Angular Acceleration

Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity $\omega$ was defined as the time rate of change of angle $\theta$

$\omega=\dfrac{\Delta \theta}{\Delta t},$

where $\theta$ is the angle of rotation as seen in Figure 10.3. The relationship between angular velocity $\omega$ and linear velocity $v$ was also defined in Rotation Angle and Angular Velocity. as

$v=r \omega$

or

$\omega=\dfrac{v}{r},$

where $r$ is the radius of curvature, also seen in Figure 10.3. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative

Figure 10.3 This figure shows uniform circular motion and some of its defined quantities.

Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer's hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which $\omega$ changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration $\alpha$ is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:

$\alpha=\dfrac{\Delta \omega}{\Delta t},$

where $\Delta \omega$ is the change in angular velocity and $\Delta t$ is the change in time. The units of angular acceleration are $(\mathrm{rad} / \mathrm{s}) / \mathrm{s}$, or $\mathrm{rad} / \mathrm{s}^{2}$. If $\omega$ increases, then $\alpha$ is positive. If $\omega$ decreases, then $\alpha$ is negative.

If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is tangent to the circle at the point of interest, as seen in Figure 10.4. Thus, linear acceleration is called tangential acceleration a_t

Figure 10.4 In circular motion, linear acceleration a, occurs as the magnitude of the velocity changes: a is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration a_t.

Figure 10.5 Centripetal acceleration $a_{\mathrm{c}}$ occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.

Now we can find the exact relationship between linear acceleration $a_{\mathrm{t}}$ and angular acceleration $\alpha$. Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in One-Dimensional Kinematics to be

$a_{\mathrm{t}}=\dfrac{\Delta v}{\Delta t}.$

For circular motion, note that $v=r \omega$, so that

$a_{\mathrm{t}}=\dfrac{\Delta(r \omega)}{\Delta t}.$

The radius $r$ is constant for circular motion, and so $\Delta(r \omega)=r(\Delta \omega)$. Thus,

$a_{\mathrm{t}}=r \dfrac{\Delta \omega}{\Delta t}.$

By definition, $\alpha=\dfrac{\Delta \omega}{\Delta t}.$ Thus,

$a_{\mathrm{t}}=r \alpha$

or

$\alpha=\dfrac{a_{\mathrm{t}}}{r}.$

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car's drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration $\alpha$.

So far, we have defined three rotational quantities $-\theta, \omega$, and $\alpha.$ These quantities are analogous to the translational quantities $x, v$, and $a.$ Table 10.1 displays rotational quantities, the analogous translational quantities, and the relationships between them.

Table 10.1 Rotational and Translational Quantities

Source: Rice University, https://openstax.org/books/college-physics/pages/10-1-angular-acceleration
This work is licensed under a Creative Commons Attribution 4.0 License.