Experimental analysis of compound pendulum oscillation periods

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?

1.3. 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.

2.3. Variables

The length of the string is the independent variable in this lab and it is measured in meters. The period of oscillation was the dependent variable and it was measured in sews. The controls needed to obtain precise and correct results are to utilize the same materials all through the entire experiment like the stop watch, meter stick, string, and pendulum. In addition, the individuals performing the counting of periods of oscillation and letting go of the string should all remain unchanged.

3. Observation and Results

The pendulum assembly shall cycle through numerous patterns over a 30 second time interval. You shall observe everything in sync: pendulum waves of different lengths rippling down the line, alternating swings, and also apparently chaotic movement.

Reason

The pendulum lengths are actually designed in a way that each one of them completes a differing whole number of swings every thirty seconds. The longest (first) pendulum, in 30 seconds, oscillates 25 times, the following one 26 times, the next one 27 times, and so on; and the shortest (last) pendulum 33 times.

All that takes place in the middle of this interval is a beautiful display of several pendulums, every one of them having a somewhat shorter period compared to the previous one. As the shorter pendulums begin getting ahead of the longer pendulums, they somewhat "lead" the pendulums next to them and develop a wave effect along the meter stick. All other pendulums will have finished a whole number of cycles at 15 seconds, whereas the remaining ones are all synced together at a "half cycle." If this is the case, half of the pendulums are categorized together on one side and the other half categorized on the other side (Stephens and Bate, 1950).

3.1. Quantitative Data

The table below is a representation of the quantity of time, in seconds, that it took for a single oscillation period in every trial. Four different trials together with the average for every string length are also shown. The experimental or relative error for the results obtained by the group was quite small. A less than 1% difference existed between the theoretical and measured values in this particular lab.

Figure 2: A Table Showing Quantitative Data Used in the Experiment

Length (m)

Trial 1

Trial 2

Trial 3

Trial 4

Averages

0.25

0.987

0.963

0.985

0.981

0.979

0.30

1.068

1.072

1.075

1.073

1.072

0.35

1.167

1.162

1.16

1.165

1.164

0.40

1.242

1.236

1.245

1.242

1.241

0.45

1.312

1.306

1.317

1.323

1.315

0.5

1.382

1.388

1.386

1.385

1.385

3.2. Qualitative Data

One feature which was realized by the sense of sight was that as the length of the string increased, it appeared to oscillate back and forth slower. There were times when it seemed like it oscillated faster than normal, which illustrated that there might have been some errors in the pulling back and releasing of the string.

3.3. Analysis

The findings of the experiment depict that the length of the string does influence the oscillation period. The oscillation period increases with increase in the length of the string. For every increment of 0.05 m of the string, the average period of oscillation increased with nearly 0.1 of a second.

Founded on this experiment's findings, the hypothesis of the group, if the length of the string reduces, there shall be a reduction in the oscillation period, was true. It was imagined that the association amid the two variables would actually be linear. The association, however, ended up being more parabolic, having Y being proportional to the square root of X.

3.4. Interpretation

The importance of the findings is that they confirm that the oscillation period is influenced by an object's length. It could be inferred that longer lengths make an object in oscillation to have a slower motion and take longer time durations to cover a particular area. On the contrary, a shorter length has no additional material moving with it, thus it is capable of oscillating at a higher speed (Caska and Finnigan, 2008).

3.4.1. Compare Expected Results With Those Obtained

The results gotten in this experiment were not quite far off from the anticipated results. A less than 1% relative error was realized. The disparities in the results could be explained by the pendulum oscillations not being accurately measured for every trial. The results might also be distorted as a result of the postponed reaction time, and not being capable of determining the precise timing of all the oscillation periods (Couder et al., 2005). Not carrying out every trial in the same exact manner is also another cause that could explain the differences.

3.4.2. Analysis of Experimental Error

One of the design's strengths was that the 0.05 m increments had actually been marked on the string using a permanent marker. This gave a better precision during modification of the string's length. The second strength of the design was a protractor was utilized in measuring the degree that the string had been pulled back to as closely as possible (Flocard and Finnigan, 2012). However, the experiment also had some weaknesses, one of them being that the arm's height was not the same all through the trials. It was actually varied in accordance to the string's length so that it could be simpler to observe when the washer passed the pencil positioned on the table. The arm's height ought to have remained the same.

Figure 3. Graph of P versus l for a compound pendulum where P is the period in seconds and l is the distance in cm between the centre of gravity of the pendulum and the point of suspension. (A solid line indicates calculated values; points with error bars indicate experimental results.)

Using the differentiation equation 1, it is easily displayed that the least value of P is

Pmin = 2?. From figure 2, we get Pmin = 1.54 ± 0.01 s; thus, k = 29.5 ± 0.5 cm, where we have applied the values of g as 981 cm/s2 for the acceleration gravity.

The1st equation, equation 1, has two solutions; 11 and 12, for a fixed P value. The equation's algebraic manipulations show that

When P = 1.58 seconds, the observed values for 1 were l1 = 42.5 ± 0.01 cm and l2 = 20.0 ± 0.1cm. These two values give the result, k = 29.2 ± 0.1 cm that is actually in reasonable agreement with the k value gotten from Pmin (equation 6).

When equation 2 is applied to a uniform bar of length, b, and width, a (see figure 1), the result obtained is k2 = a2/12 + b2/12.

In the current experiments, the value for a was 3.80 ± 0.01 cm and that of b was 100.0 ± 0.1 cm. Therefore, the calculated k value is k = 28.9 ± 0.01 cm.

4. Conclusion

From this particular experiment, we have proven that as we vary the mass or amplitude of the pendulum, the period does not change, it stays constant. However, there exists slight systematic deviation in the period when the amplitude of the oscillation is increased. Our data demonstrates that an increase in amplitude results to a small increase in the period. This small deviation, however, could be explained by air resistance. As the air resistance reduces, the amplitude also reduces. According to our analysis, the experimental data of the period, which we have recorded for the change of the amplitude or mass, is at least 98 or 99% accurate. This gives confidence that our data is reliable. Additionally, according to our experiment, period actually relies on the pendulum's length, and from our evaluation, we also get the earth's gravitation constant g, having a value of 9.355 m/s 2 with 95% confidence. Generally, the findings of our experiment support the founded knowledge that amplitude and mass do not have any influence on the period, and that the square of the period is directly proportional to the pendulum's length.

5. References

Caska, A. J. and Finnigan, T. D. (2008). Hydrodynamic characteristics of a cylindrical bottom-pivoted wave energy absorber, Ocean Eng., 35(1): 6-16.

Couder, Y. Proti'ere, S. Fort, E. and Boudaoud, A. (2005). Walking and orbiting droplets. Nature, 437:208.

Falnes, J. (2002). Optimum control of oscillation of wave-energy converters. International Journal of Offshore and Polar Engineering 12(2): 147-154.

Flocard, F. and Finnigan, T. D. (2012). Increasing power capture of a wave energy device by inertia adjustment, Appl. Ocean Res., 34, 126-134.

Hibbeler, R.C. (2007). Engineering Mechanics, 11th Edition. Singapore. Pearson Education

Newman, F.H. and Searle, V.H.L. (1951). The General Properties of Matter. Edward Arnold: London.

Qiu, S. Q., Ye, J. W., Wang, D. J. and Liang, F. L. (2011a). Capture width study on a pendulum wave energy converter, Power and Energy Engineering Conference (APPEEC), 2011 Asia-Pacific, 1-4.

Qiu, S. Q., Ye, J. W., Wang, D. J. and Liang, F. L. (2011b). Investigation on the efficiency of a pendulum energy converter in regular waves, Proc. 21st Int. Offshore Polar Eng. Conf. (ISOPE), Maui, Hawaii, USA, 650-654.

Salter, S. H. (1974). Wave power, Nature, 249, 720-724.

Stephens, R.W.B. and Bate. A.E. (1950). Wave Motion and Sound. Edward Arnold: London.

Xu, X., Wiercigroch M. and Cartmell, M. (2005). Rotating orbits of a parametrically-excited pendulum. Chaos, Solitons and Fractals 23(5): 1537- 1548.