Applications of the Equations of Motion - Lab Report Example

Applications of the Equations of motion Name: Course: Lecturer: Institution: City & State: Date: Applications of the Equations of motion Introduction The equations of motion can be described using different experiments in everyday life. Many ordinary experiments help in the calculations and applications of the laws of motion. From these experiments energy used in these experiments maybe determined for different masses, also from these experiments, it is easy to determine the pace of a given mass at a given time and distance. This pendulum uses a massless rod or a string to help the suspension of the ball on the pivot. As a result, the pendulum used in this experiment is best fit to calculate the equation and in application of the equations of motion. Occasionally, one has to stay on checking the calculations pertaining motion. This investigation describes how to measure motion using a pendulum ball. A pendulum ball suspended on a rope and hang on the wall or a fixed point, it gives a reasonable motion that produces motion. This study is usable when describing the equations of motion. This analysis generates motion when a pendulum bob is released from a given height (equilibrium position) and swings over a timer (Halliday, & Resnick, 1997). The motion generated by the pendulum ball is constant and duplicating, which is a periodic motion. In order to understand the motion of a pendulum there are a number of quantities that change in the course of the experiment. These quantities include state, velocity, energy and forces. Ideally, the use of the pendulum bob in most cases like in the application of the equation of motion helps to get easier mean that one can use to determine the necessary equation and the related factors that link with the equation. Objectives The main purpose of conducting this study is to determine the applications of the equations of motion. The other objective is that, in conducting this study helps students to understand the equations of motion and learn how to calculate the velocity at any given time. Literature review A period swing of a simple pendulum relies on the length of the string and the gravity. In extreme cases, it depends on the angle of the pendulum swing; this gives the pendulum’s amplitude. When the period of the pendulum is too close or almost equal to the preceding trial, which means it is independent of the mass of the bob (Michael et al., 2005. Pp 162-170). Therefore, the pendulum uses the amplitude to determine the period, T and the duration taken the pendulum takes for average cycles. The amplitude of the pendulum is proportional to its angle of displacement. When using the angle of displacement, it is easy to determine a constant angle to use when dealing the measurement of the velocity of the pendulum along a given path. The use of the pendulum in the analysis of the motion is well applicable when dealing with the different weights to analyze the weights (Herbert, 2007). This is necessary since it helps one in the understanding of the relationship between the motions of these weights. Therefore, in order to capture the application of motion equation, one needs a basic know that will enable them to tackle the task without much struggle. When dealing with the energy calculation, the equations of motion are necessary in the determination of the energy that a given material body has in a given time. This is only possible when one is able determining the velocity of the body at that given time. Therefore, the use of the pendulum bob experiment helps in the greater part in determining the instantaneous velocity of a body that in turn is applicable in the calculation of the kinetic energy of the body. Since kinetic energy is given as K.E = 1/2mV2, it is easy then to calculate the kinetic energy possessed by the body at a given. From this observation, one is able to determine the relationship between the motion of the body and its velocity. This shows that the kinetic energy of the body is directly proportional to the kinetic energy of the body. As the period changes the motion of the body change, since an increase in motion, that is, speed of the bob causes a resultant decrease in period taken by the bob per cycle. Equipment needed Strings of different lengths, between 2000mm to 500mm One or more weights A stopwatch (for timing the swaying weights) A secure place to hang the string to ensure that the pendulum is secure Experimental procedures The experiment to determine the application of motion using the pendulum ball uses the following set up: Figure 1: pendulum bob hangs on a fixed position Using the shortest string of 2m attach the weight of the pendulum bob, and then attach the string to support as shown in fig. 1. After attaching the string measure the length of the string. Raise the bob to about 15 degrees and release it without pushing. Record the time taken to cover at least 10 cycles using the stopwatch and divide by one to get period for trial 1. This is recorded as trial one. It is advisable not to use one cycle since there are errors in measurements when recording time for the pendulum to swing. Repeat the procedure six times to get different results. The periods obtained from this experiment do not have a greater difference since it uses a string with one string with uniform length. The graph obtained from the results is as in the graph below. Data analysis Results Experiment Trial Time (s) 1 1.3 2 1.4 3 1.4 4 1.2 5 1.3 6 1.4 Distance = 2m Average time (t) = 1.31666 s Initial speed = 0 Speed or velocity =  From the above analysis as while keeping other quantities constant the period for trial1 and trial two were different. In trials 3 through 4, the periods are different since the masses of the bob are changing due to the effect of the change in velocity and position. From the graph above, it is a straight line along the trial axis. This is because the numbers of trials have different time records. Therefore, the results obtained do not fit in the straight line since there are errors associated with the data recording when recording the time. When one applies the theoretical calculations, it is easy to achieve the exact velocity of the pendulum at the average position and average time (Meiners, 2009). If the pendulum has friction and the string or the ball experiences an external force, it gradually slows down and it stops thus giving poor results or data with error. Therefore, this cases should be less in the experiment to avoid an increased number of error in the entire experiment. Discussions The phenomenon of application of motion is described using a pendulum bob/ball. A pendulum ball of a given weight is suspended from a fixed pivot to allow it to swing freely. Displacing the pendulum from the resting equilibrium position, a force is subjected on the on the ball by gravity and its own weight and pushes it back to its equilibrium position. When released, the pendulum ball gains the restoring force that in combination with the mass of the pendulum, which makes it oscillate about its equilibrium position, swaying back and forth. This swaying creates complete cycles by the pendulum ball thus, creating a complete period. The pendulum periods depend solely on the pendulum length from the fixed point. Therefore, it is necessary to ensure that the pendulum string has the right length that allows the ball to swing freely and gain the required period. From the above experiment, the bob creates an arc as it moves from its equilibrium position to its motion status. Thus, as the bob moves it accelerates to a maximum velocity then back to equilibrium. The displacement to the left of the swinging bob gives a negative velocity while the displacement on the right side gives a positive velocity. From the analysis of the bob along the path, the velocity of the bob is least when the displacement is in maximum position. Therefore, this means that the bob moves at a slower rate thus showing a low velocity of the bob. As the ball/bob moves away from equilibrium, the slower it becomes and then moves back to equilibrium position. The ball at this position has less velocity which makes it come to rest and start accerelating in the opposite direction. Considering the displacement as a factor that may affect the period of the pendulum, other factors may control the use of the pendulum in the determination of motion. Amongst other factors that affect the period of the pendulum, include mass of bob, the angular displacement of bob, and the length of the massless rod or string. From the above data, the length, mass and the displacement angle are constant. Conclusions In essence, different trials in this experiment show that the period of the pendulum bob is proportional to the pendulum length. Therefore, as the length increase the period increases. This does not affect the plotting of the period and the number of trials; this is because as the trials changes so do the period of the pendulum change (Eidson, 2006). In addition, it is important for one use values that best suits in the plot in the above equation give a regression that relates to the variables of the data obtained. In the graph, the regression is obtained using the time (s) data, the analysis show that the data gives the best line of fit and the regression of the graph line is on the right. Therefore, this data give a period that is based on the following regression equation. Period = 16x + 1.3333 where y-intercept = 1.3333. It is clear for this experiment that the Newton’s second law of motion is applicable using the pendulum bob/ball experiment. Therefore, it is necessary to consider the accuracy when recording the data from the experiment. This helps to control the amount of error that may be available from the recording of data and one may be able to get desirable results from the experiment. It is necessary to carry out as many trials as possible in order to achieve better results that will have a positive input in the detemination of the motion in this experiment (Michael, et al., 2005). Lastly, it is necessary to assume the data used in this experiment to have been obtained from frictionless media and thus the gravitational acceleration is necessary in this experiment. References Michael R. M., et al., 2005, ‘The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives, Springer, New York. Halliday,D. & Resnick, R., 1997, ‘Fundamentals of Physics, : John Wiley & Sons, New York. Herbert, J. C., 2007, ‘ Scientific Instruments, Hutchinson's New York Eidson, J. C., 2006, Control, Measurement, and Communication using IEEE 1588, Burkhausen, New York. Meiners, H. F., 2009, Physics Demonstration Experiments, The Ronald Press Company, New York Read More

As the period changes the motion of the body change, since an increase in motion, that is, speed of the bob causes a resultant decrease in period taken by the bob per cycle. Equipment needed Strings of different lengths, between 2000mm to 500mm One or more weights A stopwatch (for timing the swaying weights) A secure place to hang the string to ensure that the pendulum is secure Experimental procedures The experiment to determine the application of motion using the pendulum ball uses the following set up: Figure 1: pendulum bob hangs on a fixed position Using the shortest string of 2m attach the weight of the pendulum bob, and then attach the string to support as shown in fig. 1. After attaching the string measure the length of the string.

Raise the bob to about 15 degrees and release it without pushing. Record the time taken to cover at least 10 cycles using the stopwatch and divide by one to get period for trial 1. This is recorded as trial one. It is advisable not to use one cycle since there are errors in measurements when recording time for the pendulum to swing. Repeat the procedure six times to get different results. The periods obtained from this experiment do not have a greater difference since it uses a string with one string with uniform length.

The graph obtained from the results is as in the graph below. Data analysis Results Experiment Trial Time (s) 1 1.3 2 1.4 3 1.4 4 1.2 5 1.3 6 1.4 Distance = 2m Average time (t) = 1.31666 s Initial speed = 0 Speed or velocity =  From the above analysis as while keeping other quantities constant the period for trial1 and trial two were different. In trials 3 through 4, the periods are different since the masses of the bob are changing due to the effect of the change in velocity and position.

From the graph above, it is a straight line along the trial axis. This is because the numbers of trials have different time records. Therefore, the results obtained do not fit in the straight line since there are errors associated with the data recording when recording the time. When one applies the theoretical calculations, it is easy to achieve the exact velocity of the pendulum at the average position and average time (Meiners, 2009). If the pendulum has friction and the string or the ball experiences an external force, it gradually slows down and it stops thus giving poor results or data with error.

Therefore, this cases should be less in the experiment to avoid an increased number of error in the entire experiment. Discussions The phenomenon of application of motion is described using a pendulum bob/ball. A pendulum ball of a given weight is suspended from a fixed pivot to allow it to swing freely. Displacing the pendulum from the resting equilibrium position, a force is subjected on the on the ball by gravity and its own weight and pushes it back to its equilibrium position. When released, the pendulum ball gains the restoring force that in combination with the mass of the pendulum, which makes it oscillate about its equilibrium position, swaying back and forth.

This swaying creates complete cycles by the pendulum ball thus, creating a complete period. The pendulum periods depend solely on the pendulum length from the fixed point. Therefore, it is necessary to ensure that the pendulum string has the right length that allows the ball to swing freely and gain the required period. From the above experiment, the bob creates an arc as it moves from its equilibrium position to its motion status. Thus, as the bob moves it accelerates to a maximum velocity then back to equilibrium.

The displacement to the left of the swinging bob gives a negative velocity while the displacement on the right side gives a positive velocity. From the analysis of the bob along the path, the velocity of the bob is least when the displacement is in maximum position. Therefore, this means that the bob moves at a slower rate thus showing a low velocity of the bob. As the ball/bob moves away from equilibrium, the slower it becomes and then moves back to equilibrium position.

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