Estimation of Value of Gravitational Acceleration G Based on Measurements from a Simple Pendulum - Coursework Example

Estimation of the value of gravitational acceleration g based on measurements from a simple pendulum ID Number Word count: 1150 The relationship between various factors in the simple harmonic motion of a pendulum was investigated. From the relationship and with reference to literature the acceleration due to gravity was also estimated to be 9.97m/s which is a 15% difference compared to 9.8m/s widely recognized. Therefore, it is possible to estimate the acceleration due to gravity using a simple pendulum. Introduction A point mass that is suspended using a massless string from a fixed point can be ideally described as a pendulum. The pendulum when set in motion swings back and forth and this movement is considered to be periodic. The period is deemed to be the duration taken for one complete oscillation. The frequency can be deemed to be the number of oscillations that the pendulum can make per unit time the inverse of which is the period i.e. f = 1/T. The amplitude is the longest distance that is traversed by the pendulum in reference to its equilibrium position. The displacement causes the exertion of force that tends to restore the pendulum to its equilibrium position (Nethercott & Walton 2013). Figure 1 Diagram showing displacement (a) and the restoring force (b) When the restoring force is directly proportional and opposite to the distance of displacement, then the pendulum satisfies the relationship below, and it is considered to be a simple harmonic motion. F = - k x The period for a simple harmonic motion is calculated using the formula The sum vector of the gravitational force of the mass of the pendulum (mg) and the tension force (T) shown in Figure 1. They constitute the restoring force whose magnitude depends on the displacement from the equilibrium position. Therefore, the restoring force F can be calculated as F = -mgsinθ The negative sign is an indication that the restoring force is in the opposite direction of the displacement. For small amplitudes, θ is small and therefore θ can be used in place of sinθ. Therefore, the resulting equation is F = mgθ In radians θ is the ratio of the arc length which is the displacement length (x) and the string length suspending the pendulum. Therefore θ = x/l and F = -mg x/l which is directly proportion to the displacement (x) since F = -kx k= mg/l Therefore The aim of this experiment is to estimate the acceleration due to gravity using a pendulum. For purposes of this experiment, the independent variable is the length of the pendulum whereas the period is the dependent variable (Bolton and Bolton 2012). Method Apparatus Split cork, pendulum bob, string, table stand with clamp, timer, Vernier calipers, meter stick Procedure The table top stand with clamp was placed on a flat working surface. The string was then passed through the pendulum bob and knotted as appropriate to hold the bob in position. The string with the pendulum attached to one end was passed through the split cork, and the length of string adjusted to 0.85m before being clamped onto the retort stand. A Vernier calipers was used to measure the diameter. The length of the string was adjusted to about .8 m. Therefore, the length of the pendulum is l = ls + r .where r is the radius of the bob. Figure 2 Assembly of the apparatus The pendulum was then displaced approximately 5º from its equilibrium position and left to swing back and forth. The time taken for 6 complete oscillations was recorded for the 0.85m pendulum. The length of the pendulum was decreased to about 0.80 m and the measurements repeated as in the previous. Successive measurements were made using varying lengths of the pendulum up to 0.1. The period of each duration of the pendulum was calculated by dividing the total time by the number of oscillations, and the respective values recorded in the data table. Results and Discussion The results obtained from the experiment were recorded in Table 1 below. The period corresponding to each string length was measured using the timer and the time taken to be one oscillation. From the results, it was observed that with a decrease in the length of string the period also decreased therefore it took much shorter time for a complete oscillation to occur. However, the increase in the period with the length is at a rate much less than a linear rate. Therefore, the period can be surmised to be dependent on the square root of the length Table 1 Results for the experiment showing the period corresponding to each string length Length T_6/s T/s T2 0.8 11.5 1.916667 3.673611 0.85 11.1 1.85 3.4225 0.7 10.4 1.733333 3.004444 0.6 9.35 1.558333 2.428403 0.5 8.9 1.483333 2.200278 0.4 8.32 1.386667 1.922844 0.3 7.22 1.203333 1.448011 0.2 6.06 1.01 1.0201 0.1 4.71 0.785 0.616225 0.05 3.88 0.646667 0.418178 The square of the period of the length was plotted and gave a linear line of the best fit with a slope of 3.97 calculated as the equation of the line is of the form y= ax + b Figure 3 The graph of t2/s2 against length Using the relation it can be seen that y= T2, x = l and a=. B in this instance is equal to 0. Since the slope of the graph a = by rearrangement we can calculate the value of g Using Therefore g = g=9.95m/s Errors in the experiment are mainly due to two factors. The first involves the measurement of the pendulum length while the second is the measurement of the period. The error in measuring the period can be minimized by taking a larger number of oscillations and dividing them with time and then getting an average. Errors can be minimized by having a calibrated string. Also, it is worth noting that it is not possible that the string can be massless, but it is assumed that it is massless. From the error bars, it is inferred that there is no statistically significant difference in the measurements. This is because most of the error bars overlap signifying a standard error of ± 1 which as per the value obtained is close to the theoretical value. Conclusion The pendulum experiment was used to estimate the acceleration due to gravity (g). The research successfully evaluated this acceleration to be 9.95m/s which is close to 9.8m/s that is universally recognized. The difference in values could be likely due to small errors in measurements. Although, the difference is only 15%, and therefore the results can be deemed to be accurate. It was noted that the time increased in direct proportionality to the pendulum length. The graph of the square of the period of the duration of the pendulum gave a linear graph which collaborates literature. Therefore, it is possible to determine the force of gravity using a simple pendulum. References Nethercott, Q., & Walton, E. (2013, March 8). Determining the Acceleration Due to Gravity with a Simple Pendulum. Retrieved January 19, 2015, from http://www.physics.utah.edu/~ewalton/lab_report.pdf Bolton, W., & Bolton, W. (2012). Mechanical Oscillations. In Higher Engineering Science (2nd Ed.). 125-126. Hoboken: Taylor and Francis. Read More