# Simple harmonic motion as projection of circular motion

Can we consider $$\omega$$ (angular frequency) in equation of simple harmonic motion (SHM) as the angular velocity of the object in circular motion, when we see simple harmonic motion as projection of circular motion?

• Short answer: yes, but if and only if the circular motion is uniform circular motion – Eagle May 13 '19 at 20:48
• Then if it's correct, it means angular frequency w=2pie/T should be equal to angular velocity w=v/r of the particle executing shm – Nikhil Pant May 13 '19 at 20:56
• Yes, you're right. – Eagle May 13 '19 at 21:06
• Have you learned about rotations in phase space? – Cosmas Zachos May 13 '19 at 22:09
• In fact, this explains why the harmonic motion is sinusoidal, but not along any other periodic function. – safesphere May 14 '19 at 6:39

Yes. The point on the circle with radius $$R$$ revolves around the centre, with angular velocity:

$$\omega=\dot{\theta}=\frac{\mathbf{d}\theta}{\mathbf{dt}}$$

The projection on the RHS axis, call it $$y$$, is:

$$y(t)=R\sin(\omega t+\phi)$$

where $$\phi$$ is the angle $$\theta$$ at $$t=0$$.

• but do u really think the ω in (ω=θ) is equal to the ω in (y(t)=Rsin(ωt+ϕ)). basically my Que is can we really think the angular frequency in a shm as angular velocity of the object when we manipulate the motion as the shadow projection of circular motion (the the object is moving in a circular motion) – Nikhil Pant May 14 '19 at 8:01
• Please note: $\omega=\dot{\theta}=\frac{\mathbf{d}\theta}{\mathbf{dt}}$ – Gert May 14 '19 at 14:22
• It's more an analogy. But mathematically the projection of the position vector (red) of a particle moving uniformly on a circle onto any axis corresponds exactly to the description of an SHO. – Gert May 14 '19 at 14:32