Working through math for circular motion

Hey everyone,

I'm hoping to get some pointers on the following problem (screenshot of question and solution attached). The formula describes the motion of a particle in space with the position defined as r = r(cosωtˆi + sinωt ˆj)

It asks me to find the trajectory. In the solution, it solves for the trajectory by taking the magnitude of r

In a different problem, the trajectory is solved for by looking at limiting cases (i.e. in the limit where t -> 0 or infinity)

I don't get this first step and the logic in using one method over the other. I think this first step (setting up the problem) is critical to solving the question.

How does one normally go about this? What techniques does one use to determine how to tee up a problem like this?

Once the setup is complete, the math is relatively straight forward (e.g. substitution, taking the derivative of sin and cos)

Thanks in advance for your help

 

"Hey everyone,

I'm hoping to get some pointers on the following problem (screenshot of question and solution attached). The formula describes the motion of a particle in space with the position defined as r = r(cosωtˆi + sinωt ˆj)"
So [math]x= r cos(\omega t)[/math] and [math]y= r sin(\omega t)[/math].

"It asks me to find the trajectory. In the solution, it solves for the trajectory by taking the ma0gnitude of r"
[math] x^2= r^2 cos^2(\omega t)[/math], [math]y^2= r^2 sin^2(\omega)[/math]
[math]x^2+ y^2= r^2 cos^2(\omega t)+ r^2 sin^2(\omega t)= r^2(cos^2(\omega t)+ sin^2(\omega t)= r^2[/math]. Yes, that is a circle with center at (0, 0) and radius r.

"In a different problem, the traje[ctory is solved for by looking at limiting cases (i.e. in the limit where t -> 0 or infinity)

I don't get this first step and the logic in using one method over the other. I think this first step (setting up the problem) is critical to solving the question.
[How does one normally go about this? What techniques does one use to determine how to tee up a problem like this?

Once the setup is complete, the math is relatively straight forward (e.g. substitution, taking the derivative of sin and cos)

Thanks in advance for your help"
You have x= f(t), y= g(t), two equations in the three variables. You want to eliminate t so you have one equation in x and y. You CAN say [math]t= f^{-1}(x)[/math] so [math]y= g(f^{-1}(x))[/math] but you can better use special properties of f and g to make it easier to eliminate t, as using [math]sin^2(\theta)+ cos^2(\theta)= 1[/math] above.