Oscillation Lab
Name: Kyle Collins
Lab Partner: Hunter McCabe
Date: 19 April 2015
Purpose: The purpose of the oscillation lab is to investigate the factors that affect a simple pendulum.
Theory: A simple pendulum is made of a mass at the end of a cord. It is assumed that the string has no mass and does not stretch. The period for a simple pendulum with a small angle is given by the formula: T=(2(pi))(sqrt(L/g)). According to this formula, the only factor that should affect a pendulum's period is its length.
Experimental Technique: Mass and amplitude will not affect the period of a pendulum. Two different masses were hung from a cord attached to the ceiling, with their centers of mass positioned at the same height. They were allowed to swing freely, and a video with a known frame rate was taken. The frames for the period were counted to determine how many seconds it swung for. The process was repeated for measuring amplitude; here, the only difference was a constant mass and amplitude measured by the level on an iPhone. Lastly was length, where the pendulum was affixed to a table with a clamp to allow for easier adjustment of height. The frame counting process was repeated here as well.
Lab Partner: Hunter McCabe
Date: 19 April 2015
Purpose: The purpose of the oscillation lab is to investigate the factors that affect a simple pendulum.
Theory: A simple pendulum is made of a mass at the end of a cord. It is assumed that the string has no mass and does not stretch. The period for a simple pendulum with a small angle is given by the formula: T=(2(pi))(sqrt(L/g)). According to this formula, the only factor that should affect a pendulum's period is its length.
Experimental Technique: Mass and amplitude will not affect the period of a pendulum. Two different masses were hung from a cord attached to the ceiling, with their centers of mass positioned at the same height. They were allowed to swing freely, and a video with a known frame rate was taken. The frames for the period were counted to determine how many seconds it swung for. The process was repeated for measuring amplitude; here, the only difference was a constant mass and amplitude measured by the level on an iPhone. Lastly was length, where the pendulum was affixed to a table with a clamp to allow for easier adjustment of height. The frame counting process was repeated here as well.
Data and Analysis: For the mass variation, one mass of 1.5kg and one of 3.5kg were used. When analyzing the footage, both had a period of 2.87 seconds.
When amplitude was varied, the amplitude was found using the angle measurement and the following calculation:
For small angles (those 15 degrees and under) the period did not change when amplitude did. However, when large angles came into play, the pattern became perfectly linear. Therefore, this can be used to represent the ideal of what a small angle is.
Length of the pendulum was adjusted ten times ranging from .4m to .85m. When plotted, the data showed a strong positive correlation, exactly what was expected.
This graph, however, is not linearized. Linearization is defined as a linear approximation of a nonlinear system that is valid in a small region around the operating point. By linearizing this slope of the graph, it becomes:
Since the y-value is T^2, it can also be written as:
This can then be used to solve for gravity in order to see how close the calculated values are to known ones. The equation to solve for g is as follows:
Next, the percent difference equation.
Conclusion: The mass variation performed as expected, as did the amplitude variation. The amplitude graph also distinguished between a small angle and a large one by showing that large angle had a strong positive correlation, whereas small ones had no effect on period. The length variation had a high percent error; likely attributable to the fact that the graph was already extremely linear before the necessary linearization. However, the graph did show that length did have an observable effect on the period of a simple pendulum.
References:
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics. Retrieved April 21, 2015 from www.lahsphysics.weebly.com
Documentation. (n.d.). Retrieved April 21, 2015, from http://www.mathworks.com/help/slcontrol/ug/linearizing-nonlinear-models.html
References:
Giancoli, D. (2009). Physics for Scientists and Engineers (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall.
Lahs Physics. Retrieved April 21, 2015 from www.lahsphysics.weebly.com
Documentation. (n.d.). Retrieved April 21, 2015, from http://www.mathworks.com/help/slcontrol/ug/linearizing-nonlinear-models.html