Graphical Analysis Essay
This part of the experiment looked at a pendulum to see the relationship between the length of string and the time it took to do ten full oscillations. Unlike the rubber band experiment the results produced a curve. The best-fit curve was produced from a power regression using the Graphical Analysis software and the Ti-83 plus calculator to cut down on systematic error. This fit was much better than the linear regression because more points lie on the curve and it also supports what the textbook says.
In both experiments the possibility of errors are a concern; a concern that must not be forgotten. Systematic error is caused by the mis-collection of data or an improper model. One type of error that is always found is random error. It is the combination of errors that are important and calculated differently depending on the circumstances. Relative error gives more meaning to the importance of a random error. It is much easier to see the influence of a particular error when it is compared to the whole to make a percentage of error.
The general function to determine error is a derivative of the function multiplied by the error. For example the error formula of y = x5 then the error formula is 5×4 ? x. The error formula for y = VX is (1 / (2VX)) ? X. Shortcuts can be used when determining the relative error of z when z = xy by adding the relative errors of x and y (? z = ? x/x + ? y/y). This can be done because of the following proof: ? z = x? y + y? x and z = xy then ? z/z = (x? y + y? x) / xy. The shortcut ? z/z = ? x/x + ? y/y is the same equation when a common denominator is calculated as xy.
The same holds true for division z = x/y with the same result being ? z/z = ? x/x + ? y/y. Traditionally the equation would initially look like ? z/z = ((x? y + y? x)/y2) / (x/y), which is also the same as saying ((x? y + y? x) / y2) X (y/x). After the y/x is multiplied through then the same equation as the former is produced which is ? z/z = (x? y + y? x) / xy therefore proving the results. Method Part D Part D of the experiment looked at the quadratic function and its unique properties. The quadratic function of y = 10 + 30t – 4. 9t2 was graphed using Graphical Analysis.
This function represents a ball thrown upwards at 30 m/s with gravity working against it causing a downward motion. Figure 3 shows the resulting graph. This is just a theoretical situation and does not represent data collected. Analysis Part D The graphed quadratic function helps to understand the quadratic relationship more closely. It is a very useful function in physics because it is often seen when using motion. The roots are a very important part of the function. These are the points at which the curve crosses through the horizontal axis when y = 0.
To figure this out the quadratic function can be rewritten as: x = -b/2a i?? Vb2 – 4ac / 2a. The graph is helpful to quickly see where the roots are. The Graphical Analysis software allowed me to zoom into the roots closer than what figure 3 allows to be seen. Visually the roots looked to be -0. 32 and 6. 44 but with a calculator the equation was a little more accurate for the first root at -0. 317. Using the proper number of significant figures then the answers would be the same whether calculated or visually enhanced.
Looking at the equation more closely shows an interesting equation within the bigger one. Before the “i?? ” symbol is -b / 2a, which is the same for both roots. This is important because the quadratic equation is symmetrical and the -b / 2a equation points to the apex of the curve, like a mid-point. Another note on this point is that the point is also where the slope = 0, which is when the ball would begin falling back to the earth. Conclusion This lab looked at some of the different physical relationships that are current theories or laws. The experiments verified these relationships.
University/College: University of Arkansas System
Type of paper: Thesis/Dissertation Chapter
Date: 5 July 2017
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