Note the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider Example 1. What is the acceleration due to gravity in a region where a simple pendulum having a length We are asked to find g given the period T and the length L of a pendulum.
This method for determining g can be very accurate. This is why length and period are given to five digits in this example. Knowing g can be important in geological exploration; for example, a map of g over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.
Use a simple pendulum to determine the acceleration due to gravity g in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long.
Attach a small object of high density to the end of the string for example, a metal nut or a car key. Calculate g. How accurate is this measurement? How might it be improved? An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg. In the animation below right, the initial angle is large.
The black pendulum is the small angle approximation, and the lighter gray pendulum initially hidden behind is the exact solution. For a large initial angle, the difference between the small angle approximation black and the exact solution light gray becomes apparent almost immediately.
Oscillation of a Simple Pendulum The Equation of Motion A simple pendulum consists of a ball point-mass m hanging from a massless string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion.
An ideal simple pendulum consists of a point mass m suspended from a support by a massless string of length L. A good approximation is a small mass, for example a sphere with a diameter much smaller than L, suspended from a light string. The equilibrium position of the mass is a distance L below the support. It is a restoring force. See graph. As it does, its height is increasing as it moves further and further away.
It reaches a maximum height as it reaches the position of maximum displacement from the equilibrium position. As the bob moves towards its equilibrium position, it decreases its height and decreases its potential energy.
Now let's put these two concepts of kinetic energy and potential energy together as we consider the motion of a pendulum bob moving along the arc shown in the diagram at the right. We will use an energy bar chart to represent the changes in the two forms of energy. The amount of each form of energy is represented by a bar.
The height of the bar is proportional to the amount of that form of energy. The TME bar represents the total amount of mechanical energy possessed by the pendulum bob. The total mechanical energy is simply the sum of the two forms of energy — kinetic plus potential energy. What do you notice? When you inspect the bar charts, it is evident that as the bob moves from A to D, the kinetic energy is increasing and the potential energy is decreasing.
However, the total amount of these two forms of energy is remaining constant. Whatever potential energy is lost in going from position A to position D appears as kinetic energy. There is a transformation of potential energy into kinetic energy as the bob moves from position A to position D. Yet the total mechanical energy remains constant. We would say that mechanical energy is conserved. As the bob moves past position D towards position G, the opposite is observed.
Kinetic energy decreases as the bob moves rightward and more importantly upward toward position G. There is an increase in potential energy to accompany this decrease in kinetic energy. Energy is being transformed from kinetic form into potential form. Yet, as illustrated by the TME bar, the total amount of mechanical energy is conserved.
This very principle of energy conservation was explained in the Energy chapter of The Physics Classroom Tutorial. Our final discussion will pertain to the period of the pendulum. As discussed previously in this lesson , the period is the time it takes for a vibrating object to complete its cycle.
In the case of pendulum, it is the time for the pendulum to start at one extreme , travel to the opposite extreme , and then return to the original location. Here we will be interested in the question What variables affect the period of a pendulum?
We will concern ourselves with possible variables. The variables are the mass of the pendulum bob, the length of the string on which it hangs, and the angular displacement. The angular displacement or arc angle is the angle that the string makes with the vertical when released from rest. These three variables and their effect on the period are easily studied and are often the focus of a physics lab in an introductory physics class.
The data table below provides representative data for such a study. In trials 1 through 5, the mass of the bob was systematically altered while keeping the other quantities constant.
By so doing, the experimenters were able to investigate the possible effect of the mass upon the period. As can be seen in these five trials, alterations in mass have little effect upon the period of the pendulum. In trials 4 and , the mass is held constant at 0. However, the length of the pendulum is varied. By so doing, the experimenters were able to investigate the possible effect of the length of the string upon the period. As can be seen in these five trials, alterations in length definitely have an effect upon the period of the pendulum.
As the string is lengthened, the period of the pendulum is increased. There is a direct relationship between the period and the length. Finally, the experimenters investigated the possible effect of the arc angle upon the period in trials 4 and The mass is held constant at 0.
As can be seen from these five trials, alterations in the arc angle have little to no effect upon the period of the pendulum. So the conclusion from such an experiment is that the one variable that effects the period of the pendulum is the length of the string. Increases in the length lead to increases in the period.
But the investigation doesn't have to stop there. The quantitative equation relating these variables can be determined if the data is plotted and linear regression analysis is performed.
The two plots below represent such an analysis. In each plot, values of period the dependent variable are placed on the vertical axis. In the plot on the left, the length of the pendulum is placed on the horizontal axis. The shape of the curve indicates some sort of power relationship between period and length. The results of the regression analysis are shown. The analysis shows that there is a better fit of the data and the regression line for the graph on the right.
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