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where we have taken the limit of the average angular acceleration, $\stackrel{\u2013}{\alpha}=\frac{\text{\Delta}\omega}{\text{\Delta}t}$ as $\text{\Delta}t\to 0$ .
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 $\overrightarrow{v}=\overrightarrow{\omega}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\overrightarrow{r}$ 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.
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