9. Example 1 Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is 4.23 ×10 7 m. If the angular separation of the two satellites is 2.00 degrees, find the arc length that separates them.
10. 8.1.1. Over the course of a day (twenty-four hours), what is the angular displacement of the second hand of a wrist watch in radians? a) 1440 rad b) 2880 rad c) 4520 rad d) 9050 rad e) 543 000 rad
12. Next, there was “velocity” How do we describe the rate at which the angular displacement is changing?
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 of the gymnast.
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16. Then, there was “acceleration” Changing angular velocity means that an angular acceleration is occurring. DEFINITION OF AVERAGE ANGULAR ACCELERATION SI Unit of Angular acceleration: radian per second per second (rad/s 2 )
17. Example 4 A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.
18. 8.2.1. The planet Mercury takes only 88 Earth days to orbit the Sun. The orbit is nearly circular, so for this exercise, assume that it is. What is the angular velocity, in radians per second, of Mercury in its orbit around the Sun? a) 8.3 × 10 7 rad/s b) 2.0 × 10 5 rad/s c) 7.3 × 10 4 rad/s d) 7.1 × 10 2 rad/s e) This cannot be determined without knowing the radius of the orbit.
19. 8.2.2. Complete the following statement: For a wheel that turns with constant angular speed, a) each point on its rim moves with constant acceleration. b) the wheel turns through “equal angles in equal times.” c) each point on the rim moves at a constant velocity. d) the angular displacement of a point on the rim is constant. e) all points on the wheel are moving at a constant velocity.
21. Making Sense of it All Five kinematic variables displacement initial velocity final velocity elapsed time acceleration x v o v t a o t
22. The equations of rotational kinematics for constant angular acceleration:
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24. Example 5 Blending with a Blender The blades are whirling with an angular velocity of +375 rad/s when the “puree” button is pushed in. When the “blend” button is pushed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44.0 rad. The angular acceleration has a constant value of +1740 rad/s 2 . Find the final angular velocity of the blades.
25. +375 rad/s ? +1740 rad/s 2 +44.0 rad t ω o ω α θ
26. 8.3.1. The propeller of an airplane is at rest when the pilot starts the engine; and its angular acceleration is a constant value. Two seconds later, the propeller is rotating at 10 rad/s. Through how many revolutions has the propeller rotated through during the first two seconds? a) 300 b) 50 c) 20 d) 10 e) 5
27. 8.3.2. A ball is spinning about an axis that passes through its center with a constant angular acceleration of rad/s 2 . During a time interval from t 1 to t 2 , the angular displacement of the ball is radians. At time t 2 , the angular velocity of the ball is 2 rad/s. What is the ball’s angular velocity at time t 1 ? a) 6.28 rad/s b) 3.14 rad/s c) 2.22 rad/s d) 1.00 rad/s e) zero rad/s
31. Example 6 A Helicopter Blade A helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s 2 . For point 1 on the blade, find the magnitude of (a) the tangential speed and (b) the tangential acceleration.
32. 8.4.1. The Earth, which has an equatorial radius of 6380 km, makes one revolution on its axis every 23.93 hours. What is the tangential speed of Nairobi, Kenya, a city near the equator? a) 37.0 m/s b) 74.0 m/s c) 148 m/s d) 232 m/s e) 465 m/s
33. 8.4.2. The original Ferris wheel had a radius of 38 m and completed a full revolution (2 radians) every two minutes when operating at its maximum speed. If the wheel were uniformly slowed from its maximum speed to a stop in 35 seconds, what would be the magnitude of the instantaneous tangential speed at the outer rim of the wheel 15 seconds after it begins its deceleration? a) 0.295 m/s b) 1.12 m/s c) 1.50 m/s d) 1.77 m/s e) 2.03 m/s
34. 8.4.3. A long, thin rod of length 4 L rotates counterclockwise with constant angular acceleration around an axis that is perpendicular to the rod and passes through a pivot point that is a length L from one end as shown. What is the ratio of the tangential acceleration at a point on the end closest to the pivot point to that at a point on the end farthest from the pivot point? a) 4 b) 3 c) 1/2 d) 1/3 e) 1/4
35. 8.4.4. A long, thin rod of length 4 L rotates counterclockwise with constant angular acceleration around an axis that is perpendicular to the rod and passes through a pivot point that is a length L from one end as shown. What is the ratio of the tangential speed (at any instant) at a point on the end closest to the pivot point to that at a point on the end farthest from the pivot point? a) 1/4 b) 1/3 c) 1/2 d) 3 e) 4
39. Example 7 A Discus Thrower Starting from rest, the thrower accelerates the discus to a final angular speed of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves in a circular arc of radius 0.810 m. Find the total acceleration.
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41. 8.5.1. An airplane starts from rest at the end of a runway and begins accelerating. The tires of the plane are rotating with an angular velocity that is uniformly increasing with time. On one of the tires, Point A is located on the part of the tire in contact with the runway surface and point B is located halfway between Point A and the axis of rotation. Which one of the following statements is true concerning this situation? a) Both points have the same tangential acceleration. b) Both points have the same centripetal acceleration. c) Both points have the same instantaneous angular velocity. d) The angular velocity at point A is greater than that of point B. e) Each second, point A turns through a greater angle than point B.
42. 8.5.2. A wheel starts from rest and rotates with a constant angular acceleration. What is the ratio of the instantaneous tangential acceleration at point A located a distance 2 r to that at point B located at r, where the radius of the wheel is R = 2 r ? a) 0.25 b) 0.50 c) 1.0 d) 2.0 e) 4.0
45. The tangential speed of a point on the outer edge of the tire is equal to the speed of the car over the ground.
46. Example 8 An Accelerating Car Starting from rest, the car accelerates for 20.0 s with a constant linear acceleration of 0.800 m/s 2 . The radius of the tires is 0.330 m. What is the angle through which each wheel has rotated?
48. 8.6.1. The wheels of a bicycle have a radius of r meters. The bicycle is traveling along a level road at a constant speed v m/s. Which one of the following expressions may be used to determine the angular speed, in rev/min, of the wheels? a) b) c) d) e)
49. 8.6.2. Josh is painting yellow stripes on a road using a paint roller. To roll the paint roller along the road, Josh applies a force of 15 N at an angle of 45 with respect to the road. The mass of the roller is 2.5 kg; and its radius is 4.0 cm. Ignoring the mass of the handle of the roller, what is the magnitude of the tangential acceleration of the roller? a) 4.2 m/s 2 b) 6.0 m/s 2 c) 15 m/s 2 d) 110 m/s 2 e) 150 m/s 2
50. 8.6.3. Which one of the following statements concerning a wheel undergoing rolling motion is true? a) The angular acceleration of the wheel must be zero m/s 2 . b) The tangential velocity is the same for all points on the wheel. c) The linear velocity for all points on the rim of the wheel is non-zero. d) The tangential velocity is the same for all points on the rim of the wheel. e) There is no slipping at the point where the wheel touches the surface on which it is rolling.
51. 8.6.4. A circular hoop rolls without slipping on a flat horizontal surface. Which one of the following is necessarily true? a) All points on the rim of the hoop have the same speed. b) All points on the rim of the hoop have the same velocity. c) Every point on the rim of the wheel has a different velocity. d) All points on the rim of the hoop have acceleration vectors that are tangent to the hoop. e) All points on the rim of the hoop have acceleration vectors that point toward the center of the hoop.
52. 8.6.5. A bicycle wheel of radius 0.70 m is turning at an angular speed of 6.3 rad/s as it rolls on a horizontal surface without slipping. What is the linear speed of the wheel? a) 1.4 m/s b) 28 m/s c) 0.11 m/s d) 4.4 m/s e) 9.1 m/s
55. 8.7.1. A packaged roll of paper towels falls from a shelf in a grocery store and rolls due south without slipping. What is the direction of the paper towels’ angular velocity? a) north b) east c) south d) west e) down
56. 8.7.2. A packaged roll of paper towels falls from a shelf in a grocery store and rolls due south without slipping. As its linear speed slows, what are the directions of the paper towels’ angular velocity and angular acceleration? a) east, east b) west, east c) south, north d) east, west e) west, west
57. 8.7.3. A top is spinning counterclockwise and moving toward the right with a linear velocity as shown in the drawing. If the angular speed is decreasing as time passes, what is the direction of the angular velocity of the top? a) upward b) downward c) left d) right
58. 8.7.4. A truck and trailer have 18 wheels. If the direction of the angular velocity vectors of the 18 wheels point 30 north of east, in what direction is the truck traveling? a) 30° east of south b) 30° west of north c) 30° north of east d) 30° south of west e) 30° south of east
59. 8.7.5. A girl is sitting on the edge of a merry-go-round at a playground as shown. Looking down from above, the merry-go-round is rotating clockwise. What is the direction of the girl’s angular velocity? a) upward b) downward c) left d) right e) There is no direction since it is the merry go round that has the angular velocity.
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