## College Physics (4th Edition)

The frequency is independent of the amplitude $A$ and the frequency is proportional to $\sqrt{\frac{k}{m}}$
Frequency is measured in Hertz which is $s^{-1}$ in SI units We can assume that $f = C~A^a~k^b~m^c$ $C$ is a dimensionless constant $A$ is amplitude in units of $m$ $k$ is spring constant in units of $N/m = kg~s^{-2}$ $m$ is mass in units of $kg$ We can equate the units of the terms: $m^a~kg^b~(s^{-2})^b~kg^c = s^{-1}$ We can consider the units $s$: $-2b = -1$ $b = \frac{1}{2}$ We can consider the units $kg$: $b+c = 0$ $c = -b$ $c = -\frac{1}{2}$ We can consider the units $m$: $a = 0$ Then: $f = C~A^0~k^{1/2}~m^{-1/2}$ $f = C~\sqrt{\frac{k}{m}}$ Therefore, the frequency is independent of the amplitude $A$ and the frequency is proportional to $\sqrt{\frac{k}{m}}$