# Lecture 18: Dimensional analysis for Scaling symmetries

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Almost always, when we build equations, the variables we include in our equations have more meaning than just numbers -- there is extra information attached that reminds us of what the variables means and how its value should be interpretted. The most common illustration of this is the association of dimensonal units with physical variables. There are many kinds of fundamental units in physics.

These units are like a type-system in computer programming, that constrain how we should use a variable. Rather than representing state explicitly, we represent state of as a combination of measurements in some units, and compare measurements rather than states directly. There are many situations where it makes sense to add two lengths together, but it doesn't make sense to add minutes and meters together in a calculation -- the number that comes out won't have any physical meaning.

Pressure in a soap bubble

In a soap bubble, the size is determined by a balance between the internal pressure and the surface tension of the bubble trying to contract. Can we come up with a formula for predicting the size? Well, in general, this would give an equation like

\[f(r, P, s)=0\]

If we use force and length as our units,

\[f(r X_L, P X_F X_L^{-2}, s X_F X_L^{-1})=0\]

Taking \(X_L = 1/r\) and \(X _ F = 1/sr\), we find

\[f(1, \frac{P r}{s}, 1)=0\] or if we solve for pressure, \[P = C \frac{s}{r}\]

Having a formula like this tells us several things already, even without knowing the constant \(C\). For a constant surface tension, pressue inside decreases as the bubble get's larger. Since the pop of a bubble depends on it's pressure difference \(P\), so this predicts small bubbles will pop more loudly than large bubbles. (Think champaign bubbles frothing vs. giant soap bubbles!)

The situation for a balloon is different -- the surface tension is not a constant, but rather increases as we inflate the balloon, creating a bigger pressure difference until things pop.

Pendulum period

The relationship between a pendulum's period and length

\[T = f(L,g,m,\theta _ 0)\]

which we can solve to find the period. To determine \(f()\), looks like we need to explore 4 parameter dimensions. However, we can greatly simplify this need for exploration based on a little reasoning.

\[f(T X_t, L X_l , g X_l X_t^2, m X_m, \theta _ 0 )\]

Let \(X_l = 1/L\), \(X_t = \sqrt{L/g}\), and \(X_m =0\).

\[f( T \sqrt{\frac{L}{g}}, 1,1,1,\theta _ 0)\] Now, solving this for the period, we find \[T = \sqrt{\frac{g}{L}} H(\theta _ 0)\] Not that the period is independent of the mass. But if we observer a swinging pendulum, we can say somthing about the gravitationa accelarations in the place we observe it (like on the moon) as long as we know the universal function \(H(\theta _ 0)\).

This matches our scaling law relationship from last class, so our theory is in good agreement with atleast one set of observations.