Looking at the triangle you should see, from simple trigonometry, that y = A sin θ at any time.
Now θ = ω t therefore y = A sin ω t
S.H.M. and U.C.M.
Imagine putting a ball on a turntable and shining a light on it so that you can see its shadow on the wall. How does its shadow move? What does this tell us about the relationship between SHM and UCM?
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This diagram shows a particle moving with
uniform circular motion of radius A, and a graph of its vertical
displacement against time. Notice that it is sinusoidal. Looking at the triangle you should see, from simple trigonometry, that y = A sin θ at any time. Now θ = ω t therefore y = A sin ω t |
Circular motion is 2 dimensional while SHM is 1 dimensional. From the above you should see that they are closely related, at least mathematically. SHM is like 1 dimensional circular motion.
One important difference is that in circular motion ω is angular velocity in radians / sec whereas in SHM it is just constant which is related to the frequency. (although it is still measured in radians / sec.)