Finding Gravity Using a Simple Pendulum - Case Study Example

Finding gravity using a simple pendulum Aim The study aims at determining the acceleration due to gravity using a simple pendulum. Data is collected on time taken by the pendulum to make an oscillation and theory used to determine the acceleration due to gravity. Introduction and theory Newton’s second law of motion provides that the rate in change in momentum of a body is proportional to a force that acts on the body and takes place in the direction of the force (Doshi 2006, p. 77). Real life phenomena however identify changes in motion of objects to indicate existence of external forces that acts on the objects. An object that is released, freely, for example, moves towards the centre of the earth with an increasing velocity. Similarly, an object that is thrown upwards moves with a decreasing velocity before it comes to rest and then accelerates downwards. Newton hypothesized existence of a force, force of gravity that attracts objects towards its centre and that explains the observation. The force acts in the same direction as the direction of motion of an object towards the centre of the earth and thus increases velocity. It however counters movements away from the centre of the earth and therefore reduces velocity of associated object (Agarwal 2011, p. 442; McGinnis2013, p. 22). In the absence of any other external force that could act on an object, the realized force is called acceleration due to gravity and is a constant for all objects within the earth’s surface and a limit of the earth’s environment (Chabay and Sherwood 2011, p. 6). When a mass, on a simple pendulum, is displaced, it moves in a vertical plane and for small displacements, the motion of the object assumes a harmonic motion. Newton’s second law of motion then explains the object’s motion from an angular perspective as shown bellow. F=ma This translates to the equation, F=-mgsin θ For small values of θ, however, sin θ= θ and this changes the equation to F=-mg θ Expressing force as a second order derivative of angular displacement and eliminating m from both sides generates the following as the resultant equation of motion. D2 θ/dt2 =-(g/L) θ With the angular frequency = (g/L) 0.5, the following is the equation for the period. (Serway and Jewett 2009, p. 448) From the equation, a direct proportionality relation is identifiable between period for an oscillation and length of a pendulum as shown in bellow. T=2πg-0.5l0.5 This can also be expressed as T2=4π2l/g And T2=(4π2/g) l This allows for graphical determination of the value of acceleration due to gravity from the gradient of the graph of period versus length of the string used in an experiment. From the graph, Gradient= 4π2/g g=4π2/ gradient Similarly, the value of g can be calculated, empirically, from the equation and for a single trial. The formula for determining g is as follow. g=4π2l/T2 The current experiment applies the formula to determine acceleration due to gravity from each trial of an experiment that uses the simple pendulum. The theoretical value of acceleration due to gravity is 9.8 m/s2 (Franklin et. al. 2010, p. 7). Methodology Apparatus The experiment used a pendulum bob, a split cork, a string, and a timer. The following diagram was used as the experimental set up. Diagram 1: Experimental set up Procedure Using the cork and the clamp of a triploid stand, a string was suspended between two halves of the cork and the clamp. The thread was set at 1 meters from the bottom of the cork to the centre of the bob. The pendulum was then displaced at a small angle and the time for ten oscillations measured. A division by 10 followed this to determine the period, time taken to complete one oscillation. The experiment was repeated for 8 different lengths, time for ten oscillations measured, and period for each trial determined. A graph for T 2 against length l was then drawn and the gradient of the graph used to calculate the experimental value of acceleration due to gravity. Results The following table shows the results of the experiment. Table 1: Results time for 10 oscillations perion (T) L T2 19.95 1.995 1 3.980025 20.94 2.094 1.1 4.384836 21.94 2.194 1.2 4.813636 22.89 2.289 1.3 5.239521 23.56 2.356 1.4 5.550736 24.41 2.441 1.5 5.958481 25.38 2.538 1.6 6.441444 25.87 2.587 1.7 6.692569 The results show a direct proportionality relationship between T2 and the length of the pendulum. This indicates that the longer the length of the pendulum, the greater the period. The proportionality also indicates existence of a constant term that determines the period based on the length of the string used. The indicated constant is a derivative of the acceleration due to gravity. The following graph shows the relationship between T2 and length of the pendulum used. Graph 1: T2 versus length of pendulum The results form a linear trend and support the hypothesis that a constant exist for the relationship between length of used pendulum and the period and square of period of a displaced pendulum. Discussion The graph of T2 against length of the pendulum is a straight line and shows existence of a constant for the relationship between the two variables. The gradient of the graph represents the constant and can be calculated using two of the points on the line of best fit. Taking the first and the last points, Gradient=change in T2/ change in length =(6.6925- 3.398)/(1.7-1) = 2.7125/ 0.7 = 3.8751 But g=4π2/ gradient Therefore, g= (39.51/3.8751) =10.19ms-2 This is the experimental value of acceleration. The experimental value differs from the theoretical value that is 9.8 ms-2. Magnitude of the realized error is calculated bellow. Percentage error= {(10.19-9.8)/9.8}*100 =3.98 % The error is small, less than 5 percent and can therefore be assumed negligible. Potential sources of error The experimental acceleration due to gravity is different from the actual value, 9.8 ms-2 and a number of factors exist that could have caused the error. Error in measuring time is the major potential source of error. The measurement involved observation of oscillations that was followed by stopping of the clock to determine time taken for ten oscillations. There is however, a possibility that some of the oscillations were not exactly ten and even though the deviation would be minimal for detection on the graph of best fit, the effects could be transferred to the final value of the calculated acceleration due to gravity. This could further be consistent with the negligible percentage error realized. Another threat to error at the measurement stage of the experiment is the potential deviation between timing, starting and stopping of the clock. Some of the measurements could have started well after the pendulum was already released or even before release of the pendulum. A similar scenario could have occurred at the stoppage of the clock to have it stopped either before the pendulum completed the tenth oscillation or just after completion. In either case, the recorded values would be different from the actual values and a negative net deviation would account for the higher acceleration due to gravity. The experiment also assumed that no other external force acts on the pendulum but this may not have been the case in reality. This is because forces such as friction exist and waves exist and could have caused the inaccuracy in the observed values towards difference in the value of acceleration due to gravity. Validity of results Even though a level of error exists in the experiment, the scope of the experiment establishes validity of the results. The empirical approach ensures objectivity and therefore eliminates possible bias in developed and communicated data. Results also consist of repeated trials and the many number of trials, eight, helped to minimize effects of error. Recommendations The major potentials source of error in the experiment is inaccuracy in determining number of oscillations in trials and point of completion of the final oscillations and times at which the pendulum is released. A higher level of accuracy is recommended in future replications of the experiment. Taking time to monitor a few oscillations in each trial can help in understanding oscillation patterns and ensure accuracy in determination of the critical points. Repeating each trial several times and determining average time for oscillations is another recommendation to improving on accuracy of the experiment. Performing the experiments in groupd can also help to facilitate accuracy because peer can identify and help in correcting error that one may make during experiments. Conclusion The experiment aimed at determining the value of acceleration due to gravity. A simple pendulum was used and the length of the pendulum and number of oscillations used to determine the value of acceleration due to gravity. Graph facilitated determination of a constant for the relationship between square of period and length and pendulum and the acceleration determined from the gradient. The experimental value of the acceleration was 10.19 ms-2, a 3.98 percent deviation from the theoretical value of acceleration due to gravity. Inaccurate readings and resistance to the movement of the pendulum could have caused the error that is however minimal. Higher level of care in taking measurements and performing the experiment in groups is recommended for minimizing the errors. The minimal error confirms that the theoretical value of acceleration due to gravity is correct. Reference list Agarwal, P 2011, I.I. T. Physics volume-1, Krishna Prakashan Media, New Delhi. Chabay, R and Sherwood, B 2011, Matter and interactions, John Wiley & Sons, Hoboken. Doshi, D 2006, New living science physics for class 9 with more numerical problems, Ratna Sagar, New Delhi. Franklin, K et. al. 2010, Introduction to biological physics for health and life sciences, John Wiley & Sons, Hoboken. McGinnis, P 2013, Biomechanics of sport and exercise, Champaign. Serway, R and Jewett, J 2009, Physics for scientists and engineers, Cengage Learning, Mason. Read More