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You guys have been a great help so far, and I'm sorry for asking so many questions, (I litterally have thousands) but I have run into a phrase I don't understand in my book.
The book reads as follows:
"Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r say r^n, and some power of v, say v^m. How can we determine the values of n and m?
Solution:
Let us take a to be
a = k(r^n)(v^m)
where k is a dimensionless constant of proportionality. Knowing the dimensions of a, r, and v, we see that the dimensional equation must be
L/T² = (L^n)((L/T)^m = (L^(n + m))/T^m "
I believe I understand all of this to this point, but this next statement confuses me.
"This dimensional equation is balanced under the conditions
n + m = 1 and m = 2
Therefore n = -1, and we can write the acceleration expression as
a = k(r^-1)(v^2) = k/((v^2)/r
When we discuss uniform circular motion later, we shall see that k = 1 if a consistent set of units is used. The constant k would not equal 1 if, for example, v were in km/h and you wanted a in m/s²."
