Ch 8 Rotational Kinematics Rotational Motion and Angular

Ch 8. Rotational Kinematics Rotational Motion and Angular Displacement Angular displacement: When a rigid body rotates about a fixed axis, the angular displacements 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) 1

Angular displacement is expressed in one of three units: 1. Degree (1 full turn 3600 degree) 2. Revolution (rev) RPM 3. Radian (rad) SI unit 2

(in radians) For 1 full rotation, 3

Example 1. Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is r = 4. 23*107 m. The orbit is in the plane of the equator, and two adjacent satellites have an angular separation of . Find the arc length s. 4

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Conceptual example 2. A Total Eclipse of the Sun 6

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. 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. 7

Since they are very far apart. 8

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total eclipse 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. 10

CONCEPTS AT A GLANCE To define angular velocity, we use two concepts previously encountered. The angular velocity is obtained by combining the angular displacement and the time during which the displacement occurs. Angular velocity is defined in a manner analogous to that used for linear velocity. Taking advantage of this analogy between the two types of velocities will help us understand rotational motion. 13

DEFINITION OF AVERAGE ANGULAR VELOCITY SI Unit of Angular Velocity: radian per second (rad/s) 14

Example 3. Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time of 1. 90 s. Find the average angular velocity (in rad/s) of the gymnast. 15

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In linear motion, a changing velocity means that an acceleration is occurring. Such is also the case in rotational motion; a changing angular velocity means that an angular acceleration is occurring. CONCEPTS AT A GLANCE The idea of angular acceleration describes how rapidly or slowly the angular velocity changes during a given time interval. 18

DEFINITION OF AVERAGE ANGULAR ACCELERATION SI Unit of Average Angular Acceleration: radian per second squared (rad/s 2) The instantaneous angular acceleration a is the angular acceleration at a given instant. 19

Example 4. A Jet Revving Its Engines A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of – 110 rad/s, where the negative sign indicates a clockwise rotation. As the plane takes off, the angular velocity of the blades reaches – 330 rad/s in a time of 14 s. Find the average angular velocity, assuming that the orientation of the rotating object is given by…. . 20

The Equations of Rotational Kinematics 21

In example 4, assume that the orientation of the rotating object is given by q� 0 = 0 rad at time t 0 = 0 s. Then, the angular displacement becomes Dq� = q� – q� 0 = q�, and the time interval becomes Dt = t – t 0 = t. 22

The Equations of Kinematics for Rational and Linear Motion Rotational Motion (a = constant) Linear Motion (a = constant) 23

Symbols Used in Rotational and Linear Kinematics Rotational Motion Quantity Linear. Motion q� w 0 w a t Displacement Initial velocity Final velocity Acceleration Time x v 0 v a t 24

Example 5. Blending with a Blender The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in. When the “blend” button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44. 0 rad (seven revolutions). The angular acceleration has a constant value of +1740 rad/s 2. Find the final angular velocity of the blades. 25

q� +44. 0 rad a +1740 rad/s 2 w ? w 0 +375 rad/s t 26

Check Your Understanding 2 The blades of a ceiling fan start from rest and, after two revolutions, have an angular speed of 0. 50 rev/s. The angular acceleration of the blades is constant. What is the angular speed after eight revolutions? What can be found next? 27

after eight revolution, 1. 0 rev/s 28

Angular Variables and Tangential Variables For every individual skater, the vector is drawn tangent to the appropriate circle and, therefore, is called the tangential velocity v. T. The magnitude of the tangential velocity is referred to as the tangential speed. This was the speed we used for uniform circular motion. 29

If time is measured relative to t 0 = 0 s, the definition of linear acceleration is given by Equation 2. 4 as a. T = (v. T – v. T 0)/t, where v. T and v. T 0 are the final and initial tangential speeds, respectively. 30

Example 6. A Helicopter Blade A helicopter blade has an angular speed of w = 6. 50 rev/s and an angular acceleration of a = 1. 30 rev/s 2. For points 1 and 2 on the blade, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations. 31

(a) (b) 32

Centripetal Acceleration and Tangential Acceleration (centripetal acceleration) 33

The centripetal acceleration can be expressed in terms of the angular speed w by using v. T = rw While the tangential speed is changing, the motion is called nonuniform circular motion. Since the direction and the magnitude of the tangential velocity are both changing, the airplane experiences two acceleration components simultaneously. a. T a. C 34

Check Your Understanding 3 The blade of a lawn mower is rotating at an angular speed of 17 rev/s. The tangential speed of the outer edge of the blade is 32 m/s. What is the radius of the blade? 0. 30 m 35

Example 7. A Discus Thrower Discus throwers often warm up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. A top view of such a warm -up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of +15. 0 rad/s in a time of 0. 270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0. 810 m. 36

Find (a) the magnitude a of the total acceleration of the discus just before it is released and (b) the angle f that the total acceleration makes with the radius at this moment. (a) 37

(b) 38

Check Your Understanding 4 A rotating object starts from rest and has a constant angular acceleration. Three seconds later the centripetal acceleration of a point on the object has a magnitude of 2. 0 m/s 2. What is the magnitude of the centripetal acceleration of this point six seconds after the motion begins? 39

after six second, (=0) 40

at six second 8. 0 m/s 2 41