Creating Graphs and Linearizing Data

Example 1 : The period of a pendulum is given by T = 2 π √ L g . In lab students perform an experiment in which they measure the period, T, and the length of the pendulum, L. From the equation, which shows the relationship between the T and L, we see that they are not directly related to each other. So, a graph of T vs L would not be linear. 1. What can the students plot to obtain a linear graph? Answer 1a: They can plot T vs √ L and in this case the slope = 2 π √ g . 2. What are the units of the slope? Unit of slope = (units of y -axis) = units of T = seconds (units of x -axis) units of √ L √ meters 3. What does the slope represent? Or what is the physical significance of the slope? It is understood that the slope represents how T changes with √ L so that is not what is being asked. The slope represents the coefficient of the variable on the x-axis. So, in this case the slope is the coefficient of √ L. Rewrite the equation, So now you can clearly see that the coefficient of √ L is 2 π √ g . So the slope = 2 π √ g Alternative solution : Alternatively, students can square both sides of the equation to obtain T 2 = 4 π 2 L g . 1. What can the students plot to obtain a linear graph? The students can plot T 2 vs L to obtain a linear graph. 2. What are the units of the slope? The units of the slope will be units of T 2 = (seconds) 2 units of L = meter 3. What does the slope represent? Or what is the physical significance of the slope? The slope will now be the coefficient of L. Therefore, the slope ¿ 4 π 2 g .