Rotational Kinematics and Energy: Understanding Angular Position, Velocity, and Acceleration

1. Chapter 10 Rotational Kinematics and Energy

2. Units of Chapter 10 • Angular Position, Velocity, and Acceleration • Rotational Kinematics • Connections Between Linear and Rotational Quantities • Rolling Motion • Rotational Kinetic Energy and the Moment of Inertia • Conservation of Energy

3. 10-1 Angular Position, Velocity, and Acceleration

4. 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions:

5. 10-1 Angular Position, Velocity, and Acceleration Arc length s, measured in radians:

6. 10-1 Angular Position, Velocity, and Acceleration

7. 10-1 Angular Position, Velocity, and Acceleration

8. 10-1 Angular Position, Velocity, and Acceleration

9. 10-1 Angular Position, Velocity, and Acceleration

10. 10-2 Rotational Kinematics Analogies between linear and rotational kinematics:

11. Angular Velocity 1) 1/2 w 2) 1/4 w 3) 3/4 w 4) 2 w 5) 4 w An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity w at time t, what was its angular velocity at the time 1/2 t?

12. Angular Velocity 1) 1/2 w 2) 1/4 w 3) 3/4 w 4) 2 w 5) 4 w An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity w at time t, what was its angular velocity at the time 1/2 t? The angular velocity is w = at (starting from rest), and there is a linear dependence on time. Therefore, in half the time, the object has accelerated up to only half the speed.

13. Example • The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. • What is the acceleration of the propeller? • How long did the change in angular speed take?

14. 10-3 Connections Between Linear and Rotational Quantities

15. Tangential Speed

16. 10-3 Connections Between Linear and Rotational Quantities

17. 10-3 Connections Between Linear and Rotational Quantities

18. 10-3 Connections Between Linear and Rotational Quantities This merry-go-round has both tangential and centripetal acceleration.

19. w Bonnie Klyde Bonnie and Klyde 1) Klyde 2) Bonnie 3) both the same 4) linear velocity is zero for both of them 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 two seconds. Who has the larger linear (tangential) velocity?

20. w Klyde Bonnie Bonnie and Klyde 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 two seconds. Who has the larger linear (tangential) velocity? 1) Klyde 2) Bonnie 3) both the same 4) linear velocity is zero for both of them Their linear speedsv will be different since v = Rw and Bonnie is located further out (larger radius R) than Klyde. Follow-up: Who has the larger centripetal acceleration?

21. 10-4 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

22. 10-4 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion:

23. 10-5 Rotational Kinetic Energy and the Moment of Inertia For this mass,

24. 10-5 Rotational Kinetic Energy and the Moment of Inertia We can also write the kinetic energy as Where I, the moment of inertia, is given by

25. 10-5 Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular objects can be calculated:

26. 10-6 Conservation of Energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.

27. 10-6 ConcepTest If these two objects, of the same mass and radius, are released simultaneously, which will reach the ground first? • Hoop • Disc

28. 10-6 ConcepTest If these two objects, of the same mass and radius, are released simultaneously, which will reach the ground first? • Hoop • Disc

29. Example – Real Atwood’s Machine • The two masses (m1 = 5.0 kg and m2 = 3.0 kg) in the Atwood’s machine shown in the figure are released from rest, with m1 at a height of 0.75 m above the floor. When m1 hits the ground its speed is 1.8 m/s. Assuming that the pulley is a uniform disc with a radius of 12 cm, find the mass of the pulley. Assume the rope does not slip on the pulley.

30. Summary of Chapter 10 • Describing rotational motion requires analogs to position, velocity, and acceleration • Average and instantaneous angular velocity: • Average and instantaneous angular acceleration:

31. Summary of Chapter 10 • Period: • Counterclockwise rotations are positive, clockwise negative • Linear and angular quantities:

32. Summary of Chapter 10 • Linear and angular equations of motion: Tangential speed: Centripetal acceleration: Tangential acceleration:

33. Summary of Chapter 10 • Rolling motion: • Kinetic energy of rotation: • Moment of inertia: • Kinetic energy of an object rolling without slipping: • When solving problems involving conservation of energy, both the rotational and linear kinetic energy must be taken into account.