10.3 Relating angular and translational quantities  (Page 3/5)

Solution

Therefore, the tangential acceleration is

The angular velocity at $t=29.0\text{s}$ is

Thus, the tangential speed at $t=29.0\text{s}$ is

We can now calculate the centripetal acceleration at $t=29.0\text{s}$ :

Since the two acceleration vectors are perpendicular to each other, the magnitude of the total linear acceleration is

Since the centrifuge has a negative angular acceleration, it is slowing down. The total acceleration vector is as shown in [link] . The angle with respect to the centripetal acceleration vector is

The negative sign means that the total acceleration vector is angled toward the clockwise direction.

Significance

Summary

• The linear kinematic equations have their rotational counterparts such that there is a mapping $x\to \theta ,v\to \omega ,a\to \alpha$ .
• A system undergoing uniform circular motion has a constant angular velocity, but points at a distance r from the rotation axis have a linear centripetal acceleration.
• A system undergoing nonuniform circular motion has an angular acceleration and therefore has both a linear centripetal and linear tangential acceleration at a point a distance r from the axis of rotation.
• The total linear acceleration is the vector sum of the centripetal acceleration vector and the tangential acceleration vector. Since the centripetal and tangential acceleration vectors are perpendicular to each other for circular motion, the magnitude of the total linear acceleration is $\left|\overrightarrow{a}\right|=\sqrt{a_{\text{c}}^2+a_{\text{t}}^2}$ .

Conceptual questions