

Guide 16. Dimensional Analysis This guide shows a method of using units to determine the functional form of a relationship between physical variables. As an example, suppose the physical situation is that of a ball of mass m thrown upward from the surface of the Earth with initial velocity v_{0}. The ball reaches a maximum height h. We wish to use dimensional analysis to determine how v_{0} depends on m, g, and h. Of course, you already know the answer. However, we provide this familiar situation as an example to illustrate the method of dimensional analysis. We first assume that the equation for v_{0} can be written in the form v_{0} = Cm^{a}g^{b}h^{c}, where a, b, and c are exponents to be determined, and C is a constant that has no units. We next rewrite the equation with the units of v_{0}, m, g, and h only: [m/s^{}]^{1} = [kg]^{a}[ms^{2}]^{b}[m]^{c}. Now we examine the units of mass, length, and time separately and form individual equations involving the exponents for each. Consider the units of length. On the lefthand side, we know that the unit of [m] is raised to the power 1. On the righthand side, we combine the two instances of length units to obtain [m]^{b+c}. Thus, we can say: 1 = b + c. Next we examine the units of time. On the left, we have [s]^{1} and on the right we have [s]^{2b}. Thus, 1 = 2b. Finally we look at the units of mass. While the unit [kg] doesn't appear on the left, we can write [kg]^{0} for that side. On the right, we have [kg]^{a}. Now we're ready to solve for the values of a, b, and c. We have these relationships:
The second relationship yields b = 1/2. Substituting that into the first relationship gives c = 1/2. Thus, we've solved for the three exponents: a = 0, b = 1/2, c = 1/2. We can now say that based on a dimensional analysis, we think that v_{0} depends on m, g, and h according to the following relationship.
Dimensional analysis allowed us to determine how the initial velocity depends on the factors m, g, and h. A limitation of the method is that we can't determine the value of the constant C. We would need other means to determine that. Suppose, for example, we did an experiment in which we measured the height to which the ball rose as a function of the initial velocity. If we then plotted v_{0} vs. h^{1/2}, we'd expect the slope of a linear fit to be Cg^{1/2}. Knowing the value of g, we could solve for the value of C. (Of course, you already know based on past experience that C = 2^{1/2}.) Note that dimensional analysis isn't a proof of the relationship but does provide a way to determine the functional form. If we overlooked a variable on which v_{0} depended, then the statement we started with, namely, v_{0} = Cm^{a}g^{b}h^{c}, would be incorrect and anything following from that would be incorrect. 

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