Defining the tangential velocity of a body

We say that a body that follows a circular path has uniform circular motion when its angular velocity is constant. Now, in circular motion, we define the tangential velocity as the instantaneous velocity of the point considered in this motion.

In the figure above, we have a weather vane that rotates causing point A to describe a circular motion. If we know the value of the radius R of the circle described by point A, it will be possible to determine the value and direction of the tangential velocity of this point, this for any instant.

Assuming that the path that point A travels during a complete revolution is (D), mathematically we have that D is the length of the circle described by point A, therefore, we have:

D=2πR

If we also know the time that point A took to complete a complete revolution, that is, if we know what the period is, we can calculate the speed of point A, so, mathematically, we have:

In this way, we have:

Therefore, we can rewrite equation (I) as:

v=ω.R

We can therefore conclude that the tangential velocity of point A, or of any other body describing a circular motion, is a vector v whose direction is always tangent to the path it describes. Notice in the figure below that the tangential velocity v always changes direction in order to remain tangent to the trajectory. With this, we can also define that the magnitude of the tangential velocity is always constant in uniform circular motion.