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Parametric equation of a particle in a circular trajectory

  1. Dec 13, 2014 #1
    1. The problem statement, all variables and given/known data

    You are given a particle, with the trajectory of a circle, with radius of 3 meters. The particle moves clock-wise, and when t=0, the x, y coordinates are (0, -3). Find parametric equations for the coordinates (x, y) in terms of the parametric t (time).

    2. Relevant equations


    3. The attempt at a solution

    My attempt at a solution was to plot the circle with the given points. I was able to determine that x = -3sin, and that y = -3cos. I know that period equals 2 pi / B, but I'm not sure what to do here for the last part of the trig equations.

    Edit: Just remembered that the period should be 4. So, B = pi/2.

    Would the correct parametric equations be

    x = -3sin(pi/2) t
    y = -3cos(pi/2) t
     
    Last edited: Dec 13, 2014
  2. jcsd
  3. Dec 13, 2014 #2

    ehild

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    Gold Member

    I got the same result.
     
  4. Dec 13, 2014 #3

    Mark44

    Staff: Mentor

    Was this something that you forgot to put in the original problem statement? Otherwise, there is nothing in the problem statement that says anything about the period.
     
  5. Dec 13, 2014 #4

    Zondrina

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    You have a circle ##x^2 + y^2 = 9## of radius ##3## which passes through the point ##(0, -3)## at ##t = 0##.

    The parametrization you have given is on the right track.

    Replace the ##\frac{\pi}{2}## you have there with the ##t## you seem to have left out of the argument.
     
  6. Dec 13, 2014 #5
    Thanks all for the help. Still slightly confused about why we replace pi/2 with t. Isn't the period equal to 4, since both trig functions complete their shapes in t = 4?
     
  7. Dec 13, 2014 #6

    Zondrina

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    You're probably better off plotting in radians rather than plain numbers.

    Try finding values of ##x(t)## and ##y(t)## for the points ##t = 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}, 2 \pi##.

    Do those points match up to your circle over in the ##(x, y)## plane?

    What can you conclude?

    Edit: Out of boredom I created this useful MATLAB session to assist the visualization. Here is a parametric plot of ##x(t)##, ##y(t)## where I've chosen some useful points to plot:

    Screen Shot 2014-12-13 at 7.02.58 PM.png

    Simply rotating the picture so that the ##(x, y)## plane is in your front view, you can see the points in the plane:

    Screen Shot 2014-12-13 at 7.00.41 PM.png

    In fact, allowing the interval ##[0, 2 \pi]## to become dense, you obtain something like this:

    Screen Shot 2014-12-13 at 7.08.36 PM.png
     
    Last edited: Dec 13, 2014
  8. Dec 13, 2014 #7
    Thank you! That makes a lot more sense. The visuals help a lot.
     
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