1.1.3 Homogeneity of Physical Equations

Figuring out the units of quantities in an equation by the principle of homogeneity of physical equation is an extremely useful skill.

For example, the Bernoulli’s equation states that

$\displaystyle p+\frac{1}{2}\rho {{v}^{2}}+\rho gh={{p}_{0}}$

where p is the fluid pressure, v is the fluid velocity, g is the acceleration of free fall, h is the fluid height, and p₀ is some constant. I am not going to tell you what ρ is. But can you figure it out?

The key is to realize that we can only add, subtract or equate quantities if they are of the same kind! This means that in the equation above, p, $\displaystyle\frac{1}{2}\rho {{v}^{2}}$ , $\displaystyle\rho gh$ and p₀ must all be pressures, and thus have the same unit pascal (Pa).

$\displaystyle \begin{aligned}\text{units of }\frac{1}{2}\rho {{v}^{2}}&=\text{units of }p\\\text{units of }\rho &=\text{units of }\frac{p}{{{{v}^{2}}}}\\&=\frac{{\text{N }{{\text{m}}^{{-2}}}}}{{{{{(\text{m }{{\text{s}}^{{-1}}})}}^{2}}}}\\&=\frac{{\text{(kg m }{{\text{s}}^{{-2}}})({{\text{m}}^{{-2}}})}}{{{{{(\text{m }{{\text{s}}^{{-1}}})}}^{2}}}}\\&=\frac{{\text{kg}}}{{{{\text{m}}^{3}}}}\end{aligned}$