A particle P moves with constant angular speed ? (radians per unit time) around

Question

A particle P moves with constant angular speed ? (radians per unit time) around a circle whose center is the origin with radius R. Such a particle is in uniform circular motion. Assume that at t = 0 the particle is at the point (R, 0). (a) Give a parametric equation for the motion of the particle. (b) Find the velocity vector for the motion of the particle. (c) Show that the velocity is orthogonal to the position vector of the particle. (d) The period is the amount of time it takes for the particle to complete one full revolution. What is the period of the particle? (e) Find the acceleration vector of the particle. What is the magnitude and direction of this vector? (f) Assuming that the particle has mass m, find the force vector which produces this motion. This vector is called the centripetal force. (g) Now suppose a particle

Explanation / Answer

a)parametric equation is

x=Rcos(wt)

y=Rsin(wt)

position vector r=Rcoswt i + Rsinwt j

b)velocity vector V= dx/dt i + dy/dt j = Rw(-sinwt i +coswt j)=w(-y i + xj)

c)dot product of position vector and velocity vector is r.v = (x i+yj).(-wy i +wx j) = -w*x*y+w*x*y =0

so position vector and velocityy vector are always perpendicular

d) period of the particle = 2*pi/w sec

e) acceleration vector a = dv/dt = -Rw^2(coswt i +sinwt j)

so its magnitude = Rw^2

and direction is opposite to dirction of position vector..ie., radially inwards

f)F=ma=-mRw^2(coswt i +sinwt j)N

e)no because if angular velocity changes then angular acceleration comes into picture.so net acceleration wont be perpendicular to velocity

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