In `U.C.M.` (Uniform Circular Motion), prove the relation `vecupsilon = vecomega xx vecr`, where symbols have their usual meanings.

Explanation

Analytical method : Consider a particle revolving in the anticlockwise sense along the circumference of a circle of radius r with centre O. Let `deltat` be the time required for particle performing circular motion to move from initial position A to final position B as shown in the figure below.
Let `deltas` (arc AB) be the distance travelled by particle performing circular motion in time `deltat`.

Let, `vecomega` = angular velocity of the particle
`vecv =` linear velocity of the particle
`vecr =` radius vector of the particle
In vector form, the linear displacement is
`vecdeltas = deltatheta xx vecr`
Dividing both side by `deltat`, we get
`(vecdeltas)/(deltat) = (vecdeltatheta)/(deltat) xx vecr`
`underset(deltatrarr 0)("lim") (vecdeltas)/(deltat) = underset(deltatrarr0)("lim")(vecdeltatheta)/(deltat) xx vecr`
`:. (vecds)/(dt) = (vecd theta)/(dt) xx vecr`
But `(vecds)/(dt) = vecv =` Linear velocity and `(vecd theta)/(dt) = omega` = Angular velocity
`:. vecv = vecomega xx vecr`