Question

relation between linear velocity and angular velocity ​

Answer 1

Answer:

Given:

An object is travelling along a circular trajectory with constant velocity (assumed uniform circular motion)

To find:

Relationship between Linear velocity and angular Velocity

Definitions and Directions:

1. Linear velocity denotes the instantaneous direction of velocity at any point along the circular path.

The direction coincides with the tangent the that point.

2. Angular velocity is an axial velocity directed perpendicular to the plane of the circular trajectory.

Relationship:

In vector notation, angular Velocity is obtained by cross product between

linear Velocity vector and radial vector.

v = ω × r

v => linear Velocity

ω => Angular velocity

r => radial velocity

Answer 2

To establish relationship between angular and linear velocity

Consider a particle involved in rotational dynamics i.e.,moving in a circular path. The tangent to the circle would give the linear velocity while the path of circle signifies angular velocity.

Consider a section of the circle and let the length of the arc be "ds"

Now,the arc would subtend some angle with the centre, assuming it to be ∅

Implies,

$\large{ \hookrightarrow \: \tt{ds =d \theta \times dr}} \\ \\ \large{ \hookrightarrow \: \tt{ \frac{ds}{dt} = dr \times \frac{d \theta}{dt} }} \\ \\ \large{ \hookrightarrow \: \boxed{ \boxed{\tt{v = r \omega}}}}$

Here,

• v is linear velocity
• w is angular velocity
• r is radius of the path followed