Relationship Between Linear and Angular Quantities

  

Figure 7.2: Circular Motion

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, $\Delta$s , travelled divided by the time interval, $\Delta$t :

$$\overline{v}_{t}^{}$$ (11)

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

$$\lim_{\Delta t\to 0}^{} {\Delta s\over \Delta t}$$ (12)

Using the fact that

$$\Delta \Delta \theta$$ (13)

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

$$\lim_{\Delta t\to 0}^{} {\Delta \theta \over \Delta t} \omega$$ (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:

$$\overline{a}_{t}^{} {\Delta v_t\over \Delta t}$$ =

$${\Delta \omega \over \Delta t} \overline{\alpha}$$ = (15)

where $\overline{\alpha}$ is the average angular acceleration. The instantaneous tangential acceleration is given by:

$$\lim_{\Delta t\to 0}^{} {\Delta v_t\over \Delta t}$$ a_t =

$$\alpha$$ = (16)

where $\alpha$ 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.