**Relation between Angular
Velocity and Linear Velocity**

**What
is Angular Velocity?**

Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. It is also known as rotational velocity. We can also define the angular velocity of a particle as the rate at which the particle rotates around a center point i.e., the time rate of change of its angular displacement relative to the origin.

**What is Linear Velocity?**

Linear velocity is the rate of change of the position of an object that is traveling along a straight path. Because any moving object has a linear velocity, this measurement shows up very often in everyday life. For instance, if a person goes for a walk, drive, run, or bike ride, there is linear velocity.

**Relation between Angular Velocity and
Linear 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 then angular displacement (d), θ is expressed as dθ = ds/r, But ds = v dt.

\(\frac{d\theta }{dt}=\frac{v}{r}\) i.e. Angular velocity (ω) = v/ r

v = ωr

In vector notation: \(\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}\).

Thus, for a given angular velocity, the linear velocity of the particle is directly proportional to the distance of the particle from the center of the circular path.