40 Rotation Angle and Angular Velocity

In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

Rotation Angle

When objects rotate about some axis—for example, when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature: [latex]\displaystyle\Delta\theta=\frac{\Delta{s}}{r}\\[/latex]

The arc length Δs is the distance traveled along a circular path as shown in Figure 2 Note that r is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius r. The circumference of a circle is 2πr. Thus for one complete revolution the rotation angle is

[latex]\displaystyle\Delta\theta=\frac{2\pi{r}}{r}=2\pi\\[/latex].

This result is the basis for defining the units used to measure rotation angles, Δθ to be radians (rad), defined so that 2π rad = 1 revolution.

A comparison of some useful angles expressed in both degrees and radians is shown in Table 1.

If Δθ = 2π rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad = 360º so that

[latex]1\text{ rad}=\frac{360^{\circ}}{2\pi}\approx57.3^{\circ}\\[/latex].

Angular Velocity

How fast is an object rotating? We define angular velocity ω as the rate of change of an angle. In symbols, this is [latex]\omega=\frac{\Delta\theta}{\Delta{t}}\\[/latex], where an angular rotation Δθ takes place in a time Δt. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ω is analogous to linear velocity v. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δs in a time Δt, and so it has a linear velocity [latex]v=\frac{\Delta{s}}{\Delta{t}}\\[/latex].

From [latex]\Delta\theta=\frac{\Delta{s}}{r}\\[/latex] we see that Δs = rΔθ. Substituting this into the expression for v gives [latex]v=\frac{r\Delta\theta}{\Delta{t}}=r\omega\\[/latex].

We write this relationship in two different ways and gain two different insights:

[latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex].

The first relationship in [latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex] states that the linear velocity v is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest r), as you might expect. We can also call this linear speed v of a point on the rim the tangential speed. The second relationship in[latex]v=r\omega\text{ or }\omega\frac{v}{r}\\[/latex] can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed v of the car. See Figure 4. So the faster the car moves, the faster the tire spins—large v means a large ω, because v=rω. Similarly, a larger-radius tire rotating at the same angular velocity (ω) will produce a greater linear speed (v) for the car.

Both ω and v have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 5.

PhET Explorations: Ladybug Revolution

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug’s x, y position, velocity, and acceleration using vectors or graphs.

Section Summary

• Uniform circular motion is motion in a circle at constant speed. The rotation angle [latex]\Delta\theta\\[/latex] is defined as the ratio of the arc length to the radius of curvature: [latex]\Delta\theta=\frac{\Delta{s}}{r}\\[/latex], where arc length Δs is distance traveled along a circular path and r is the radius of curvature of the circular path. The quantity [latex]\Delta\theta\\[/latex] is measured in units of radians (rad), for which [latex]2\pi\text{rad}=360^{\circ}= 1\text{ revolution}\\[/latex].
• The conversion between radians and degrees is [latex]1\text{ rad}=57.3^{\circ}\\[/latex].
• Angular velocity ω is the rate of change of an angle, [latex]\omega=\frac{\Delta\theta}{\Delta{t}}\\[/latex], where a rotation [latex]\Delta\theta\\[/latex] takes place in a time [latex]\Delta{t}\\[/latex]. The units of angular velocity are radians per second (rad/s). Linear velocity v and angular velocity ω are related by [latex]v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}\text{.}[/latex]

Glossary

arc length: Δs, the distance traveled by an object along a circular path

pit: a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD

rotation angle: the ratio of the arc length to the radius of curvature on a circular path: [latex]\Delta\theta=\frac{\Delta{s}}{r}\\[/latex]

radius of curvature: radius of a circular path

radians: a unit of angle measurement

angular velocity: ω, the rate of change of the angle with which an object moves on a circular path