8.1

Rotational Motion and Angular Displacement

In the simplest kind of rotation, points on a rigid object move on circular paths. In Figure 8-1, for example, we see the circular paths for points A, B, and C on a spinning skater. The centers of all such circular paths define a line, called the axis of rotation.

Figure 8-1 When an object rotates, points on the object, such as A, B, or C, move on circular paths. The centers of the circles form a line that is the axis of rotation.

The angle through which a rigid object rotates about a fixed axis is called the angular displacement. Figure 8-2 shows how the angular displacement is measured for a rotating compact disc (CD). Here, the axis of rotation passes through the center of the disc and is perpendicular to its surface. On the surface of the CD we draw a radial line, which is a line that intersects the axis of rotation perpendicularly. As the CD turns, we observe the angle through which this line moves relative to a convenient reference line that does not rotate. The radial line moves from its initial orientation at angle to a final orientation at angle (Greek letter theta). In the process, the line sweeps out the angle . As with other differences that we have encountered (), it is customary to denote the difference between the final and initial angles by the notation (read as “delta theta”): . The angle is the angular displacement. A rotating object may rotate either counterclockwise or clockwise, and standard convention calls a counterclockwise displacement positive and a clockwise displacement negative.

Figure 8-2 The angular displacement of a CD is the angle swept out by a radial line as the disc turns about its axis of rotation.

| Definition of Angular Displacement
| When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise.
SI Unit of Angular Displacement: radian (rad)*

Angular displacement is often expressed in one of three units. The first is the familiar degree, and it is well known that there are 360 degrees in a circle. The second unit is the revolution (rev), one revolution representing one complete turn of 360°. The most useful unit from a scientific viewpoint, however, is the SI unit called the radian (rad). Figure 8-3 shows how the radian is defined, again using a CD as an example. The picture focuses attention on a point P on the disc. This point starts out on the stationary reference line, so that , and the angular displacement is . As the disc rotates, the point traces out an arc of length s, which is measured along a circle of radius r. Equation 8.1 defines the angle in radians:

| (8.1)

According to this definition, an angle in radians is the ratio of two lengths; for example, meters/meters. In calculations, therefore, the radian is treated as a number without units and has no effect on other units that it multiplies or divides.

Figure 8-3 In radian measure, the angle is defined to be the arc length s divided by the radius r.

To convert between degrees and radians, it is only necessary to remember that the arc length of an entire circle of radius r is the circumference . Therefore, according to Equation 8.1, the number of radians that corresponds to 360°, or one revolution, is

Since 2π rad corresponds to 360°, the number of degrees in one radian is

It is useful to express an angle in radians, because then the arc length s subtended at any radius r can be calculated by multiplying by r. Example 1 illustrates this point and also shows how to convert between degrees and radians.

| Example 1 | | | Adjacent Synchronous Satellites

Synchronous or “stationary” communications satellites are put into an orbit whose radius is . The orbit is in the plane of the equator, and two adjacent satellites have an angular separation of , as Figure 8-4 illustrates. Find the arc length s (see the drawing) that separates the satellites.

Figure 8-4 Two adjacent synchronous satellites have an angular separation of . The distances and angles have been exaggerated for clarity.

The physics of communications satellites.

Reasoning Since the radius r and the angle are known, we may find the arc length s by using the relation (in radians) = s/r. But first, the angle must be converted to radians from degrees.
Solution To convert 2.00° into radians, we use the fact that radians is equivalent to 360°:

From Equation 8.1, it follows that the arc length between the satellites is

The radian, being a unitless quantity, is dropped from the final result, leaving the answer expressed in meters.

Conceptual Example 2 takes advantage of the radian as a unit for measuring angles and explains the spectacular phenomenon of a total solar eclipse.

| Conceptual Example 2 | | | A Total Eclipse of the Sun

The diameter of the sun is about 400 times greater than that of the moon. By coincidence, the sun is also about 400 times farther from the earth than is the moon. For an observer on earth, compare the angle subtended by the moon to the angle subtended by the sun, and explain why this result leads to a total solar eclipse.
Reasoning and Solution Figure 8-5a shows a person on earth viewing the moon and the sun. The angle subtended by the moon is given by Equation 8.1 as the arc length smoon divided by the distance rmoon from the earth to the moon: . The earth and moon are sufficiently far apart that the arc length smoon is very nearly equal to the moon’s diameter. Similar reasoning applies for the sun, so that the angle subtended by the sun is . But since (where the symbol “” means “approximately equal to”) and , the two angles are approximately equal: . Figure 8-5b shows what happens when the moon comes between the sun and the earth. Since the angle subtended by the moon is nearly equal to the angle subtended by the sun, the moon blocks most of the sun’s light from reaching the observer’s eyes, and a total solar eclipse like that in Figure 8-5c occurs.

Figure 8-5 (a) The angles subtended by the moon and sun at the eyes of the observer are and . (The distances and angles are exaggerated for the sake of clarity.) (b) Since the moon and sun subtend approximately the same angle, the moon blocks nearly all the sun’s light from reaching the observer’s eyes. (c) The result is a total solar eclipse. (c. Roger Ressmeyer/Corbis Images)

The physics of a total solar eclipse.

Related Homework: Problem 68

Check Your Understanding 1

Three objects are visible in the night sky. They have the following diameters (in multiples of d) and subtend the following angles (in multiples of ) at the eye of the observer. Object A has a diameter of 4d and subtends an angle of . Object B has a diameter of 3d and subtends an angle of . Object C has a diameter of d/2 and subtends an angle of . Rank them in descending order (greatest first) according to their distance from the observer.
(The answer is given at the end of the book.)
Background: The radian is a unit for measuring angular displacement, and it lies at the heart of this problem.
For similar questions (including calculational counterparts), consult Self-Assessment Test 8.1. This test is described at the end of Section 8.3.

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