Home / Topics / Measurement & Math Tools
Deriving formulas by dimensions
Advanced
Use dimensions to find how a quantity depends on others, up to a constant.
T ∝ √(l/g)Assume a quantity depends on a product of powers of the relevant variables, then choose the exponents so the dimensions match. This gives the functional form but not the constant.
For a pendulum, guess T = k·m^a l^b g^c. Matching [T] gives a = 0, b = ½, c = −½, so T = k√(l/g).
You must include every variable that matters. Leaving one out gives a wrong result without any warning.
In more detail
Dimensional analysis can suggest the form of a formula when you know which quantities it depends on. If the period of a pendulum depends on length and g, then combining them to get a time gives T proportional to sqrt(L/g).
Real-world uses
- Physicists estimate the period of many systems before solving the exact equations.
- It gives a quick first guess for the speed of waves in a medium from its tension and density.
- Dimensional analysis suggests that the period of a pendulum depends on its length and gravity, which leads to the correct form of the formula up to a numerical factor.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
Does dimensional analysis give the exact formula?
No. It finds the combination of variables but not the numerical constant, which needs experiment or full theory.
What is the Buckingham pi theorem?
A result stating that a physical relation among n variables can be rewritten using n minus k dimensionless groups, where k is the number of independent dimensions.
How do I find the numerical factor?
Dimensional analysis cannot give it. You need an experiment, a full derivation or a known solution to fix the constant.
Related topics in Measurement & Math Tools
Dimensions of a quantity · Dimensional homogeneity · Checking equations by dimensions · Limitations of dimensional analysis · Dimensionless numbers · Order-of-magnitude estimates