To find the velocity and acceleration of an object undergoing uniform circular m

Question

To find the velocity and acceleration of an object undergoing uniform circular motion. Suppose that a particle's position is given by the following expression: where a; is the constant value of the particle's angular velocity. How would you describe the particle's motion? The particle's motion can be described by an ellipse starting at time t = 0 on the positive x axis. an ellipse starting at time t = 0 on the positive y axis. a circle starting at time t = 0 on the positive x axis. a circle starting at time t = 0 on the positive y axis. Part B - Time required by the particle to cross the negative x axis for the first time When does the particle first cross the negative x axis? Express your answer in terms of some or all of the variables w, R, and pi. Now. consider the velocity and speed of the particle. Find the particle's velocity as a function of time. v(t). Express your answer using unit vectors (e.g., Ai where A and D are functions of some or all of the variables w, R, t, and pi). Part D - Speed of the particle Find the particle's speed at time t. Express your answer in terms of some or all of the variables w, R and pi. Part E - Acceleration of the particle as a function of time Find the particle's acceleration as a function of time. Express your answer using unit vectors (e.g., Ai + Bj, where A and D are functions of some or all the variables w,R, t, and pi). Part F - Acceleration of the particle in terms of position Your calculation is actually a derivation of the centripetal acceleration. To see this, express the particle's acceleration in terms of its position, r(t). Express your answer in terms of some or all of the variables r (t) and w. Part H - Magnitude of the particle's acceleration in terms of the radius of rotation and the particle's linear speed Express the magnitude of the particle's acceleration vector in terms of R and v using the expression you obtained for the particle's speed (from Part D). Express your answer in terms of some or all of the variables R and v.

Explanation / Answer

A) a circle starting on positive x axis

B)

time is half a period so T/2

T = 2 pi/w

so time = pi/w

c) v = dr/dt = - R w sin(wt) i + R w cos(wt) j

d) speed = mag of v = sqrt( R^2 w^2 sin^2 + R^2 w^2 cos^2) = R w

e) a = dv/dt = - R w^2 cos(wt) i - R w^2 sin(wt) j

f) a = - w^2 r(t)

h) mag a = R w^2
use that v = R w
so w = v/R

a = v^2/R

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