Rotational Variables

Angular Velocity

Uniform circular motion (discussed previously in Motion in Two and Three Dimensions) is motion in a circle at constant speed. Although this is the simplest case of rotational motion, it is very useful for many situations, and we use it here to introduce rotational variables.

In (Figure), we show a particle moving in a circle. The coordinate system is fixed and serves as a frame of reference to define the particle’s position. Its position vector from the origin of the circle to the particle sweeps out the angle [latex]\theta[/latex], which increases in the counterclockwise direction as the particle moves along its circular path. The angle [latex]\theta[/latex] is called the angular position of the particle. As the particle moves in its circular path, it also traces an arc length s.

The angle is related to the radius of the circle and the arc length by

The angle [latex]\theta[/latex], the angular position of the particle along its path, has units of radians (rad). There are [latex]2\pi[/latex] radians in [latex]360\text{°}.[/latex] Note that the radian measure is a ratio of length measurements, and therefore is a dimensionless quantity. As the particle moves along its circular path, its angular position changes and it undergoes angular displacements [latex]\text{Δ}\theta .[/latex]

We can assign vectors to the quantities in (Figure). The angle [latex]\stackrel{\to }{\theta }[/latex] is a vector out of the page in (Figure). The angular position vector [latex]\stackrel{\to }{r}[/latex] and the arc length [latex]\stackrel{\to }{s}[/latex] both lie in the plane of the page. These three vectors are related to each other by

That is, the arc length is the cross product of the angle vector and the position vector, as shown in (Figure).

The magnitude of the angular velocity, denoted by [latex]\omega[/latex], is the time rate of change of the angle [latex]\theta[/latex] as the particle moves in its circular path. The instantaneous angular velocity is defined as the limit in which [latex]\text{Δ}t\to 0[/latex] in the average angular velocity [latex]\stackrel{–}{\omega }=\frac{\text{Δ}\theta }{\text{Δ}t}[/latex]:

where [latex]\theta[/latex] is the angle of rotation ((Figure)). The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in radians per second. In many situations, we are given the rotation rate in revolutions/s or cycles/s. To find the angular velocity, we must multiply revolutions/s by [latex]2\pi[/latex], since there are [latex]2\pi[/latex] radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.

We can see how angular velocity is related to the tangential speed of the particle by differentiating (Figure) with respect to time. We rewrite (Figure) as

Taking the derivative with respect to time and noting that the radius r is a constant, we have

where [latex]\theta \frac{dr}{dt}=0[/latex]. Here [latex]\frac{ds}{dt}[/latex] is just the tangential speed [latex]{v}_{\text{t}}[/latex] of the particle in (Figure). Thus, by using (Figure), we arrive at

That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From (Figure), we see that the tangential speed of the particle increases with its distance from the axis of rotation for a constant angular velocity. This effect is shown in (Figure). Two particles are placed at different radii on a rotating disk with a constant angular velocity. As the disk rotates, the tangential speed increases linearly with the radius from the axis of rotation. In (Figure), we see that [latex]{v}_{1}={r}_{1}{\omega }_{1}[/latex] and [latex]{v}_{2}={r}_{2}{\omega }_{2}[/latex]. But the disk has a constant angular velocity, so [latex]{\omega }_{1}={\omega }_{2}[/latex]. This means [latex]\frac{{v}_{1}}{{r}_{1}}=\frac{{v}_{2}}{{r}_{2}}[/latex] or [latex]{v}_{2}=\left(\frac{{r}_{2}}{{r}_{1}}\right){v}_{1}[/latex]. Thus, since [latex]{r}_{2}>{r}_{1}[/latex], [latex]{v}_{2}>{v}_{1}[/latex].

Up until now, we have discussed the magnitude of the angular velocity [latex]\omega =d\theta \text{/}dt,[/latex] which is a scalar quantity—the change in angular position with respect to time. The vector [latex]\stackrel{\to }{\omega }[/latex] is the vector associated with the angular velocity and points along the axis of rotation. This is useful because when a rigid body is rotating, we want to know both the axis of rotation and the direction that the body is rotating about the axis, clockwise or counterclockwise. The angular velocity [latex]\stackrel{\to }{\omega }[/latex] gives us this information. The angular velocity [latex]\stackrel{\to }{\omega }[/latex] has a direction determined by what is called the right-hand rule. The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x-axis (the direction in which [latex]\theta[/latex] increases) toward the y-axis, your thumb points in the direction of the positive z-axis ((Figure)). An angular velocity [latex]\stackrel{\to }{\omega }[/latex] that points along the positive z-axis therefore corresponds to a counterclockwise rotation, whereas an angular velocity [latex]\stackrel{\to }{\omega }[/latex] that points along the negative z-axis corresponds to a clockwise rotation.

We can verify the right-hand-rule using the vector expression for the arc length [latex]\stackrel{\to }{s}=\stackrel{\to }{\theta }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}[/latex], (Figure). If we differentiate this equation with respect to time, we find

Since [latex]\stackrel{\to }{r}[/latex] is constant, the term [latex]\stackrel{\to }{\theta }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{d\stackrel{\to }{r}}{dt}=0[/latex]. Since [latex]\stackrel{\to }{v}=\frac{d\stackrel{\to }{s}}{dt}[/latex] is the tangential velocity and [latex]\stackrel{\to }{\omega }=\frac{d\stackrel{\to }{\theta }}{dt}[/latex] is the angular velocity, we have

That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in (Figure). From part (a) of this figure, we see that with the angular velocity in the positive z-direction, the rotation in the xy-plane is counterclockwise. In part (b), the angular velocity is in the negative z-direction, giving a clockwise rotation in the xy-plane.

Angular Acceleration

We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an angular acceleration for describing situations where [latex]\omega[/latex] changes. The faster the change in [latex]\omega[/latex], the greater the angular acceleration. We define the instantaneous angular acceleration [latex]\alpha[/latex] as the derivative of angular velocity with respect to time:

where we have taken the limit of the average angular acceleration, [latex]\stackrel{–}{\alpha }=\frac{\text{Δ}\omega }{\text{Δ}t}[/latex] as [latex]\text{Δ}t\to 0[/latex].

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

In the same way as we defined the vector associated with angular velocity [latex]\stackrel{\to }{\omega }[/latex], we can define [latex]\stackrel{\to }{\alpha }[/latex], the vector associated with angular acceleration ((Figure)). If the angular velocity is along the positive z-axis, as in (Figure), and [latex]\frac{d\omega }{dt}[/latex] is positive, then the angular acceleration [latex]\stackrel{\to }{\alpha }[/latex] is positive and points along the [latex]+z\text{-}[/latex] axis. Similarly, if the angular velocity [latex]\stackrel{\to }{\omega }[/latex] is along the positive z-axis and [latex]\frac{d\omega }{dt}[/latex] is negative, then the angular acceleration is negative and points along the [latex]+z-[/latex] 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 [latex]\stackrel{\to }{v}=\stackrel{\to }{\omega }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}[/latex] and is left as an exercise:

The vector relationships for the angular acceleration and tangential acceleration are shown in (Figure).

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 (Figure) with respect to time, noting that the radius r is constant, we obtain

Thus, the tangential acceleration [latex]{a}_{t}[/latex] is the radius times the angular acceleration. (Figure) and (Figure) 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.

Now let’s apply this problem-solving strategy to a few specific examples.

We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section.