Dimensional analysis
A way to obtain information about solutions to equations without solving them: rescale variables to remove constants, or associate dimensions and require with dimensionless arguments. For example, a pendulum period has the form .
We first list all the dimensions of the relevant quantities:
Then we let
This gives
Therefore, the period is
Pendulum period by rescaling
We will derive this equation for a pendulum later:
Suppose we release the pendulum from rest at some angle . We want to find the period of oscillation where
We can remove from the equation by rescaling
We are effectively writing and by chain rule,
This equation does not depend on , and so the solution is some function . Therefore, the period of may depend on the initial angle but can’t depend on . Hence,
giving
Hence,
Therefore, without solving the equation, we have found that the period of a pendulum is proportional to .
Basic dimensions and principles
Associate dimensions to all constants and variables as bookkeeping for how quantities appear in Newton’s equations. The basic dimensions are length , time , and mass ; there can be others (such as charge), depending on the problem. For example,
The fundamental principles are
-
,
-
all arguments of nontrivial functions (involving sums of different powers) must be dimensionless.
If the exponents in an ansatz like are not all fixed, a dimensionless ratio remains.
Related
Stated in
- Example 2.8 (Pendulum, Revisited)§2.6 Dimensional Analysis
- Example 2.7§2.6 Dimensional Analysis