Estimation of the Value of Gravitation Force Using Simple Pendulum - Coursework Example

Estimation of the Value of Gravitation Force Using Simple Pendulum al affiliation Estimation of the Value of Gravitation Force Using Simple Pendulum Abstract The estimation of the value of acceleration due to the gravitation force can be done using the simple pendulum experiment. Such an experiment can be successfully conducted using a bob, timer, thread, and split cork. With all this apparatus in place, the pendulum is set to swing as its period of oscillation is recorded at different lengths of the thread. From the data obtained, a graph is drawn, and its slope is used to compute the gravitation force. The value of gravity computed is compared with the accepted value, and the difference is determined to assess the level of accuracy in the experiment. Introduction The use of a simple pendulum experiment in the study of motion helps to provide valuable insights into the acceleration of objects due to the gravitation force. In this experiment, a mass is suspended from one end of a piece of string and set in motion to determine the number of oscillations in a particular period. Such an oscillatory motion (to and from motion) is referred to as simple harmonic motion. The time a pendulum takes to swing forth and back is affected by factors such as the pendulum’s length and the acceleration due to the gravitation. A shorter pendulum has a shorter period to complete a single oscillation than a longer pendulum. In view of this, this simple pendulum experiment used the relation between the length applied in the pendulum and the time of oscillation to estimate the value of acceleration due to the gravitation force (Avison & Caribbean Examinations Council, 1988). The objective of the experiment The simple pendulum experiment was mainly conducted to facilitate the understanding of the relationship between different parameters in an oscillatory system. In addition, the experiment seeks to use its data analysis to facilitate the calculation of a value for the gravitational acceleration (g) and compare this value with the widely accepted value of 9.81 m/s2. Experimental equations and literature review If a mass of m hangs from the string in a simple pendulum experiment and sets to swing with small amplitude, the mass will oscillate back and forth in a simple harmonic motion. In this scenario, the mass of the bob becomes the inertia as the tangential component changes the direction every time the bob (mass) passes the center of its swing and hence acting to restore the mass to its midpoint. For this reason, the restoring force, F = - mg sin (ᴓ). However, if the angle ᴓ is very small, then it is assumed that sin (ᴓ) ≈ ᴓ, hence, F = - magᴓ….. Equation (1). The angle ᴓ of displacementᴓ can be determined from the equilibrium using the arc length, x, and the string’s length, L, to obtain ᴓ = x/L. Thus, F = - (mg/L) x…… equation (2). The time for a simple harmonic motion, T= 2π√m/k…… equation (3). However, according to Hooke’s law, F = - k x, whereby k = mg/L (a constant force). For this reason, by substituting k =mg/L in the equation (3), the period (T) of a simple pendulum becomes, T= 2π√L/g…… equation (4). For this reason, T2= (4π2/g) L. Experimental apparatus and procedures The apparatus used in the experiment included the pendulum bob, string, split cork, and timer. The thread of the pendulum was placed between two halves of a split cork and clamped to a firm support. Additionally, the length of the thread was set at a detachment from the bottom of the split cork to the center of the bob. The pendulum was configured to swing through a small angle as the measurement for the period of the oscillations was recorded in a table. The measurements for different lengths of the pendulum thread were repeated and recorded separately in the table. Data and results analysis During the experiment, the pendulum’s period was measured in lengths that ranged from 0.12 to 0.5 meters while maintaining both masses and angle of release at 0.400 kilograms and 10 degrees respectively. As such, the below data and results were obtained. Results test no length t_8/s u(T_10)/S t/s u(T)/s t^2/s^2 u(t^2) 1 0.12 7.9 1 0.9875 0.1 0.975156 0.1975 2 0.17 8.65 1 1.08125 0.1 1.169102 0.21625 3 0.22 9.01 1 1.12625 0.1 1.268439 0.22525 4 0.27 9.79 1 1.22375 0.1 1.497564 0.24475 5 0.37 11.13 1 1.39125 0.1 1.935577 0.27825 6 0.4 11.55 1 1.44375 0.1 2.084414 0.28875 7 0.45 12.04 1 1.505 0.1 2.265025 0.301 8 0.5 12.51 1 1.56375 0.1 2.445314 0.31275 9 0.56 13.21 1 1.65125 0.1 2.726627 0.33025 10 0.65 14.12 1 1.765 0.1 3.115225 0.353 Error 0.274775 Table 1: Results from a simple pendulum experiment Graph Graph 1: Length against Period On plotting a graph of Length against time, all dots did not align in a straight line due to experimental errors. For this reason, the best line of fit – a line that passed through most of the points was drawn. The line sloped down from the right to the left showing that the period of the oscillation increased as the length of the pendulum increased. This means that the period the pendulum took to complete an oscillation is dependent on the length of the thread from which the bob was hanged. Computation of gravitational force and percentage error The computation of acceleration due to gravitation force is done using the slope of the best line of fit, m = 4π2/g (the gradient of the line). From the graph, the black line is the best line of fit. Therefore, taking points (0.12, 0.9) and (0.64, 3.1); The gradient will, be = 4.2307 And gravitation force, g = 4π^2 / 4.2307 = 9.334 m/s^2. The experimental error in gravity computation; The error = 9.81- 9.334 = 0.48 The percentage error from experimental calculation of gravity; % error = (9.81- 9.334) /9.81 × 100 = 4.89% Error analysis The sources of errors in this simple pendulum experiment came from the stopwatch used to measure the period of an oscillation as well as the tools used to measure the length. Some of the errors from measuring the period include the reaction time error and precision error. According to the calculation made from data in the table, these errors added up to 0.274775. However, the percentage error in the computation of gravitation force is less than 5% indicating that this experiment is 95% accurate. Conclusion This experiment shows that the computation of acceleration due to the gravitational force depends greatly on the accuracy of the measurements taken from the length and period of the pendulum. The value of the gravitational force obtained using the gradient of the slope has a less error value due to the use of the best line of fit. The experiment also shows that the period square increases as the length of pendulum increases. Reference Avison, J. H., & Caribbean Examinations Council. (1988). Physics for CXC. Surrey: Nelson. Read More