IBPhysicsDPHLRevision NotesToolsTool 3: MathematicsUsing Dimensional Analysis

Using Dimensional Analysis (DP IB Physics): Revision Note

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Updated on

DP Physics: HLDP Physics: SL

Using dimensional analysis

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

  • This is also known as dimensional analysis

  • The 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 v of sound waves in a gas is given by

v space equals space square root of fraction numerator gamma p over denominator rho end fraction end root

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

space p space equals space F over A

  • Where

    • the unit of force F is N

    • the unit of area A is m2

  • Force is defined by

F space equals space m a

  • Where

    • the unit of mass m is kg

    • the unit of acceleration a is m s-2

  • Therefore, the unit of pressure is

space p space equals space straight N over straight m squared space equals space fraction numerator kg space straight m space straight s to the power of negative 2 end exponent over denominator straight m squared end fraction = 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 space equals space straight m space straight s to the power of negative 1 end exponent

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

p over rho space equals space fraction numerator kg space straight m to the power of negative 1 end exponent space straight s to the power of negative 2 end exponent over denominator kg space straight m to the power of negative 3 end exponent end fraction space equals space straight m squared space straight s to the power of negative 2 end exponent

square root of p over rho end root space equals space square root of straight m squared space straight s to the power of negative 2 end exponent end root space equals space straight m space straight s to the power of negative 1 end exponent

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

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    Author
    Reviewer
    Katie M

    Author: Katie M

    Expertise: Physics Content Creator

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

    Caroline Carroll

    Reviewer: Caroline Carroll

    Expertise: Physics & Chemistry Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.