Consider a particle--the small red block in the figure--that is constrained to m

Question

Consider a particle--the small red block in the figure--that is constrained to move in a circle of radius R. We can specify its position solely by (t), the angle that the vector from the origin to the block makes with our chosen reference axis at time t. Following the standard conventions we measure (t) in the counterclockwise direction from the positive x axis.

a) What is the position vector r (t) as a function of angle (t). For later remember that (t) is itself a function of time.

Give your answer in terms of R, (t), and unit vectors i^ and j^ corresponding to the coordinate system in the figure.

b) For uniform circular motion, find (t) at an arbitrary time t.

c) What does r (t) become now?

Express your answer in terms of R, , t, and unit vectors i^ and j^.

d) Find r , a position vector at time t=0.

Give your answer in terms of R and unit vectors i^ and/or j^.

e)

Determine an expression for the position vector of a particle that starts on the positive y axis at t=0 (i.e., at t=0, (x0,y0)=(0,R)) and subsequently moves with constant .

Express your answer in terms of R, , t, and unit vectors i^ and j^.

Explanation / Answer

a) r(t) = R*cos(theta(t)) i + R*sin(theta(t)) j

b) theta(t) = w*t

c) r(t) = R*cos(w*t) i + R*sin(w*t) j

d) at t = 0 ,

r(0) = R*cos(w*0) i + R*sin(w*0) j

= R i

e) r(t) = R*sin(w*t) i + R*cos(w*t) j

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