Angle of Rotation and Angular Velocity Quiz

PDF page 212: The arc length, , is the distance traveled along a circular path. The radius of curvature, r, is the radius of the circular path. Both are shown in Figure 6.3.

PDF page 218: a. The radius of curvature is the radius of a circular path.

PDF page 213: . The arc length, , is the distance covered along the circumference. Consider a line from the center of the CD to its edge. In a given time, each pit (used to record information) on this line moves through the same angle. The angle of rotation is the amount of rotation and is the angular analog of distance. The angle of is the arc length divided by the radius of curvature. rotation

PDF page 213: The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers radians (or 360 degrees), and therefore has an angle of rotation of radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship 1 revolution = rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

PDF page 213: The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers radians (or 360 degrees), and therefore has an angle of rotation of radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship 1 revolution = rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

PDF page 213: The angle of rotation is often measured by using a unit called the radian. (Radians are actually dimensionless, because a radian is defined as the ratio of two distances, radius and arc length.) A revolution is one complete rotation, where every point on the circle returns to its original position. One revolution covers radians (or 360 degrees), and therefore has an angle of rotation of radians, and an arc length that is the same as the circumference of the circle. We can convert between radians, revolutions, and degrees using the relationship 1 revolution = rad = 360°. See Table 6.1 for the conversion of degrees to radians for some common angles.

PDF page 217: Discussion We were able to drop the radians from the final solution to part (b) because radians are actually dimensionless. This is because the radian is defined as the ratio of two distances (radius and arc length). Thus, the formula gives an answer in units of meters, as expected for an arc length.

PDF page 213: occurs in a time, . If an object rotates through a greater angle of rotation in a which means that an angular rotation given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s). Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the

PDF page 214: angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation. Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length in a short time so its tangential speed is

PDF page 215: Angular velocity ω and tangential velocity v are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in Figure 6.6.

PDF page 214: The equation says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.

PDF page 214: The equation says that the tangential speed v is proportional to the distance r from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger r) than for a point closer to the center of the CD (with smaller r). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.

PDF page 214: Now, consider another example: the tire of a moving car (see Figure 6.5). The faster the tire spins, the faster the car moves—large means large v because . Similarly, a larger-radius tire rotating at the same angular velocity, , will produce a greater linear (tangential) velocity, v, for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.

PDF page 219: b. Tangential velocity vector is always perpendicular to the radius of the circular path along which the object moves.

PDF page 214: angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation. Angular velocity (ω) is the angular version of linear velocity v. Tangential velocity is the instantaneous linear velocity of an object in rotational motion. To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length in a short time so its tangential speed is