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# Comparison with Circular Motion

Idea: There is a very strong analogy between circular motion and simple harmonic motion. Consider a particle moving with constant angular velocity in a circle of radius R , as shown in Fig.13.2.

Its x-coordinate is given as a function of time by:

 x = Rcos(t) (9)

and the x-component of its tangential velocity is:

 vx = - vtsin(t) = - Rsin(t). (10)

From this we deduce that

 x 2 + vx2 = R 2 (11)

which can be solved for vx :

 vx = . (12)

This is precisely the same as Eq.(13.7) relating the speed of an SHO to its position, providing we identify the radius with the amplitude and the angular velocity with . Since the period (and frequency) are known for circular motion ( T = 2/ , f = 1/T ), this analogy allows us to deduce expressions for the period (and frequency) of the corresponding simple harmonic oscillator:

 T = = 2. (13)

Similarly, the displacement and velocity as functions of time can also be deduced:
 x = Acos(t) = Acos(2ft) (14) v = - Asin(t) = - Asin(2ft). (15)
These expressions are plotted in the following figures.

Note: ()t = 2ft gives the argument of the sine and cosine functions in radians. Make sure that your calculator is set to radians when doing these problems.

Next: The Simple Pendulum Up: Vibrations and Waves Previous: Elastic Potential Energy