Graphing. y m = cx n (3) where c is constant. What was true about Equation 2 is applicable here; the ratio. y m x n. = c

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1 Graphing Theory At its most basic, physics is nothing more than the mathematical relationships that have been found to exist between different physical quantities. It is important that you be able to identify these relationships, as you will be attempting to verify them in lab. You are most likely familiar with the linear relationship y = mx + b (1) betwen dependent variable y and independent variable x. This plots as a straight line with slope m and y-intercept b. If the line goes through the origin (b = 0) then this becomes a proportional relationship so named since the ratio (or proportion) of the variables is constant. y = mx (2) y x = m Equation 2 is one example of the general power function y m = cx n (3) where c is constant. What was true about Equation 2 is applicable here; the ratio y m x n is constant and we can say that y m x n (read y m is proportional to x n ). A graph of y m vs. x n should produce a straight line through the origin with a slope of c. The table below summarizes this method with some of the equations you will encounter in PHY 251 and PHY 252. = c 1

2 Name Equation m n Relationship Plot Slope Free Fall y = 1 2 gt2 1 2 y t 2 y vs.t g Second Law F = ma 1 1 F a F vs.a m Gravitation F = Gm 1m 2 r F 1 r 2 F vs. 1 r 2 Gm 1 m 2 Hooke s Law F = kx 1 1 F x F vs.x k Simple Pendulum L T = 2π g T 2 L T 2 vs.l 4π 2 g Ohm s Law V = IR 1 1 V I V vs.i R Capacitive Reactance X C = 1 2πfC 1-1 X C 1 f X C vs. 1 f 1 2πC A Word of Warning In the lab, sometimes you will be working with an equation in a different form than is presented in your text. The table above displays Newton s second law as F = ma which implies F a. A graph of F vs. a should produce a straight line through the origin with a slope of m. However, in the lab you would typically do an experiment in which you vary F (independent variable) and measure the response in a (dependent variable). Therefore you would actually be investigating a = ( 1 m ) F or a F. A graph of a vs. F is again a straight line but the slope is now 1. Though both F a and m a F are true, in this experiment you are testing whether or not a F - your analysis should reflect this. Plotting Experimental Data Graphs are always plotted with the dependent variable as a function of (vs.) the independent variable as defined in the experiment - in Microsoft Excel. It produces graphs in a fraction of the time it would take to do them by hand, and it gives you the equation of the best-fit line through your data points. See the Excel tutorial section. Does My Data Confirm the Relationship? If the purpose of your graph is to confirm a proportion between the variables, then your plotted data should fit a straight line through the origin. Below is a typical Excel graph of some distance and time data. 2

3 Obviously, all of the plotted points do not fall on the trendline. But how do you know whether or not this is a good fit? The R 2 value is called the coefficient of determination and will be in the range [0,1]. It tells you how much the variance in the dependent variable can be explained by the independent variable; i.e., how well the data fits the trendline. You are testing relationships already known to exist, so you should expect values very close to 1. We will take R as confirmation that the linear trendline is a good fit for the data. In the graph above R 2 = so this data is linear. Just as obvious is the fact that the trendline does not go through the origin. There is an option in Excel to force the line through the origin, but we do not do this. Why? We want the trend in the collected data - which will not include the point (0,0) - to be through the origin of its own accord. Obtain the slope m and y-intercept b along with their uncertainties using Excel s LINEST() function; if 0 is in the range b ± 3δ b then the trendline does go through the origin. Here is the output for the data above: m (cm/s) b (cm) δ m (cm/s) δ b (cm) Since 0 is in the range b = (0.5 ± 0.6)cm it does go through the origin. Therefore, this graph shows experimentally that distance is proportional to time. Remember that in the lab you are testing relationships already known to exist, so you should expect your data to be linear. If it is not, then you have done something wrong. Take a look at what you are plotting; is it what you need to plot in order to get a straight line? If it is, then something is wrong with your data - check the calculations you did for the graphical data. If these are correct, then your experimental data is incorrect, and you will need to repeat the procedure. If the data is linear but does not go through the origin; i.e., within the bounds of the allowed error, then there is systematic error present. This skews the data up or down by a constant amount; assuming 1 This value is arbitrary - not theoretical. 3

4 no bias in your measured values this would indicate that something assumed negligible in the system (e.g., friction) was actually not. Although the graph does not confirm the proportionality, it does show that the dependent variable plotted is a linear function of the independent variable. The important thing is that the ratio of these variables (the slope) is still constant; as a matter of fact it would be the same as that of a line that did go through the origin. Slopes and Areas Slopes Graphing data means plotting one physical quantity versus another physical quantity. Since these quantities most likely have units, the slope of the line on the graph will have units (and likely a physical significance). For example, consider the distance versus time graph graph in the previous section. As you can see, Excel gives the slope of the line as , but with no units it is contingent upon you to provide the appropriate units. Remember that slope is defined as slope = rise run = x t cm s Therefore, the slope here represents velocity, and would be reported as cm/s. This is always the case; i.e., slope units are the ratio of the dependent variable units to the independent variable units. Areas Just as a slope can have units (and physical significance), the area under a curve can as well. For example, consider the velocity versus time graph below. The area under the curve specifically, the area under the line and above the x-axis between t = 0s and t = 10s (the shaded rectangle) is given by 4

5 area = (base)(height) = (10s)(3.0m/s) = 30m and represents the total distance traveled during this time. This is always the case; i.e., area units are the product of the dependent variable units and the independent variable units. 5