Rotation Angle and Angular Velocity Quiz

PDF page 259: When objects rotate about some axis—for example, when the CD (compact disc) in Figure 6.2 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 to be the ratio of the arc length to the radius of curvature:

PDF page 260: The arc length is the distance traveled along a circular path as shown in Figure 6.3 Note that 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 . The circumference of a circle is . Thus for one complete revolution the rotation angle is

PDF page 260: This result is the basis for defining the units used to measure rotation angles, to be radians (rad), defined so that 6.3 A comparison of some useful angles expressed in both degrees and radians is shown in Table 6.1.

PDF page 261: FIGURE 6.4 Points 1 and 2 rotate through the same angle ( ), but point 2 moves through a greater arc length

PDF page 261: because it is at a greater distance from the center of rotation . If rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are in a circle or one revolution, the relationship between radians and degrees is thus 6.4 so that

PDF page 261: How fast is an object rotating? We define angular velocity as the rate of change of an angle. In symbols, this is

PDF page 261: where an angular rotation takes place in a time . 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 is analogous to linear velocity . 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 in a time , and so it has a linear velocity

PDF page 261: where an angular rotation takes place in a time . 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 is analogous to linear velocity . 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 in a time , and so it has a linear velocity

PDF page 261: where an angular rotation takes place in a time . 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 is analogous to linear velocity . 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 in a time , and so it has a linear velocity

PDF page 261: The first relationship in states that the linear velocity is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest ), as you might expect. We can also call this linear speed of a point on the rim the tangential speed. The second relationship in 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 of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large means a large , because . Similarly, a larger-radius tire rotating at the same angular velocity ( ) will produce a greater linear speed ( ) for the car.

PDF page 261: The first relationship in states that the linear velocity is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest ), as you might expect. We can also call this linear speed of a point on the rim the tangential speed. The second relationship in 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 of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large means a large , because . Similarly, a larger-radius tire rotating at the same angular velocity ( ) will produce a greater linear speed ( ) for the car.

PDF page 261: The first relationship in states that the linear velocity is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest ), as you might expect. We can also call this linear speed of a point on the rim the tangential speed. The second relationship in 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 of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large means a large , because . Similarly, a larger-radius tire rotating at the same angular velocity ( ) will produce a greater linear speed ( ) for the car.

PDF page 261: The first relationship in states that the linear velocity is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest ), as you might expect. We can also call this linear speed of a point on the rim the tangential speed. The second relationship in 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 of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large means a large , because . Similarly, a larger-radius tire rotating at the same angular velocity ( ) will produce a greater linear speed ( ) for the car.