The position vector of a particle executing uniform circular motion in the -plane is…

PHYSICS · Multi-part · CBSE · Physics · Motion in a Plane · Motion in a Plane

Problem

The position vector of a particle executing uniform circular motion in the

XY

-plane is given in Cartesian form by

\mathbf r(t)=r\cos\omega t\,\hat\imath + r\sin\omega t\,\hat\jmath,

where

r

and

\omega

are constants. (i) Differentiate

\mathbf r(t)

with respect to time to obtain the velocity vector

\mathbf v(t)

and show that

\mathbf v(t)

is perpendicular to

\mathbf r(t)

at every instant. (ii) Differentiate

\mathbf v(t)

to find the acceleration vector

\mathbf a(t)

. Prove that

\mathbf a(t)

points towards the centre of the circle and that its magnitude is

v^{2}/r

. (iii) Write explicitly the components of

\mathbf v

and

\mathbf a

at the instant

t = T/4

, where

T

is the time period of revolution.