第10章固定軸まわりの回転

So far in this text, we have mainly studied translational motion, including the variables that describe it: displacement, velocity, and acceleration. Now we expand our description of motion to rotation—specifically, rotational motion about a fixed axis. We will find that rotational motion is described by a set of related variables similar to those we used in translational motion.

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 10.2, 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 $\theta$, which increases in the counterclockwise direction as the particle moves along its circular path. The angle $\theta$ is called the angular position of the particle. As the particle moves in its circular path, it also traces an arc length s. The particle may complete more than one revolution around the circle, and so the angle $\theta$ may be greater than $2\pi$, and the arc length s may be greater than the circumference, $2\pi r$.

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

The angle $\theta$, the angular position of the particle along its path, has units of radians (rad). There are $2\pi$ radians in $360\text{°}.$ 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 $\text{Δ}\theta .$

We can assign vectors to the quantities in Equation 10.1. The angle $\overrightarrow{\theta }$ is a vector out of the page in Figure 10.2. The angular position vector $\overrightarrow{r}$ and the arc length $\overrightarrow{s}$ 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 10.3.

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

where $\theta$ is the angle of rotation (Figure 10.2). 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 $2\pi$, since there are $2\pi$ 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 Equation 10.1 with respect to time. We rewrite Equation 10.1 as

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

where $\theta \frac{dr}{dt}=0$. Here $\frac{ds}{dt}$ is just the tangential speed $v_{\text{t}}$ of the particle in Figure 10.2. Thus, by using Equation 10.3, we arrive at

That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From Equation 10.4, 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 10.4. 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 10.4, we see that $v_1=r_1{\omega }_1$ and $v_2=r_2{\omega }_2$. But the disk has a constant angular velocity, so ${\omega }_1={\omega }_2$. This means $\frac{v_1}{r_1}=\frac{v_2}{r_2}$ or $v_2=\left(\frac{r_2}{r_1}\right)v_1$. Thus, since $r_2>r_1$, $v_2>v_1$.

Up until now, we have discussed the magnitude of the angular velocity $\omega =d\theta \text{/}dt,$ which is a scalar quantity—the change in angular position with respect to time. The vector $\overrightarrow{\omega }$ 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 $\overrightarrow{\omega }$ gives us this information. The angular velocity $\overrightarrow{\omega }$ 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 $\theta$ increases) toward the y-axis, your thumb points in the direction of the positive z-axis (Figure 10.5). An angular velocity $\overrightarrow{\omega }$ that points along the positive z-axis therefore corresponds to a counterclockwise rotation, whereas an angular velocity $\overrightarrow{\omega }$ that points along the negative z-axis corresponds to a clockwise rotation.

Similar to Equation 10.2, one can state a cross product relation to the vector of the tangential velocity as stated in Equation 10.4. Therefore, we have

That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in Figure 10.6. 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.