Worksheet 2.1 Problem 3

This problem focuses on using dimensional analysis to derive an equation. I was particularly fond of this problem because it demonstrates how elegant and useful effective dimensional analysis can be. Additionally, it helps to break down a problem and illustrate the actual units dimensions that you are working with in a given equation, that may otherwise seem arbitrary.

This problem as to determine the relationship of pressure and density for a gas to determine the speed of sound traveling through the medium.

First, I approached this problem by determining the units for each of the constraints:
\[C_s = \frac{distance}{time} = \frac{cm}{s} \]
\[P = \frac{force}{area} = \frac{N}{cm^2} = \frac{kg * \frac{cm}{s^2}}{{cm}^2} = \frac{kg}{cm s^2}\]
\[\rho = \frac{mass}{volume} = \frac{kg}{{cm}^3} \]

With each of these terms defined by their units, we can start to tackle the relationship between pressure and density for determining the speed of sound in the gas.

First, lets consider our desired units: \[speed = \frac{distance}{time}\]

Next, lets consider the units of pressure from above: \[pressure = \frac{mass}{distance*{time}^2}\]

Finally, lets consider the units of density: \[density = \frac{mass}{volume}\]

Since we know that our desired units don't have mass, we should look to cancel out the mass units. Additionally, we know that time is on the bottom and distance is on the top. By a quick inspection, we can see that the dividing pressure by density will get us close to this.

\[ \frac{P}{\rho} = \frac{\frac{mass}{distance*{time}^2}}{\frac{mass}{{distance}^3}} = \frac{{distance}^3}{distance*{time}^2} = \frac{{distance}^2}{{time}^2}\]
From this, we see that the only difference between this and our final desired units is that both the top and bottom units are squared. To handle this, we can take the square root of both the top and bottom and achieve our desired results.
\[\sqrt{\frac{{distance}^2}{{time}^2}} = \frac{distance}{time}\]

Therefore...

\[C_s = \sqrt{\frac{P}{\rho}}\]

Collaborators on this problem included Willie Pirc and Johnathan Budd.