Homogeneity of Physical Equations & Powers of Ten (Cambridge (CIE) AS Physics): Revision Note

The speed of sound v  in a gas is given by

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

where p  is gas pressure and ρ  is gas density.

Show that γ has no units.

Answer:

Step 1: Determine the units on the left:

• The only term on the left is speed, which has units of m s⁻¹

Step 2: Apply the homogeneity of physical equations:

• This equation describes the speed of sound waves and is therefore physical

  • This means it must be homogeneous, so the units on the left must be equal to the combined units on the right

Step 3: Determine the combined SI base units of pressure:

• Pressure is defined as the force F (units of N or kg m s⁻²) per unit area (units of m²) with the equation:

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

• Written in terms of units:

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

Step 4: Determine the combined SI base units of density:

• Density is defined as mass m (units of kg) per unit volume V (units of m³) with the equation:

$\rho =\frac{m}{V}$

• Written in terms of units:

$\mathrm{units}\mathrm{of}\mathrm{density}=\frac{\mathrm{kg}}{{\mathrm{m}}^3}=\mathrm{kg}{\mathrm{m}}^{-3}$

Step 5: Equate the units of both sides of the equation:

• The units of γ will be labelled as G

$\mathrm{m}{\mathrm{s}}^{-1}=\sqrt{\frac{\mathrm{G}\cancel{\mathrm{kg}}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}}{\cancel{\mathrm{kg}}{\mathrm{m}}^{-3}}}=\sqrt{\mathrm{G}{\mathrm{m}}^2{\mathrm{s}}^{-2}}=\sqrt{\mathrm{G}}\mathrm{m}{\mathrm{s}}^{-1}$

• This shows us that the square root of G is equal to 1, so G is equal to 1

• Therefore γ has no units - this is sometimes called being dimensionless

There were multiple ways of answering this - you could have rearranged the equation to make γ the subject and shown that the other side of the equation and no units, or you could have found that, without γ in the equation, the equation was homogenous.

You will often see very large or very small numbers categorised by powers of ten, so it is very important you become familiar with these as getting these prefixes wrong is a very common exam mistake!