Homogeneity of physical equations
- 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
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:
- Written in terms of units:
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:
- Written in terms of units:
Step 5: Equate the units of both sides of the equation:
- The units of γ will be labelled as G
- 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
Examiner Tip
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.