Help w/ Explanation of Basic Physics Dimensional Analysis Problem?

Question

I'm having difficulty understanding the steps involved in solving a basic physics question I was assigned. The answers for this is in the back of the book, but I want to really understand how it was derived.

1. Use dimensional analysis to find acceleration as a function of velocity (m/s), and radius in an object moving at a constant speed around a circle of radius r. The SI unit of acceleration is m/s^2.

The answer is acceleration = K v^2/r
Where K is a dimensionless constant.

How do I come up with this? Where do I start, and how do I know I have K? Can the value of K be determined somehow?

Thanks!

Answer 1

Take a look at refs. 1 and 2, which explain how you break the problem down to dimensions and basic SI units. For your problem, acceleration with SI units of m/s^2 and resulting dimension L/T^2 is equivalent to v^2/r whose SI units are (m/s)^2/m with dimension (L/T)^2/L = L/T^2.
The actual analytical derivation of centripetal acceleration is explained in ref. 3.
K is a dimensionless factor to account for any units difference (e.g., English and metric units used in the same equation), sometimes called a "unit" factor. In this case, since everything is SI, K=1, so it can be removed from the equation. Unit factors are explained in ref. 4.
I know it seems mysterious, but all the steps are logical and dimensional analysis in particular is extremely useful in making sure your analysis is consistent; dimensions on the left side of an equation should always be the same as those on the right.

Answer 2

what