Parametric equation of a particle in a circular trajectory

In summary, the conversation discussed finding parametric equations for the coordinates (x, y) of a particle moving in a circular trajectory with a radius of 3 meters, starting at the point (0, -3) at t=0. The solution involved using trigonometric functions and replacing the period of the functions with the parametric t. The visuals provided further understanding of the parametric equations.
  • #1
leo255
57
2

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:
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  • #2
I got the same result.
 
  • #3
leo255 said:
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
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
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
leo255 said:
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:
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  • #7
Thank you! That makes a lot more sense. The visuals help a lot.
 

What is a parametric equation?

A parametric equation is a mathematical expression that describes the relationship between two or more variables. It is often used to represent the position, velocity, or acceleration of a moving object over time.

How is a particle's circular trajectory described by a parametric equation?

A particle's circular trajectory can be described by a set of parametric equations that define its position in terms of a parameter, such as time. In the case of a circular trajectory, the equations will typically involve sine and cosine functions to represent the circular motion.

What are the variables typically used in a parametric equation for a particle's circular trajectory?

The variables used in a parametric equation for a particle's circular trajectory are typically the x and y coordinates of the particle's position, as well as a parameter such as time or angle.

How can a parametric equation be used to calculate a particle's velocity and acceleration in a circular trajectory?

By taking the derivative of the parametric equations for position, the velocity and acceleration of a particle in a circular trajectory can be calculated. This allows for a more precise understanding of the particle's motion and can be used to make predictions about its future path.

What are the advantages of using a parametric equation to describe a particle's circular trajectory?

Using a parametric equation allows for a more flexible and precise description of a particle's circular trajectory compared to using traditional equations. It also allows for easier visualization of the particle's path and can be used to make predictions about its future motion.

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