10.3 Relating angular and translational quantities  (Page 2/5)

The centripetal acceleration is due to the change in the direction of tangential velocity, whereas the tangential acceleration is due to any change in the magnitude of the tangential velocity. The tangential and centripetal acceleration vectors ${\overrightarrow{a}}_{\text{t}}$ and ${\overrightarrow{a}}_{\text{c}}$ are always perpendicular to each other, as seen in [link] . To complete this description, we can assign a total linear acceleration    vector to a point on a rotating rigid body or a particle executing circular motion at a radius r from a fixed axis. The total linear acceleration vector $\overrightarrow{a}$ is the vector sum of the centripetal and tangential accelerations,

The total linear acceleration vector in the case of nonuniform circular motion points at an angle between the centripetal and tangential acceleration vectors, as shown in [link] . Since ${\overrightarrow{a}}_{\text{c}}\perp {\overrightarrow{a}}_{\text{t}}$ , the magnitude of the total linear acceleration is

Note that if the angular acceleration is zero, the total linear acceleration is equal to the centripetal acceleration.

Relationships between rotational and translational motion

We can look at two relationships between rotational and translational motion.

1. Generally speaking, the linear kinematic equations have their rotational counterparts. [link] lists the four linear kinematic equations and the corresponding rotational counterpart. The two sets of equations look similar to each other, but describe two different physical situations, that is, rotation and translation.
   

   Rotational and translational kinematic equations

   Rotational	Translational
   ${\theta }_{\text{f}}={\theta }_0+\bar{\omega }t$	$x=x_0+\bar{v}t$
   ${\omega }_{\text{f}}={\omega }_0+\alpha t$	$v_{\text{f}}=v_0+at$
   ${\theta }_{\text{f}}={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$	$x_{\text{f}}=x_0+v_{0}t+\frac{1}{2}a{t}^2$
   ${\omega }_{\text{f}}^2={\omega }_0^2+2\alpha (\text{Δ}\theta )$	$v_{\text{f}}^2=v_0^2+2a(\text{Δ}x)$

2. The second correspondence has to do with relating linear and rotational variables in the special case of circular motion. This is shown in [link] , where in the third column, we have listed the connecting equation that relates the linear variable to the rotational variable. The rotational variables of angular velocity and acceleration have subscripts that indicate their definition in circular motion.
   

   Rotational and translational quantities: circular motion

   Rotational	Translational	Relationship ( $r=\text{radius}$ )
   $\theta$	s	$\theta =\frac{s}{r}$
   $\omega$	$v_{\text{t}}$	$\omega =\frac{v_{\text{t}}}{r}$
   $\alpha$	$a_{\text{t}}$	$\alpha =\frac{a_{\text{t}}}{r}$
   	$a_{\text{c}}$	$a_{\text{c}}=\frac{v_{\text{t}}^2}{r}$

Linear acceleration of a centrifuge

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