1. Ch8. 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)

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

3. (in radians)
For 1 full rotation,

4. Example 1. Adjacent Synchronous Satellites
Synchronous satellites are put into an orbit whose radius is r = 4.23*107m. The orbit is in the plane of the equator, and two adjacent satellites have an angular separation of Find the arc length s.

6. Conceptual example 2. A Total Eclipse of the Sun

7. 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.

8. Since they are very far apart.

10. 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.

11. 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.

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

13. Example 3. Gymnast on a High Bar
A gymnast on a high bar swings through two revolutions in a time of 1.90s. Find the average angular velocity (in rad/s) of the gymnast.

15. 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.

16. DEFINITION OF AVERAGE ANGULAR ACCELERATION
SI Unit of Average Angular Acceleration: radian per second squared (rad/s2)
The instantaneous angular acceleration a is the angular acceleration at a given instant.

17. 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…..
                                                                                                                                

18. The Equations of Rotational Kinematics

19. In example 4, assume that the orientation of the rotating object is given by q 0 = 0 rad at time t0 = 0 s. Then, the angular displacement becomes Dq  = q  – q 0 = q , and the time interval becomes Dt = t – t0 = t.

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

21. q Displacement x w0 Initial velocity v0 w Final velocity v a
Symbols Used in Rotational and Linear Kinematics
Rotational Motion 
Quantity 
LinearMotion 
 q  
Displacement x
w0 
Initial velocity 
v0  w
Final velocity  v a 
Acceleration  a t
Time 

22. 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 rad (seven revolutions). The angular acceleration has a constant value of rad/s2. Find the final angular velocity of the blades.

23.  q  
 a   w
 w0   t
   rad
   rad/s2  ?
 +375  rad/s   
                                                                                                                                                                                             

24. 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?

25. after eight revolution,
1.0 rev/s

26. 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 vT. The magnitude of the tangential velocity is referred to as the tangential speed. This was the speed we used for uniform circular motion.

27. If time is measured relative to t0 = 0 s, the definition of linear acceleration is given by Equation 2.4 as aT = (vT – vT0)/t, where vT and vT0 are the final and initial tangential speeds, respectively.

28. 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/s2. For points 1 and 2 on the blade, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations.

29. (a)
                                                                               
                                                                               
(b)

30. Centripetal Acceleration and Tangential Acceleration

31. The centripetal acceleration can be expressed in terms of the angular speed w by using vT = 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. aT aC

32. 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

33. 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 rad/s in a time of s before releasing it. During the acceleration, the discus moves on a circular arc of radius m.

34. 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)
                                                                                                                                                                                

35. (b)

36. 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/s2. What is the magnitude of the centripetal acceleration of this point six seconds after the motion begins?

37. (=0)
after six second,

38. at six second
8.0 m/s2