Angular Acceleration

Previously, we looked at uniform circular motion, where an object moves in a circular path at a constant speed (but changing directions, so it undergoes centripetal acceleration). We can also describe this in terms of a rotation at a constant rate. For example, an object completing a circle every second can be said to rotate at a constant rate of 360 degrees per second about the center of the circle, since there are 360 degrees in one complete circle.

For the formulas and equations you will see in this chapter, we will introduce a mathematically convenient unit of angle called radians. An angular measure in radian is defined by the arc length covered by the angle divided by the radius of the circle that the arc length is part of. Do you see how, if the angular size remains the same, as we consider larger circles, with matching longer arc length, the ratio of arc length to the radius of the circle remains constant? Consider this example. The circumference $S$ of a circle of radius $r$ is given by $S=2\pi r$. So the angular size of a full circle is this circumference (a special kind of arc length) divided by the radius, or $2\pi$ (so you could say $2\pi$ radians is equal to 360 degrees).

So, imagine an object completing a uniform circular motion of radius $r$ at speed $v$, and let's say it takes time $\mathrm{\Delta }t$ to complete one full circle. Then using what we learned in kinematics, we know this holds:

where $\mathrm{\Delta }s$ is the arc length, or the circumference, in this case. We can also describe this in terms of angular velocity, or the rate of change of angle,

where change of angle $\mathrm{\Delta }\theta$ in this case is the full circle, or $2\pi$ radians. We use $\omega$ (lower-case omega) to denote angular velocity.

Carefully comparing Equation 6.2.1 with Equation 6.2.2, we see that following relationships hold:

and

We are going to use this relationship to relate linear motion with rotation (or angular motion) whenever necessary. A summary with figure is in Figure 6.2.1. In Figure 6.2.1, also note that, given the same angular velocity $\omega$, an object at a larger radius has a larger linear speed (does this make intuitive sense?).

Angular velocity is not constant when a skater pulls in her arms (speeding up for some reason, which we'll look at why later), 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:

where $\mathrm{\Delta }\omega$ is the change in angular velocity and $\mathrm{\Delta }t$ is the change in time. The units of angular acceleration are $\left(\text{rad/s}\right)\text{/s}$, or ${\text{rad/s}}^2$. If $\omega$ increases, then $\alpha$ is positive. If $\omega$ decreases, then $\alpha$ is negative. Hopefully all this sounds rather familiar from a similar discussion in linear kinematics.

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 6.2.2. Thus, linear acceleration is called tangential acceleration $a_{\text{t}}$. This is not to be confused with centripetal acceleration we covered earlier (see: Figure 6.2.3).

This tangential acceleration $a_{\text{t}}$ is related to angular acceleration $\alpha$ in a similar way as how linear velocity is related to angular velocity. So, taking the cues from Equation 6.2.3 and Equation 6.2.4, tangential acceleration and angular acceleration are related by,

and

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— $\theta \mathrm{, }\omega$, and $\alpha$. These quantities are analogous to the translational quantities $x\mathrm{, }v$, and $a$. Table 6.2.1 displays rotational quantities, the analogous translational quantities, and the relationships between them.