Question

Devrived the relation between linear and speed and angular velocity​

Answer 1

Angular velocity

 

The rate of change of angular displacement is called the angular velocity of the particle.

Let dθ be the angular displacement made by the particle in time dt , then the angular velocity of the particle is dθ /dt ω = . Its unit is rad s-1 and dimensional formula is T-1.

For one complete revolution, the angle swept by the radius vector is 360o or 2π radians. If T is the time taken for one complete revolution, known as period, then the angular velocity of the particle is

ω= θ/t = 2 π/T .

If the particle makes n revolutions per second, then ω=2π(1/T) = 2πn

where n = 1/T  is the frequency of revolution.

 

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.