# Simple Harmonic Motion- From Uniform Circular Motion

by chantalprince
Tags: circular, harmonic, motion, simple, uniform
 Emeritus Sci Advisor PF Gold P: 4,980 This is simply because of the fact that: $$\frac{1}{\frac{a}{b}}=\frac{b}{a}$$
 P: 122 Ah I see what you're getting at now. There's a mistake in the equation for frequency: $$\omega_{n} = \sqrt{\frac {k}{m}}$$ (1) where frequency is in radians per second. But what if you want to express the frequency in Hertz? Well, we know that 1 Hz is equal to one cycle per second. In the case of circular motion, one cycle is equal to $$2\pi$$ radians. So to convert from radians per second to Hertz, one must divide by $$2\pi$$. Hence: $$\omega = \frac {1}{2\pi} \sqrt{\frac{k}{m}}$$ (2) where frequency is now in Hertz. Now let's express this in terms of the period of one cycle, T. Bear in mind that if you were simply to reciprocate the expression for frequency when expressed in radians per second (equation 1), you would be stating the length of time of rotation for one radian alone. Hence you have to multiply the expression by $$2\pi$$ now to obtain the period for a single cycle. This is now the same equation as you would obtain by reciprocating equation 2. Hope this helps.