# Dimensional Analysis

Can anyone explain how to do these types of questions for me? I am having trouble wrapping my head around it. Thanks.

A spherical soap bubble is seen to undergo oscillations when slightly deformed. Assuming that the frequency of these oscillations is a function of the surface tension S of the soap bubble, the size of the soap bubble as given by the radius r of the original spherical bubble and the density of soap, say p. Use dimensional analysis to construct a possible dependence for f in terms of S, r and p.

A spherical soap bubble of radius 1cm oscillates at 1 Hz. What would be the anticipated frequency* of oscillation of a soap bubble of radius 2cm?

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## 1 Answer

Dimensional Analysis says ignore the variables you have and manipulate what you've got in terms of dimensions. For example, let any measure of $mass=M$, let any measure of $length=L$, let any measure of $time=T$. $Force=F=MLT^-2$.

Then surface tension is $\frac{MT^2}{L}$. Radius is simple, it is L whilst density is mass divided by area. Area in this case is $\pi*L^2$ and mass $M$ so density is $\frac{M}{\pi L^2}$.

Frequency is period, or $\frac{1}{T}$, correct? Then, you simply need to attempt to equate both sides.

So, in your case, $Frequency = f(S, r, p)$. Well, $\frac{1}{T} = \sqrt{\frac{L}{MT^2}}*\sqrt{\frac{M}{\pi L^2}} * \sqrt{L} = \sqrt{\frac{L^2M}{\pi L^2T^2M}}$.

Hopefully you can see these expressions are equivalent except for a constant $\pi$. Constants don't have dimension, so they don't matter. However, you might also see we have three expressions we've already defined, as such, $F = \sqrt{S*r*p}$. We know this to be true because the units work as expected and you can check it based on the values being what you expect.

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