vectors and acceleration in Uniform circular motion...?

Question

A particles position is given by the following expression:

r(t) = Rcos(wt)i + Rsin(wt)j

The magnitude of the velocity, |v(t)| (speed) is given by the expression:

sqrt((-Rwsin(t))^2 + (Rwcos(t))^2)

?Express the magnitude of the particle's acceleration in terms of R and v using the expression obtained for the speed of a particle?

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Please try to be concise and neat. Explain your steps if you can. Thanks in advance.

Answer 1

r(t) = Rcos(wt)i + Rsin(wt)j
v = dr/dt = -Rwsin(wt) i + Rwcos(wt) j
Please note that |v(t)| (speed) should be written as:
sqrt((-Rwsin(wt))^2 + (Rwcos(wt))^2) = Rw
a = dv/dt = -R(w^2)cos(wt) i - R(w^2)sin(wt) j
|a| = sqrt{[R(w^2)cos(wt)]^2 + [R(w^2)sin(wt)]^2}
= R(w^2)*sqrt{[cos(wt)]^2 + [sin(wt)]^2}
= R(w^2)
Thus: |a| = |v(t)|^2/R