Homogeneity of Physical Equations & Powers of Ten (Cambridge International A-Level Physics)

Homogeneity of Physical Equations

Spec point — 1.2.3

• An important skill is to be able to check the homogeneity of physical equations using the SI base units

• The combined units on either side of the equation should be the same

• To check the homogeneity of physical equations:

  • Check the units on both sides of an equation
  • Determine if they are equal
  • If they do not match, the equation will need to be adjusted

Worked Example The speed of sound v in a gas is given by

$v=\sqrt{\frac{γp}{ρ}}$

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:

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

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:

$ρ=\frac{m}{V}$

• Written in terms of units:

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

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

• The units of γ will be labelled as G

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

• 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

Exam Hint 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.

Powers of Ten

• Physical quantities can span a huge range of values
• For example, the diameter of an atom is about 10^–10 m (0.0000000001 m), whereas the width of a galaxy may be about 10²¹ m (1000000000000000000000 m)
• This is a difference of 31 powers of ten
• Powers of ten are numbers that can be achieved by multiplying 10 times itself
• It is useful to know the prefixes for certain powers of ten

Powers of ten

Exam Hint 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!