Rotational Kinematics

1. Rotational Kinematics

2. Angular Position

3. θ > 0 θ < 0 Angular Position Degrees and revolutions:

4. 1 complete revolution = 2 π radians C = 2 π r C / D = π 1 rad = 360o / (2π) = 57.3o Arc Length Arc length s, from angle measured in radians: s = r θ - What is the relationship between the circumference of a circle and its diameter? - Arc length for a full rotation (360o) of a radius=1m circle? s = 2 π (1 m) = 2 π meters

5. Why use radians? • Doesn’t involve arbitrary choice of 360 degrees. There is another unit, the gon or gradian that is used in surveying: • Radians useful for small angles:

6. Angular Velocity

7. Period = How long it takes to go 1 full revolution Period T: SI unit: second, s Instantaneous Angular Velocity

8. Linear and Angular Velocity

9. Greater translation for same rotation

10. Bonnie and Klyde II a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?  Bonnie Klyde

11.  Bonnie and Klyde II Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity? a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them Their linear speeds v will be different because v = r  and Bonnie is located farther out (larger radius r) than Klyde. Klyde Bonnie

12. Angular Acceleration

13. Instantaneous Angular Acceleration

14. Rotational Kinematics, Constant Acceleration If the angular acceleration is constant: If the acceleration is constant: v = v0 + at

15. Analogies between linear and rotational kinematics:

16. Angular Displacement I a)  b)  c)  d) 2  e) 4  ½ ¼ An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle  in the time t, through what angle did it rotate in the time t? ¾ ½

17. Angular Displacement I a)  b)  c)  d) 2  e) 4  ½ ¼ An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle  in the time t, through what angle did it rotate in the time t ? ¾ ½ The angular displacement is  = t 2 (starting from rest), and there is a quadratic dependence on time. Therefore, in half the time, the object has rotated through one-quarter the angle.

18. Which child experiences a greater acceleration? (assume constant angular speed)

19. Larger r: - larger v for same ω - larger ac for same ω ac is required for circular motion. An object may have at as well, which implies angular acceleration

20. Angular acceleration and total linear acceleration

21. Angular and linear acceleration

22. Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

23. Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion: = +

25. b) c) This is the force that is responsible for keeping Jeff in circular motion: the vine. a) Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time. (b) What centripetal acceleration does Jeff experience at the bottom of his swing? (c) What exerts the force that is responsible for Jeff’s centripetal acceleration?