**Dimensional
Analysis and Scaling Arguments: **

**Getting the
Right Answer (Without The Sweat)**

What
determines the size of raindrops? Just
how much faster can a boat go with that extra rower? Why do champagne bubbles “pop” louder than beer bubbles? Each of these questions can be answered
through dimensional analysis or simple scaling arguments. Both methods give rough answers to physical
questions by producing simple relationships between the relevant physical variables. Dimensional analysis proceeds by looking for
multiplicative combinations of the variables that are dimensionless. Often, these dimensionless quantities can be
related to each other by simple physical argument or empirical data allowing us
to deduce relationships between the original physical variables. Scaling arguments arrive at similar results
but begin by examining the fundamental physics of the problem. Based on the dominant balances (*e.g.*
forces, torques, volume flux) in the problem, a simple relationship between the
physical variables can be derived.

I propose to present a set of short lectures that will provide students with the mathematical and physical foundations to quickly find a rough solution many physical (and some not so physical) problems. After each lecture, I will present several simple problems for the students to discuss and work on in small groups. To help guide the course, I will prepare a few interesting “challenge” problems that will be presented at the beginning of the course and returned to at the end of the course. I plan to focus on fluids because it is easy to get a physical intuition for the correct “answer” through “home” experimentation and because fluid mechanics is not usually a topic in the high school curriculum.

Goals for the Course:

1. Students will improve their physical intuition and skills in distilling the dominant features of a physical problem.

2. Students will understand several features of fluids that are part of ordinary experience.

3. Students will see the fun and excitement in explaining the phenomena from everyday life.

Main Course Topics:

1. Fancy Dimensional Analysis: Buckingham’s Theorem and Applications

a. The Pendulum

b. Champagne Bubbles

2. Scaling Arguments: Finding the Dominant Physics

a. Indestructibility of Small Animals: effects of size on terminal velocity

b. McMahon’s Rowers

3. Some Basic Fluid Mechanics

a. Archimede’s Principle

b. Bernoulli’s Principle

c. Reynold’s Number

d. Surface Tension

4. More Scaling Arguments

a. Size of a pendant water drop

b. More size effects in the animal kingdom

5. Demo Day: Fun Fluids Tricks

a. The Soap Boat

b. A Boat in a Vortex

c. Tears of Wine

d. Fluid polygons/Fluid fish bones – images from Professor Bush’s Fluid Dynamics Picture Gallery

Sample Problems:

1. Dimensional Analysis

a. Proof of the Pythagorean Theorem

b. Taylor’s Blast – Determine the energy of an atomic blast from photographs of the explosion.

c. How big should a country’s military be?

2. Scaling

a. What determines the size of water striders?

b. How does a Basilisk Lizard run on water?

c. What determines the size of a raindrop?

d. Why is the stream of water from the faucet narrower at the bottom than at the top?

3. Fluid Mechanics

a. What drives a soap boat?

b. Does a boat in a vortex move inward or outward?

c. What is the origin of the tears of wine?

4. Connections with Other Fields of Science

a. Why are the tears of wine smaller in older wines?

#### References

1. Course Notes from 18.355 (Fluid Mechanics). Taught by Professor John Bush at MIT’s Applied Mathematics Department.