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What is the relation between circular motion and SHM?

7 years ago

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The connection between uniform circular motion and SHM


It might seem like we've started a topic that is completely unrelated to what we've done previously; however, there is a close connection between circular motion and simple harmonic motion. Consider an object experiencing uniform circular motion, such as a mass sitting on the edge of a rotating turntable. This is two-dimensional motion, and the x and y position of the object at any time can be found by applying the equations:

 





 

The motion is uniform circular motion, meaning that the angular velocity is constant, and the angular displacement is related to the angular velocity by the equation:




 

Plugging this in to the x and y positions makes it clear that these are the equations giving the coordinates of the object at any point in time, assuming the object was at the position x = r on the x-axis at time = 0:




 

How does this relate to simple harmonic motion? An object experiencing simple harmonic motion is traveling in one dimension, and its one-dimensional motion is given by an equation of the form




 

The amplitude is simply the maximum displacement of the object from the equilibrium position.

So, in other words, the same equation applies to the position of an object experiencing simple harmonic motion and one dimension of the position of an object experiencing uniform circular motion. Note that the  in the SHM displacement equation is known as the angular frequency. It is related to the frequency (f) of the motion, and inversely related to the period (T):




 

The frequency is how many oscillations there are per second, having units of hertz (Hz); the period is how long it takes to make one oscillation.

7 years ago

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