Relationship Between Linear and Angular Quantities

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Relationship Between Linear and Angular Quantities

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

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 :

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

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

v _t = $\displaystyle\lim_{\Delta t\to 0}^{}$ $\displaystyle{\Delta s\over \Delta t}$ . (12)

Using the fact that

$\displaystyle\Delta$ s = r $\displaystyle\Delta$ $\displaystyle\theta$ (13)

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

v_t = r $\displaystyle\lim_{\Delta t\to 0}^{}$ $\displaystyle{\Delta<br/> \theta \over<br/> \Delta t}$ = r $\displaystyle\omega$ . (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:

$\displaystyle\overline{a}_{t}^{}$	=	$\displaystyle{\Delta v_t\over \Delta t}$
	=	r $\displaystyle{\Delta<br/> \omega \over \Delta t}$ = r $\displaystyle\overline{\alpha}$	(15)

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

a_t	=	$\displaystyle\lim_{\Delta t\to 0}^{}$ $\displaystyle{\Delta v_t\over \Delta t}$
	=	r $\displaystyle\alpha$	(16)

where $\alpha$ is the instantaneous angular acceleration.

Centripetal Acceleration

Consider an object moving in a circle of radius r with constant angular velocity. The tangential speed is constant, but the direction of the tangential velocity vector changes as the object rotates.

Definition: Centripetal Acceleration

Centripetal acceleration is the rate of change of tangential velocity:

$\displaystyle\vec{a}_{c}^{}$ = $\displaystyle\lim_{\Delta t\to 0}^{}$ $\displaystyle{\Delta \vec{v}_t \over \Delta t}$ (17)

Note:

The direction of the centripital acceleration is always inwards along the radius vector of the circular motion.

The magnitude of the centripetal acceleration is related to the tangential speed and angular velocity as follows:

a_c = $\displaystyle{v_t^2 \over r}$ = $\displaystyle\omega^{2}_{}$ r. (18)

In general, a particle moving in a circle experiences both angular acceleration and centripetal accelaration. Since the two are always perpendicular, by definition, the magnitude of the net acceleration a_total is:

a_total = $\displaystyle\sqrt{a_c^2 + a_t^2}$ = $\displaystyle\sqrt{r^2\omega^4 +<br/> r^2\alpha^2}$ . (19)

Definition: Centripetal Force

Centripetal force is the net force causing the centripetal acceleration of an object in circular motion. By Newton's Second Law:

$\displaystyle\vec{F}_{c}^{}$ = m $\displaystyle\vec{a}_{c}^{}$ . (20)

Its direction is always inward along the radius vector, and its magnitude is given by:

F_c = ma_c = m $\displaystyle{v_t^2 \over r}$ = m $\displaystyle\omega^{2}_{}$ r. (21)

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Relationship Between Linear and Angular Quantities

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