Tue Sep 16

MAIN POINTS
Dimensional analysis is based on the premise that physical quantities have dimensions, and physical laws are not changed by a change in units. Dimensions are multiplied along with their associated values, i.e mass, velocity, time, space. Each has an exponent, and when all of them have an exponent of 0, they are called 'dimensionless.' It can be seen immediately if an equation is possible by checking if the dimensions are compatible with each other. This is used to show how the dimensions of a pendulum work out. It is also demonstrated that you can infer what kind of dependence is necessary in an equation if you just know the unit of the answer.
Buckingham's theorem states that an equation is dimensionally homogeneous if it can be stated as a function of dimensionless products. Applying this yields a system of equations that can be used (using linear algebra, for instance), to set up an equation for the function.

CHALLENGES
I don't quite understand the concept of "dimensionally homogeneous," and how it is important that the laws of physics be this way. For instance, I'd like to know, if physics weren't dimensionally homogeneous, what kind of contradictions would we see?
I'd like to see another problem worked out too!

REFLECTION
In science classes we got used to the idea of treating units just like numbers, and multiplying/dividing/adding them so far as they were compatible. That's what dimensional analysis means to me— taking a known equation and seeing that the units cancel out. This approaches that problem from the other end.