Pendulum Waves Science Fair Report Richard Berg from the University of Maryland explained in his journal how to build a set of uncoupled pendula, which display "pendulum waves" back in 1991. The patterns exhibited by this demonstration are quite breathtaking and the manner in which the patterns actually cycle is nothing less than spectacular. This demonstration is available in The Video Encyclopedia of Physics and it is also somehow simple to construct from scratch. The aim of this report is to discuss how the wave-like patterns created by the swinging pendula could be explained by a simple extension to the standard decryption of transverse oscillating waves in a single dimension (Couder, Proti'ere, Fort and Boudaoud, 2005). Apart from the math being very graceful in its right, it is also useful to know that the recurring patterns observed in the pendula in fact surface from aliasing of the fundamental continual function, a role which does not cycle, instead becomes increasingly intricate with the passage of time.

A compound pendulum is simply a solid body swinging in a vertical plane about a horizontal axis that passes through the body. Pendulums can be utilized in various areas such as determining gravitational field strength and timekeeping. The simple pendulum is discussed in several basic physics books; however, it is an idealization that does not incorporate the mass of the arm that offers support to the swinging bob. A hypothetical treatment of the compound pendulum has been developed by Newman and Searle (1951, 22-23) and it is founded on Newton's laws. So as to get to know if this simple theory shall permit correct evaluation of actual pendulums, we conducted an experimental investigation of a compound pendulum, concentrating on the reliance of P on 1, whereby P represents the period for the pendulum's tint oscillations and 1 represents the distance separating the rotational axis and the centre of gravity (Falnes, 2002). This laboratory report presents the findings of the experiment and compares our experimental findings to the Newman's and Searle's theory.

The aim of this particular experiment was to get an understanding of how different variables are capable of influencing the oscillation period of a pendulum. The precision of the period of oscillation was analyzed using the equation that involves tension of a pendulum. The equation is T=27N (l/g) and it was applied to compare data of the group to what is actually accurate and accepted (Xu, Wiercigroch and Cartmell, 2005).

1.1. Theory

A simple pendulum can be ideally described as a point mass that is suspended by a string with no mass from a particular point about which it can oscillate. A simple pendulum could be approximated by a tiny metal sphere that is of a tiny radius and a huge mass when somewhat compared to the mass and length of the weightless string from which it is hanging. The motion of a pendulum is regarded as periodic if after being set in motion, it oscillates back and forth. The duration that the pendulum takes to finish one entire oscillation is called the period T. The oscillation's frequency is also another important quantity that is described in the periodic motion. The frequency f of an oscillation can be described as the number of oscillations that take place per unit time and it is actually the inverse of the period, f = 1/T (Hibbeler, 2007). Likewise, the period is also the frequency's inverse, T = 1/f. The greatest distance that the mass gets displaced from its equilibrium state is known as the amplitude. Once a simple pendulum has been displaced from its state of equilibrium, there shall be a strong restoring force, which shifts the pendulum back to its position of equilibrium. As the pendulum's motion carries it past the position of equilibrium, there occurs a shift in direction of the restoring force so that it remains directed to the position of equilibrium. Let us say that the restoring force F G is directly proportional and opposite to the dislocation of x from its equilibrium state, so that it actually meets the relationship F = - k x (1); then the pendulum's motion shall be considered as a simple harmonic motion and it could be determined by applying the equation for the period of a simple harmonic motion (Caska and Finnigan, 2008; Hibbeler, 2007).

1.2. Statement of Purpose

How does the length of the pendulum's string affect the period of oscillation?

Hypothesis
The hypothesis of the group was, if the length of the pendulum's string reduces, the oscillation period shall also reduce. The guess regarding the association of the two variables was that it was linear. It was imagined that an increase in the length of the string implied an increase in the period of oscillation, and thus a linear relationship between the two variables.

2. Research

2.1. Materials

The materials utilized in this experiment were: a meter stick, pendulum stand, stop watch, string, pencil, washer, protractor, and marker. In the initial set up of this experiment, the pendulum was positioned on top of a table. A washer was tied on one end of the string and the string marked using a marker at increments of 0.05 meters from 0.25 m to 0.50 m (Flocard and Finnigan, 2012). Later on, the string was connected to the arm of the pendulum, beginning at the 0.25 m mark, utilizing the screw that was connected to the arm to fasten it. A pencil was positioned right below the arm of the pendulum in order to be capable of counting the periods of oscillations even better.

2.2. Procedure

After the first set-up of the materials, the first trial was commenced by having one individual hold the protractor in line with the arm of the pendulum so that the string was actually aligned with the 90° mark. A different member of the group pulled back the string to the 100° mark on the protractor, cautious to maintain the tension on the string. It is important to note that the string should not have any slack when it is being pulled back. The string was then released and the stop watch began as soon as the washer at the end of the string passed the location on the table that was marked by the pencil. The oscillation periods were counted as the string oscillated back and forth. A single period was counted after the washer went back to the starting point and then passed the pencil and proceeded forward again. After ten periods of oscillations had been counted, the stop watch was stopped. So as to determine the time of one period of oscillation, the obtained result was divided by ten. The data was then fed into a chart. This same procedure was repeated on the same length of the string three additional times and the average of the four results then obtained by summing up the different times and dividing by four (Qiu et al., 2011a). The screw on the pendulum's arm was then slackened and refastened at the next increment of 0.05 m that was 0.30 m. The same procedure carried out on the first set was repeated here as well. The experiment proceeded in this manner all through all the marked lengths of the string up to 0.50 m. The findings were fed into the chart and the test was finished.

In this lab, we shall utilize a photogate timer in coming up with very accurate measurements of the period T of a simple pendulum as a function of amplitude ?o and length L. A photogate is comprised of an infra-red diode, known as the source diode that releases an invisible beam of infra-red light. This particular beam is sensed by a different diode, the detector diode. When the mass m passes amid the detector and the source, there is an interruption on the infra-red light, leading to the production of an electric signal, which is utilized to start or stop an electronic timer. When in the "period mode," the timer begins when the mass passes through the gate for the first time, and does not stop till the mass passes via the same gate a third time as illustrated below. Therefore, the timer determines one entire period of the pendulum. In fact, our timers are very accurate and read to 10-4 s = 0.1 ms (Qiu et al., 2011b). Make sure that the timer is functioning well by passing a pencil through the photogate three times. If properly functioning, the timer should start on the very first pass and stop on the third pass.

Figure 1. The Compound Pendulum Used in the Experiment

The oscillation period was gotten by timing twenty oscillations using a stopwatch. To acquire data regarding errors, numerous such readings were gotten, and the period P determined by averaging. The oscillation's amplitude was maintained below 10° so as to make sure that the period of oscillation fell within 0.2% of the period for the very tiny oscillations. A meter ruler was utilized to measure the distance, 1, separating the axes of holes A and B, allowing us to calculate 1…