# Relationship Between Linear and Angular Quantities

Consider an object that moves from point P to P' along a circular trajectory of radius r , as shown in Figure 7.2.

Definition: Tangential Speed

The average tangential speed of such an object is defined to be the length of arc, s , travelled divided by the time interval, t :

 = . (11)

The instantaneous tangential speed is obtained by taking t to zero:

 v t = . (12)

Using the fact that

 s = r (13)

we obtain the relationship between the angular velocity of an object in circular motion and its tangential velocity:

 vt = r = r. (14)

This relation holds for both average and instantaneous speeds.

Note:
• The instantaneous tangential velocity vector is always perpendicular to the radius vector for circular motion.

Definition: Tangential Acceleration

Tangential acceleration is the rate of change of tangential speed. The average tangential acceleration is:

 = = r = r (15)
where is the average angular acceleration. The instantaneous tangential acceleration is given by:
 at = = r (16)
where is the instantaneous angular acceleration.

Note:
• The above formula is only valid if the angular velocity is expressed in radians per second.
• The direction of the tangential acceleration vector is always parallel to the tangential velocity, and perpendicular to the radius vector of the circular motion.

Next: Centripetal Acceleration Up: Circular Motion and the Previous: Formulae for Constant Angular