Ch8. Rotational Kinematics Rotational Motion and Angular Displacement

1. Ch8. Rotational KinematicsRotational 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. 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. Check Your Understanding 1 Three objects are visible in the night sky. They have the following diameters (in multiples of d and subtend the following angles (in multiples of q 0) at the eye of the observer. Object A has a diameter of 4d and subtends an angle of 2q 0. Object B has a diameter of 3d and subtends an angle of q 0/2. Object C has a diameter of d/2 and subtends an angle of q 0/8. Rank them in descending order (greatest first) according to their distance from the observer.

12. B, C, A

13. 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.

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

15. 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.

17. Instantaneous angular velocityw is the angular velocity that exists at any given instant. The magnitude of the instantaneous angular velocity, without reference to whether it is a positive or negative quantity, is called the instantaneous angular speed. If a rotating object has a constant angular velocity, the instantaneous value and the average value are the same.

18. 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.

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

20. 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 averageangular velocity, assuming that the orientation of the rotating object is given by…..

21. The Equations of Rotational Kinematics

22. 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.

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

24. Symbols Used in Rotational and Linear Kinematics

25. 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/s2. Find the final angular velocity of the blades.

26. q  a w w0 t +44.0  rad +1740  rad/s2 ? +375  rad/s

27. 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?

28. after eight revolution, 1.0 rev/s

29. 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.

30. 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.

31. 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.

32. (b) (a)

33. Centripetal Acceleration and Tangential Acceleration (centripetal acceleration)

34. The centripetalacceleration 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 tangentialvelocity are both changing, the airplane experiences two acceleration components simultaneously. aT aC

35. 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

36. 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.

37. 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)

38. (b)

39. Check Your Understanding 4 A rotating object starts from rest and has a constant angular acceleration. Three seconds later the centripetalacceleration 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?

40. (=0) after six second,

41. at six second 8.0 m/s2

42. Rolling Motion

43. Linear speed Tangential speed, vT Linear acceleration Tangential acceleration, aT

44. Example 8.  An Accelerating Car An automobile starts from rest and for 20.0 s has a constant linear acceleration of 0.800 m/s2 to the right. During this period, the tires do not slip. The radius of the tires is 0.330 m. At the end of the 20.0-s interval, what is the angle through which each wheel has rotated?

45. w w0 t ?  –2.42  rad/s2 0  rad/s 20.0  s

46. The Vector Nature of Angular Variables Right-Hand Rule Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocityvector. Angular acceleration arises when the angular velocity changes, and the accelerationvector also points along the axis of rotation. The acceleration vector has the same direction as the change in the angular velocity.

47. Concepts & Calculations Example 9.  Riding a Mountain Bike A rider on a mountain bike is traveling to the left. Each wheel has an angular velocity of +21.7 rad/s, where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction.

48. To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from +21.7 to +28.5 rad/s in a time of 3.50 s. • After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from +28.5 to +15.3 rad/s in a time of 10.7 s. In both instances, determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels. (a) The angular acceleration is positive (counterclockwise).

49. (b) The angular acceleration is negative (clockwise).

50. Concepts & Calculations Example 10.  A Circular Roadway and the Acceleration of Your Car