1. Rotational Kinematics

2.  Everything that we have done in linear kinematics we will now do for objects rotating

3.  This is called rotational kinematics

4.  Everything that we have done in linear kinematics we will now do for objects rotating  This is called rotational kinematics

5. What we know…..  In linear kinematics, we had measurements and variables which we used to describe the motion of an object

6. What we know…..  In linear kinematics, we had measurements and variables which we used to describe the motion of an object  We need to ‘redefine’ and ‘rename’ these since now they will be for rotational kinematics

7. Linear Kinematics

8. Measurement Variable Unit  Time t s  Displacement ∆x, ∆y m  Velocity v i, v f m/s  Acceleration a m/s 2

9. Rotational Kinematics Measurement Variable Unit  Time t s  Displacement θ (theta) rad (radians)  Velocity ω (omega) rad/s  Acceleration α (alpha) rad/s 2

10. Brand new…but not really  Everything we do in this chapter will be what we have done before, except the names change  All the concepts stay the same

11. Displacement - θ

12.  Measured in radians

13. Displacement - θ  Measured in radians

14. Displacement - θ  Measured in radians  Just like linear, we have to include direction counterclockwise is positive clockwise is negative

15. Displacement - θ  Measured in radians  Just like linear, we have to include direction counterclockwise is positive clockwise is negative  One revolution is then 2 Π radians (also known as 360 o ) 1 rev = 360 degrees = 2 Π rad

16. Displacement - θ  θ (in radians) = Arc length / Radius

17. Displacement - θ  θ (in radians) = Arc length / Radius = s / r think of s as x s θ r

18. Velocity - ω  Measured in rad/s

19. Velocity - ω  Measured in rad/s  Angular velocity = angular displacement / time

20. Velocity - ω  Measured in rad/s  Angular velocity = angular displacement / time  ω = θ / t  Remember this is for NO acceleration, or constant angular velocity just like v = ∆ x/t

21. Acceleration - α  Measured in rad/s 2

22. Acceleration - α  Measured in rad/s 2  Angular acceleration = change in angular velocity / time

23. Acceleration - α  Measured in rad/s 2  Angular acceleration = change in angular velocity / time  α = ∆ω / t

24. Acceleration - α  Measured in rad/s 2  Angular acceleration = change in angular velocity / time  α = ∆ω / t this is our definition of acceleration just like a = ∆v/t

25. Kinematic Equations…..  Remember them….

26. Kinematic Equations…..  Remember them…. v f = v i + at

27. Kinematic Equations…..  Remember them…. v f = v i + at ∆x = v i t + ½ at 2

28. Kinematic Equations…..  Remember them…. v f = v i + at ∆x = v i t + ½ at 2 vf 2 = vi 2 + 2a(∆x)

29. …Transform…… Autobots.....

30. Rotational Kinematic Equations ω f = ω i + αt θ = ω i t + ½ αt 2 ω f 2 = ω i 2 + 2 α θ θ = ½( ω f + ω i ) t

31. Rotational Kinematic Equations  We can now do kinematic problems like we used to, except for the variable change  To succeed here, you need to remember our new variables and terms  An easy way to do this is by units ex. You know velocity (v) is m/s, so remember angular velocity ( ω) is rad/s  This is one of the few things you can memorize in this class and be successful

32. minus chain or

33. minus chain or LINK!!!!

34. The Link Up  There is a way to link rotational and linear kinematics

35. The Link Up  There is a way to link rotational and linear kinematics v tan = r ω

36. The Link Up  There is a way to link rotational and linear kinematics v tan = r ω  This equation uses the angular velocity and ‘converts’ it to linear velocity

37. The Link Up

38. What about Acceleration?  Not used as often, but there is: a tan = r α

39. Hammer Time

41. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites.

43. Determine: S

44. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o

45. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o Formula: θ = s/r

46. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o Formula: θ = s/r 20 o must be converted to radians 20 o = 2 π radians / 360 o

47. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o Formula: θ = s/r 20 o must be converted to radians 20 o = 2 π radians / 360 o =.349 radians

48. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o Formula: θ = s/r 20 o must be converted to radians 20 o = 2 π radians / 360 o =.349 radians So θ = s/r.349 = s / 4.23 x 10 7

49. Stationary communications satellites are put into an orbit whose radius is r = 4.23 x 10 7 m. The orbit is in the plane of the equator and two adjacent satellites have an angular separation of 20 degrees. Find the arc length, s, that separates the satellites. Determine: S Given: r = 4.23 x 10 7 m, 20 o Formula: θ = s/r 20 o must be converted to radians 20 o = 2 π radians / 360 o =.349 radians So θ = s/r.349 = s / 4.23 x 10 7 s = 1.48 x 10 7 m

50. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast.

51. Determine: ω

52. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds

53. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds θ = 2 revolutions t = 1.9 s

54. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds θ = 2 revolutions = 2 x 2 π t = 1.9 s

55. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds θ = 2 revolutions = 2 x 2 π t = 1.9 s So……. ω = θ / t

56. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds θ = 2 revolutions = 2 x 2 π t = 1.9 s So……. ω = θ / t = 2(2 π) / 1.9

57. A gymnast on a high bar swings through two revolutions in a time of 1.9 seconds. Find the angular velocity of the gymnast. Determine: ω Given: 2 revolutions, 1.9 seconds θ = 2 revolutions = 2 x 2 π t = 1.9 s So……. ω = θ / t = 2(2 π) / 1.9 = 6.63 rad/s

58. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades.

59. Given:

60. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise)

61. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s

62. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s t = 14 seconds

63. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s t = 14 seconds Determine: α

64. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s t = 14 seconds Determine: α So…… α = Δ ω / t or ω f = ω i + αt

65. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s t = 14 seconds Determine: α So…… α = Δ ω / t or ω f = ω i + αt = (-330 – -110)/14

66. 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 clockwise with an angular velocity of 110 rad/s. As the plane takes off, the angular velocity of the blades reaches 330 rad/s in a time of 14 seconds. Find the angular acceleration of the blades. Given: ω i = -110 rad/s (negative since clockwise) ω f = -330 rad/s t = 14 seconds Determine: α So…… α = Δ ω / t or ω f = ω i + αt = (-330 – -110)/14 = -16 rad/s 2

67. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades.

68. Givens:

69. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+)

70. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π )

71. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π ) α = 1740 rad/s 2

72. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π ) α = 1740 rad/s 2 Determine: ω f

73. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π ) α = 1740 rad/s 2 Determine: ω f Equation: ω f 2 = ω i 2 + 2 α θ

74. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π ) α = 1740 rad/s 2 Determine: ω f Equation: ω f 2 = ω i 2 + 2 α θ ω f 2 = 375 2 + 2 (1740) (44)

75. The blades of an electric blender are whirling with an angular velocity of 375 rad/s counter clockwise while 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 7 rotations. The angular acceleration has a value of 1740 rad/s 2. Find the final angular velocity of the blades. Givens: ω i = 375 rad/s (+) θ = 7 rotations = 44 rad (7x2 π ) α = 1740 rad/s 2 Determine: ω f Equation: ω f 2 = ω i 2 + 2 α θ ω f 2 = 375 2 + 2 (1740) (44) ω f = 542 rad/s