Dimensional Equations

Dimensions play an important role -> equation validity!

It doesn't make sense to say that: 1 meter = 25 seconds!

Conversely, we already said that 3.2808 feet = 1 meter is valid. Why? [hint]

Dimensional Homogeneity

The dimensions on both sides of the equals sign must be the same for an equation to be valid. Another way to say this is that valid equations must be dimensionally homogeneous.

It is a good practice to make units the same (via conversion).

It is also good practice (and a course requirement!) to show all units throughout a problem to test equation validity!

OUTCOME:

Identify an invalid equation, based on dimensional arguments

TEST YOURSELF!

Which of these equations is dimensionally homogeneous?

$x(m) = x_o(m) + 0.3048(m/ft)v(ft/s)t(s) + \frac{1}{2}a(m/s^2)[t(s)]^2$

$P\left (\frac{kg}{ms^2} \right ) = 101325.0 \left (\frac{Pa}{atm} \right )1\left (\frac{\frac{kg}{ms^2}}{Pa} \right ) P_o (atm) + \rho (kg/m^3)v(m/s)$

NOTE:

It is sometimes simpler to write the units as "vertical fractions" to facilitate canceling.

IMPORTANT:

Just because an equation is dimensionally homogeneous does not mean that it is valid! Dimensional considerations act as a first test for validity only!