Rotational Motion Quiz

PDF page 226: In the section on uniform circular motion, we discussed motion in a circle at constant speed and, therefore, constant angular velocity. However, there are times when angular velocity is not constant—rotational motion can speed up, slow down, or reverse directions. Angular velocity is not constant when a spinning skater pulls in her arms, when a child pushes a merry-go-round to make it rotate, or when a CD slows to a halt when switched off. In all these cases, angular acceleration occurs because the angular velocity changes. The faster the change occurs, the greater is the angular acceleration. Angular acceleration is the rate of change of angular velocity. In equation form, angular acceleration is

PDF page 232: b. Angular acceleration is the rate of change of the angular velocity.

PDF page 226: where is the change in angular velocity and is the change in time. The units of angular acceleration are (rad/s)/s, or rad/

PDF page 226: s . If increases, then is positive. If decreases, then is negative. Keep in mind that, by convention, counterclockwise is

PDF page 226: s . If increases, then is positive. If decreases, then is negative. Keep in mind that, by convention, counterclockwise is

PDF page 226: the positive direction and clockwise is the negative direction. For example, the skater in Figure 6.9 is rotating counterclockwise as seen from above, so her angular velocity is positive. Acceleration would be negative, for example, when an object that is rotating counterclockwise slows down. It would be positive when an object that is rotating counterclockwise speeds up.

PDF page 226: the positive direction and clockwise is the negative direction. For example, the skater in Figure 6.9 is rotating counterclockwise as seen from above, so her angular velocity is positive. Acceleration would be negative, for example, when an object that is rotating counterclockwise slows down. It would be positive when an object that is rotating counterclockwise speeds up.

PDF page 227: These equations mean that the magnitudes of tangential acceleration and angular acceleration are directly proportional to each other. The greater the angular acceleration, the larger the change in tangential acceleration, and vice versa. For example, consider riders in their pods on a Ferris wheel at rest. A Ferris wheel with greater angular acceleration will give the riders greater tangential acceleration because, as the Ferris wheel increases its rate of spinning, it also increases its tangential velocity. Note that the radius of the spinning object also matters. For example, for a given angular acceleration , a smaller Ferris wheel leads to a smaller tangential acceleration for the riders.

PDF page 227: Tangential acceleration is sometimes denoted at. It is a linear acceleration in a direction tangent to the circle at the point of interest in circular or rotational motion. Remember that tangential acceleration is parallel to the tangential velocity (either in the same direction or in the opposite direction.) Centripetal acceleration is always perpendicular to the tangential velocity.

PDF page 227: Tangential acceleration is sometimes denoted at. It is a linear acceleration in a direction tangent to the circle at the point of interest in circular or rotational motion. Remember that tangential acceleration is parallel to the tangential velocity (either in the same direction or in the opposite direction.) Centripetal acceleration is always perpendicular to the tangential velocity.

PDF page 227: So far, we have defined three rotational variables: , , and . These are the angular versions of the linear variables x, v, and a.

PDF page 228: We can now begin to see how rotational quantities like , , and are related to each other. For example, if a motorcycle wheel that starts at rest has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. Putting this in terms of the variables, if the wheel’s angular acceleration is large for a long period of time t, then the final angular velocity and angle of rotation are large. In the case of linear motion, if an object starts at rest and undergoes a large linear acceleration, then it has a large final velocity and will have traveled a large distance. The kinematics of rotational motion describes the relationships between the angle of rotation, angular velocity, angular acceleration, and time. It only describes motion—it does not include any forces or masses that may affect rotation (these are

PDF page 234: angle of rotation the ratio of the arc length to the radius of curvature of a circular path angular acceleration the rate of change of angular velocity with time angular velocity ( ) the rate of change in the angular position of an object following a circular path arc length ( ) the distance traveled by an object along a circular path centrifugal force a fictitious force that acts in the direction opposite the centripetal acceleration centripetal acceleration the acceleration of an object moving in a circle, directed toward the center of the circle centripetal force any force causing uniform circular motion circular motion the motion of an object along a circular path kinematics of rotational motion the relationships between rotation angle, angular velocity, angular acceleration, and

PDF page 228: (constant a). As in linear kinematics, we assume a is constant, which means that angular acceleration is also a constant, because . The equation for the kinematics relationship between , , and t is

PDF page 228: where is the initial angular velocity. Notice that the equation is identical to the linear version, except with angular analogs of the linear variables. In fact, all of the linear kinematics equations have rotational analogs, which are given in Table 6.3. These equations can be used to solve rotational or linear kinematics problem in which a and are constant.

PDF page 229: a. The eye of the storm is the center of rotation. Winds are weaker at the eye of a tornado because tangential velocity is

PDF page 229: directly proportional to radius of curvature.

PDF page 230: massive the door is, the more slowly it opens; this is because angular acceleration is inversely proportional to mass. These relationships are very similar to the relationships between force, mass, and acceleration from Newton’s second law of motion. Since we have already covered the angular versions of distance, velocity and time, you may wonder what the angular version of force is, and how it relates to linear force. The angular version of force is torque , which is the turning effectiveness of a force. See Figure 6.11. The equation for the magnitude of torque is

PDF page 232: a. It is the rotational equivalent of a force.