The Key Principle in the Mechanism of the Oscillation of Systems - Lab Report Example

Applied Math: Using a Simple Pendulum to Arrive at an Estimated Value for Gravitational Acceleration g Table of Contents I. 4 II. Introduction 4 III. Method 5 IV. Results, Analysis 6 V. Discussion 8 VI. Conclusion 9 References 11 I. Abstract The experiment calls for the determination of the value of gravitational acceleration from a good understanding of how a simple pendulum’s parameters relate to one another. The setup involves using a weight suspended on a piece of string, in this case a golf ball. Ten oscillations were measured by the use of a smart phone timer, and varying lengths of the pendulum were determined with the use of the ruler. For varying lengths of the string, the times for ten oscillations are measured. The experiment shows that the expected value for g falls within the derived value after factoring in the uncertainty, corroborating the insight that the time period of the oscillation is dependent solely on string length (Ji and Bell 2015; Jardine-Wright 2010). II. Introduction The experiment calls for the calculation of g or the gravitational acceleration via the use of a simplified mechanism, and then undertaking a comparison between the value arrived at from that mechanism with the standard value of g, which is pegged at 9.81. The key principle in the mechanism of the oscillation of systems such as the simplified pendulum lies in being able to grasp the nature of the forces that act on the mechanism. In the pendulum, it is the force of g or the gravitational acceleration which impinge on the frequency of the oscillations, effecting a dampening effect on those oscillations through time. The oscillations of the pendulum mechanism can be construed as being simple harmonic in nature and is periodic. Here the force of restoration is in direct proportion to the pendulum displacement. For the purposes of this paper, the relevant formula is the one that prescribes the value of the time period, which is given thus (Ji and Bell 2015; Jardine-Wright 2010): . In the equation above, the value of T is dependent only on the string length l and g. Therefore, given T and l, g can be derived. The insight from the experiment is in being able to understand the way the oscillating system variables affect each other, as stated in the instructions; how the value of g is derived; and what the measurement errors and the causes of those errors are (Ji and Bell 2015; Jardine-Wright 2010). III. Method Two people made up a group, and the basic materials were the cotton rope, which was used as the string to hold the weight, which was first a plastic ball and later changed to a golf ball for more weight. The measuring devices were a ruler and an iPhone used as timing device. The rope was tied to the golf ball, and the time to ten oscillations were measured, with the golf ball suspended from the string and allowed to oscillate. The ruler was used to measure rope length, progressively shorter until the tenth iteration. In that tenth iteration the length of the rope was reduced to 0.1m. The results of the time measurements for ten oscillations for the varying lengths of the cotton rope were tabulated (Ji and Bell 2015; Jardine-Wright 2010). IV. Results, Analysis The following are the data for the experiment, tabulated: Meanwhile, the plot below illustrates how the square of the period relates to the length of the pendulum/the length of the rope (Ji and Bell 2015; Jardine-Wright 2010): In the plot above, the equation above defines the equation for the line that best fits the data, and the slope for that line corresponds to the square of the period. In this case, with y = mx + b and m being the slope, m is 4.420. We use this derived value to compute for g, in the equation , g is 4* pi2 /4.420. We therefore get the value for g = 8.93177 ms-2 (Jardine-Wright 2010). Calculating the gradient for the red line, we get m = (2.4 – 0.2)/ (0.5 – 0.1) = 5.5, while the gradient for the violet line is m = (1.95 – 0.5)/ (0.5 – 0.1) = 3.625 These two gradients define the upper and lower bounds of the gradient value, f which the gradient for the best fit line is a value that lies in between. These two latter values for m define the magnitude of the uncertainty in the derived of g, where the two m values are used to derive the value of g for the two lines. For m = 5.5, g = 7.1779 ms-2, while for m = 3.625 the value of g is 10.8906 ms-2. The level of uncertainty is the average of the difference between the two values, or (10.8906 – 7.1779)/ 2 or 1.8564 ms-2 (Ji and Bell 2015; Jardine-Wright 2010). Therefore, the derived value for g from the data is 8.93177 ms-2 +/- 1.8564 ms-2 V. Discussion It is clear from the preceding analysis that there is a consistency between the expected value for g, which is the standard value in the textbooks, and the derived value for g from the data on the oscillation time period measured for varying lengths of rope. Even with the expected errors in the measurement, the magnitude of the derived value falls within the range of the uncertainty margin as derived, albeit it falls on the upper range of the derived value for g. There is consistency therefore in the expected values and in the measured values for the oscillation, and this confirms the prognosis from the start that the value of g is dependent only on the length of the string. That said, it is clear too that errors are introduced inevitably from the way the oscillations are timed, with the use of a manual timer, and from the errors in measuring the length of the rope as well as the stretching that might have occurred while the golf ball was oscillating. The errors are therefore attributable to the nature of the measurement protocols and the limitations of the measurement devices for length and time (Ji and Bell 2015; Jardine-Wright 2010). VI. Conclusion The gravitational acceleration being a constant, the period of oscillation is dependent only on the length of the rope, so that if the period can be measured, then g can be determined experimentally, and this was indeed the case in this exercise. The exercise shows us how the period and the length of the rope used were related in such a way that where empirical data is available for both, then the value of g can likewise be empirically validated. Assuming that the standard value of g is the gold standard for the measurement of gravitational acceleration, the accuracy of the experimental data can be determined from how the final value of g conforms or deviates from the standard value of 9.8 ms-2. In this case, the error figure arrived at is also a measure of how well the experimenters were able to accurately measure the period and the length (Ji and Bell 2015; Jardine-Wright 2010). References Jardine-Wright, L. (2010). Practical 2: Investigating the Universe- Gravity. Cavendish Laboratory. [Online] Available from: http://www-outreach.phy.cam.ac.uk/workshop/Dec2010/resources/Experiment2.pdf [Accessed 26 January 2015] Ji, T. and Bell, A. (2015). Dynamics: Pendulum Systems, The University of Manchester. [Online] Available from: http://www.mace.manchester.ac.uk/project/teaching/civil/structuralconcepts/Dynamics/pendulum/pendulum_con.php [Accessed 26 January 2015] Read More