Using Dimensional Analysis (DP IB Physics: SL): Revision Note

The speed $v$ of sound waves in a gas is given by

$v=\sqrt{\frac{\gamma p}{\rho }}$

where $p$ is the pressure of the gas, $\rho$ is the density of the gas, and $\gamma$ is a constant.

Show that $\gamma$ has no units.

Answer:

Step 1: Write down the units of each quantity

• The unit of speed $v$ is m s^-1

• The unit of pressure $p$ is Pa

• The unit of density $\rho$ is kg m^-3

Step 2: Determine the fundamental SI units of pressure

• Pressure is defined by

$p=\frac{F}{A}$

• Where

  • the unit of force $F$ is N

  • the unit of area $A$ is m²

• Force is defined by

$F=ma$

• Where

  • the unit of mass $m$ is kg

  • the unit of acceleration $a$ is m s^-2

• Therefore, the unit of pressure is

$p=\frac{\mathrm{N}}{{\mathrm{m}}^2}=\frac{\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}}{{\mathrm{m}}^2}$ = kg m^-1 s^-2

Step 3: Check the units on each side of the equation

• The units on the left-hand side of the equation are:

$v=\mathrm{m}{\mathrm{s}}^{-1}$

• The units on the right-hand side of the equation are:

$\frac{p}{\rho }=\frac{\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}}{\mathrm{kg}{\mathrm{m}}^{-3}}={\mathrm{m}}^2{\mathrm{s}}^{-2}$

$\sqrt{\frac{p}{\rho }}=\sqrt{{\mathrm{m}}^2{\mathrm{s}}^{-2}}=\mathrm{m}{\mathrm{s}}^{-1}$

• The units on each side are equivalent, therefore, $\gamma$ has no units