Figure 16.18The shadow of a ball rotating at constant angular velocityωon a turntable goes back and forth in precise simple harmonic motion.
Figure 16.19shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle atconstant angular velocityω. The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis
undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has positionxand
moves to the left with velocityv. The velocity of the point P around the circle equals v
̄
max.The projection of v
̄
maxon thex-axis is the velocity
vof the simple harmonic motion along thex-axis.
Figure 16.19A point P moving on a circular path with a constant angular velocityωis undergoing uniform circular motion. Its projection on the x-axis undergoes simple
harmonic motion. Also shown is the velocity of this point around the circle, v ̄max, and its projection, which isv. Note that these velocities form a similar triangle to the
displacement triangle.To see that the projection undergoes simple harmonic motion, note that its positionxis given by
x=Xcosθ, (16.48)
whereθ=ωt,ωis the constant angular velocity, andXis the radius of the circular path. Thus,
x=Xcosωt. (16.49)
The angular velocityωis in radians per unit time; in this case2πradians is the time for one revolutionT. That is,ω= 2π /T. Substituting this
expression forω, we see that the positionxis given by:
(16.50)
x(t) = cos
⎛
⎝
2πt
T
⎞
⎠.
This expression is the same one we had for the position of a simple harmonic oscillator inSimple Harmonic Motion: A Special Periodic Motion. If
we make a graph of position versus time as inFigure 16.20, we see again the wavelike character (typical of simple harmonic motion) of theprojection of uniform circular motion onto thex-axis.
566 CHAPTER 16 | OSCILLATORY MOTION AND WAVES
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