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Parametric equation of a particle in a circular trajectory

  • Thread starter leo255
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  • #1
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Homework Statement


[/B]
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).

Homework Equations




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:

Answers and Replies

  • #2
ehild
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I got the same result.
 
  • #3
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Edit: Just remembered that the period should be 4
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.
 
  • #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.
 
  • #5
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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?
 
  • #6
Zondrina
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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?
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:
  • #7
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Thank you! That makes a lot more sense. The visuals help a lot.
 

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