10.1 Rotational variables  (Page 4/11)

where we have taken the limit of the average angular acceleration, $\bar{\alpha }=\frac{\text{Δ}\omega }{\text{Δ}t}$ as $\text{Δ}t\to 0$ .

The units of angular acceleration are (rad/s)/s, or ${\text{rad/s}}^2$ .

In the same way as we defined the vector associated with angular velocity $\overrightarrow{\omega }$ , we can define $\overrightarrow{\alpha }$ , the vector associated with angular acceleration ( [link] ). If the angular velocity is along the positive z- axis, as in [link] , and $\frac{d\omega }{dt}$ is positive, then the angular acceleration $\overrightarrow{\alpha }$ is positive and points along the $+z\text{-}$ axis. Similarly, if the angular velocity $\overrightarrow{\omega }$ is along the positive z- axis and $\frac{d\omega }{dt}$ is negative, then the angular acceleration is negative and points along the $+z-$ axis.

We can express the tangential acceleration vector as a cross product of the angular acceleration and the position vector. This expression can be found by taking the time derivative of $\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$ and is left as an exercise:

The vector relationships for the angular acceleration and tangential acceleration are shown in [link] .

We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we related the tangential speed to the angular velocity. If we differentiate [link] with respect to time, noting that the radius r is constant, we obtain

Thus, the tangential acceleration $a_t$ is the radius times the angular acceleration. [link] and [link] are important for the discussion of rolling motion (see Angular Momentum ).

Let’s apply these ideas to the analysis of a few simple fixed-axis rotation scenarios. Before doing so, we present a problem-solving strategy that can be applied to rotational kinematics: the description of rotational motion.