Question

Suppose we are told that the speed, v , of a rotating particle depends on the radius, r , and the frequency f , in the form v=2πr^a / f^b . If this is to be a dimensionally homogeneous equation, the values of a and b are...

Answer 1

The speed has the dimensions $m/s$, the radius - $m$, and the frequency - $Hz = 1/s$. Thus, have the following equation:

$$\dfrac{m^a}{(1/s)^b} = \dfrac{m}{s}$$

Comparing the powers on the both sides of the equation, one can derive the values of $a$ and $b$:

$$a =1\\ b = -1$$

Answer. a = 1, b = -1.