How do you do this problem? (look at details).?

Question

Find parametric equations and a parameter interval for the motion of a particle that starts at the point (-2,0) in the xy-plane and traces the circle x^2 + Y^2 = 4 three times clockwise. (There are many ways to do this.)

Answer 1

Ok so at t=0 it starts at P: x(0)=-2, y(0)=0
then traces the circle x²+y²=4 (radius r=2, center (0,0)) three times before ending up back there for t=1.
So the period for one circle is T=1/3
i.e. it passes through P at the three values t =1/3, 2/3, 1
By convention the domain of t is [0,1)
Circular frequency ω=2π/T = 2π/ (1/3) = 6π

x = r cos(ωt + φ), y = r sin(ωt + φ)
Since we start out at (-2,0), obviously φ=π

Thus the parametric eqns are:
x = 2 cos(6πt + π), y = 2 sin(6πt + π)
And the φ=π angle displacement effectively shifts the sin and cos functions two quadrants, so your answer is:

x = -2 cos(6πt), y = -2 sin(6πt)

Answer 2

The basic gimmick is that the equations will look like x = a cos(omega t + theta), y = a sin(omega t + theta). The variable a will be the radius of the circle, in this case 2. Omega is the speed of rotation, not given nor relevant here so could simply be taken as 1 -- except that the clockwise rotation inserts a minus sign. The initial point will give the value of theta (you figure it out). And the range of t will then be from 0 to 6 pi.

Answer 3

Probably the easiest is to note that the particle is going around in a circle (of radius 2), and its position can be given (in polar coordinates r,Θ) by
r = 2(sin²(Θ)+cos²(Θ))
Letting Θ go from π to -5π radians yields 3 clockwise turns.

HTH

Doug