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					    Unit 5
Harmonic Motion
      4.8

            JMerrill, 2007
   Learning Goal
To describe simple harmonic
motion with a trigonometric
         function.
Simple Harmonic Motion

 SIMPLE HARMONIC MOTION: motions that
 are periodic in nature such as oscillations of a
 pendulum, the orbit of a satellite, the
 vibrations of a tuning fork, etc., which can be
 described by sine or cosine curves
             Definition
• A point that moves on a coordinate line is
  said to be in simple harmonic motion if its
  distance from the origin at time t is given
  by
  – d = a sin ωt    or   d = a cos ωt
• where a and ω are real numbers such that
  ω > 0, with amplitude a, period 2π/ω, and
  frequency ω/(2π)
             Example
• A ball is bobbing up and down on a spring.
  10cm is the maximum distance the ball
  moves vertically upward or downward from
  its equilibrium point. It takes the ball
  t = 4 seconds to move from its maximum
  displacement above zero to its maximum
  displacement below zero and back again
  (assume no friction or resistance)
               Example
• When the spring is at equilibrium, d = 0,
  when t = 0, so we will use the equation
  d = a sin ωt.
• The maximum displacement from zero is 10
  and the period is 4.
• Amplitude = 10
              2               
• Period =          4, so  
                              2
• So, d  10 sin  t
           2
   Frequency
             
Frequency 
            2
             
               1
          2  cycle / sec
            2 4
         Port Aransas Pier
The graph indicates the depth of the water off a certain pier in Port
Aransas, Texas, beginning at noon, March 19th.

Note:    t = hours after noon on March 19th
         y = depth of water in meters
                     Pier Problem




At what time and date is the first high tide?   Noon, March 19th

                    The second high tide?       2 am, March 20th

                      The first low tide?       7 pm, March 19th
             Pier Problem




How long does it take the tide to complete one cycle?

          14 hours
Pier Problem
                     Amplitude = 3
                     A=3

                     14 = 2π/B
                     B = π/7




             
  y  3cos       t
             7
             Pier Problem




What is the depth of the water at midnight March 19?
         Since midnight is 12 hours after noon, let t = 12.
        Plug 12 into the equation for the function.
        y = 1.870 meters
            Pier Problem




At what time is the depth of the tide 1-meter?
   Graph y1 = equation                           t = 2.743 and 11.257
   Graph y2 = 1                                  2:45 pm and 11:15 pm
    Find intersection(s).
       Weight on Spring
    video



A weight is at rest hanging from a spring. It is then pulled down 6
cm and released. The weight oscillates up and down, completing
one cycle every 3 seconds.
                       Sketch
Distance above/below resting point, in cm


    6



                                                Time, in
                                                seconds
                                            3
   -6
         Equation
                                  Amplitude = 6
6
                                  A= 6
                                  3 = 2π/B
                                  B = 2π/3
                              3
-6




              2
     y  6sin    ( x  1.5)
               3
                      Positions
Determine the position of the weight at 1.5 seconds.
         Let x = 1.5; plug into equation for function.
         y = 0 cm (back at original position)


Use the graph to find the time when y = 3.5 for the first time.
         Graph y1 = equation you wrote; graph y2 = 3.5.
         Find intersection.
         x = 1.797 seconds


3.5 is the 3.5 cm distance above the original position of the weight.

				
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posted:5/8/2013
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