# "Proof" for the period of a simple pendulum using only dimensional analysis

This is not a mathematical proof for the period of a simple pendulum. For that you can find the basics in any first year physics book. This is just a thought experiment which I think illustrates the power of simple dimensional analysis.

Take the following Simple Pendulum. Its Period (**t**) is the time it takes for the bob to sweep through one complete cycle. Let us make the assumption that the value of **t** has a deterministic relation with some arrangement of measurable factors. Under this assumption we list the measurable factors.

**l**= Length of the string, e.g. meters**m**= Mass of the bob, e.g. kilograms**a**= Angle to which the bob is initially lifted, e.g. radians**g**= gravitational force, length per time squared e.g. meters per second squared

Only **g** has a dimensional component of time and its only other component is of distance. Only **l** has a dimension of distance. Thus, dimensional analysis indicates **m** and **a** cannot be related to **t**.

Since we know period must have the dimension of time it's a trivial task to arrange the remaining factors **l** and **g** into a formula that produces a time result. If we use seconds (s) and meters (m) as our units of time and length respectively we have:

Removing unit labels and adding a constant of proportionality (**k**) we are left with:

If we were to assume our new formula was valid we could experimentally measure the value of **k** and that value would resolve to approximately 2Pi. Thus, using only simple dimensional analysis we've generated the classical formula for the period of a Simple Pendulum.