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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.
Figure 13.2:
Analogue Between Circular and Simple Harmonic Motion

Its xcoordinate is given as a function of time by:

x = Rcos(t)
 (9)

and the xcomponent of its tangential velocity is:

v_{x} =  v_{t}sin(t) =  Rsin(t).
 (10)

From this we deduce that

x^{ 2} + v_{x}^{2} = R^{ 2}
 (11)

which can be solved for v_{x} :

v_{x} = .
 (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:
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.
Figure 13.3:
Position and Velocity as Functions of Time

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
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10/9/1997