the same height for each trial. Releasing the pendulum at slightly different heights resulted in differences of potential energy during each trial.

**5. Conclusion**

In conclusion a pendulum will exhibit a period that varies depending on its length, according to the given equation: T^{2} =4π^{2} (L/g). We also learned that pendulums move by constantly transferring energy from one form to another. Pendulums, like all simple harmonic oscillators, are great demonstrators of the conservation of energy: the idea that energy cannot be created or destroyed, only transferred. The energy you end up with has to equal the energy you start with. The reason pendulums don’t move forever is because no system is perfect. Eventually, all the energy you provided is lost to the environment and the pendulum will stop swinging.

The purpose of this lab was to test the simple harmonic motion exhibited by a pendulum, and to see how the different variables affected the motion of said pendulum. The independent variables we tested were the length of the string, and the angle of release. This experiment focused on simple harmonic motion using a pendulum. A simple harmonic motion accurately models the motion that a pendulum exhibits when it swings from side to side. A simple harmonic motion will remain in motion as long as the system does not experience any type of external force such as friction, or an opposite applied force. This experiment demonstrated how a pendulum behave in a simple harmonic motion.

**6. Guide Questions**

**Is the period dependent on the length of the pendulum? Explain.**

Changing the length of a pendulum while keeping other factors constant changes the length of the period of the pendulum. Longer pendulums swing with a lower frequency than shorter pendulums, and thus have a longer period. The period of the simple pendulum oscillations increases as the length of the pendulum increases. The period depends only on the length L of the string and the value of the gravitational field strength g, according to:

T = 2 . The length of a pendulum affects its swing because longer pendulums swing at lower frequencies. A lower frequency causes a longer period and a slower rate of swing.

**2. Would the mass of the bob affect the period of the pendulum? Explain.**

Changing the mass of the pendulum bob does not affect the frequency of the pendulum. The period of the simple pendulum oscillations does not depend on the mass of the load, nor on the angle of revolution. The mass has no effect on the period of the pendulum. Since the force of gravity is proportional to the object it is acting on, the mass will end up having no effect whatsoever on the period (or frequency) of the pendulum. In F = ma, force is directly proportional to mass. As mass increases, so does the force on the pendulum, but acceleration remains the same. (It is due to the effect of gravity.) Because acceleration remains the same, so does the time over which the acceleration occurs which is the period.

**3. A 100 gram sphere executes a simple harmonic motion with the frequency of 20 Hz and amplitude of 0.5 cm. What is:**

**a. The constant k for the restoring force acting on it?**

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**b. The maximum acceleration?**

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**c. The total energy at any point of the motion?**

**7. References**

- Osmond, I. Thornton. (1905). Treatment of Simple Harmonic Motion. [Online]. Available: jstor.org
- Patterson, Louise Diehl. (1952) Pendulums of Wren and Hooke. [Online]. Available: jstor.org
- Tutorvista, n.d. Simple pendulum[online] Available at: http://www.tutorvista.com/content/physics/physics-i/measurement-and-experimentation/simple-pendulum.php?laws-of-simple-pendulum
- PhET, 2011. Pendulum lab 2.03 [online] Available at: http://phet.colorado.edu/sims/pendulum-lab/pendulum-lab_en.html
- http://www.physicsclassroom.com/class/waves/Lesson-0/Pendulum-Motion
- http://www.newagepublishers.com/samplechapter/000853.pdf