Parametric equation of a particle in a circular trajectory

[leo255]

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

 

[ehild]

I got the same result.

 

[Mark44]

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.

 

[STEMucator]

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.

 

[leo255]

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?

 

[STEMucator]

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:

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

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

 

[leo255]

Thank you! That makes a lot more sense. The visuals help a lot.