Uniform circular motion of a particle

[mclame22]

Homework Statement

A particle's position is given by the formula
r(t) = Rcos(ωt)î + Rsin(ωt)ĵ

The particle's motion at t=0 can be described as a circle starting at time t=0 on the positive x axis.

a) When does the particle first cross the negative x axis?
b) Find the speed of the particle at time t.
(Express your answers in terms of some or all of the variables ω, R, and π.)

c) Find the magnitude of the acceleration of the particle as a function of time.
(Express your answer in terms of some or all of the variables R, ω, and t.)

Homework Equations

r(t) = Rcos(ωt)î + Rsin(ωt)ĵ
v(t) = -Rωsin(ωt)î + Rωcos(ωt)ĵ
a(t) = -Rω²[cos(ωt)î + sin(ωt)ĵ] = -ω²r(t)

The Attempt at a Solution

I'm really unsure about what to do for part a. I know it has something to do with the particle moving pi radians along the arc of the circle. Would π be the distance it travels? Would it then be
π = Rcos(ωt)î + Rsin(ωt)ĵ ?
But how would I solve this for t?

For part b, I found the velocity of the particle to be the v(t) formula above by taking the derivative of the given position function. Would I take the absolute value of the velocity function to find the speed? How would I go about doing that?

Finally, for part c, I found that the acceleration of the particle is the a(t) function above after taking the derivative of the velocity function, but once again I don't know how to get the "magnitude" of the acceleration. Would I take the absolute value?

 

[zhermes]

I'm really unsure about what to do for part a. I know it has something to do with the particle moving pi radians along the arc of the circle. Would π be the distance it travels? Would it then be
π = Rcos(ωt)î + Rsin(ωt)ĵ ?
But how would I solve this for t?

That should be an 'r' not an 'n'. Notice that this equation is given in terms of the x and y (i and j) coordinates. What constraint can you make on the y-value for part a?

For part b, I found the velocity of the particle to be the v(t) formula above by taking the derivative of the given position function. Would I take the absolute value of the velocity function to find the speed? How would I go about doing that?

Again, this equation is parametric---describing a vector. How can you find the magnitude of that vector?---"absolute value" is insufficient, although it is often written as magnitude of v(t) = |v(t)|