Question

why the angular velocity of an object is equal for all time?​

Answer 1

$\huge\mathbb\red{ANSWER}$

Explanation:

Relation between linear velocity and angular velocity

Let us consider a body P moving along the circumference of a circle of radius r with linear velocity v and angular velocity ω as shown in Fig.. Let it move from P to Q in time dt and dθ be the angle swept by the radius vector.

Let PQ = ds, be the arc length covered by the particle moving along the circle, then the angular displacement d θ is expressed as dθ = ds/r. But ds=vdt.

d θ/dt=v/r

(i.e) Angular velocity ω = v/r or v =ω r

In vector notation,

Vector v = Vector ω x Vector r

Thus, for a given angular velocity ω, the linear velocity v of the particle is directly proportional to the distance of the particle from the centre of the circular path (i.e) for a body in a uniform circular motion, the angular velocity is the same for all points in the body but linear velocity is different for different points of the body.