1. Copyright © 2010 Pearson Education, Inc. Lecture Outline Chapter 10 Physics, 4 th Edition James S. Walker

2. Copyright © 2010 Pearson Education, Inc. Chapter 10 Rotational Kinematics and Energy

3. Copyright © 2010 Pearson Education, Inc. 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

4. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration

5. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions:

6. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Arc length s, measured in radians:

7. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration

8. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration

9. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration

10. Copyright © 2010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration

11. Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics If the angular acceleration is constant:

12. Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics Analogies between linear and rotational kinematics:

13. Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics To throw a curve ball, a pitcher gives the ball an initial angular speed of 36.0 rad/s. When the catcher gloves the ball 0.595 s later, its angular speed has decreased (due to air resistance) to 34.2 rad /s. (a) What is the ball’s angular acceleration, assuming it to be constant? (b) How many revolutions does the ball make before being caught? ω = ω o + αt  α = (ω - ωo) / t α = (34.2 rad / s - 36.0 rad /s) / 0.595 s = -3.03 rad /s 2 (b). Calculate the angular displacement θ - θ o = ω o + ½ α t 2 = (36.0 rad /s)(0.595 s) + ½( -3.03 rad/ s 2 ) (0.595 s) 2 = 20.9 rad Convert the angular displacement to revolutions : 20.9 rad = 20.9 rad ( 1 rev / 2π rad) = 3.33 rev

14. Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics On a certain show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.40 rad / s. it then rotates through one – one – quarter revolutions and comes to rest on the BANKRUPT space. (a) Find the angular acceleration of the wheel, assuming it to be constant. (b). How long does it take for the wheel to come to rest? ω 2 = ω o 2 + 2α (θ - θ o )  α = (ω 2 - ω o 2 ) / 2 (θ - θ o ) θ - θ o = 1.25 rev = 1.25 rev (2π rad / 1 rev) = 7.85 rad α = (ω 2 - ω o 2 ) / 2 (θ - θ o ) = 0 – (3.40 rad/s) 2 / 2 (7.85 rad) = - 0.736 rad /s 2 ω = ω o + αt ==  t = (ω - ωo) / α ==== 0 – 3.40 rad / s) / ( - 0.736 rad / s 2 ) = 4.62 s

15. Copyright © 2010 Pearson Education, Inc. 10-3 Connections Between Linear and Rotational Quantities

16. Copyright © 2010 Pearson Education, Inc. 10-3 Connections Between Linear and Rotational Quantities

17. Copyright © 2010 Pearson Education, Inc. 10-3 Connections Between Linear and Rotational Quantities

18. Copyright © 2010 Pearson Education, Inc. 10-3 Connections Between Linear and Rotational Quantities This merry-go-round has both tangential and centripetal acceleration.

19. Copyright © 2010 Pearson Education, Inc. 10-4 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

20. Copyright © 2010 Pearson Education, Inc. 10-4 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion:

21. Copyright © 2010 Pearson Education, Inc. 10-5 Rotational Kinetic Energy and the Moment of Inertia For this mass,

22. Copyright © 2010 Pearson Education, Inc. 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

23. Copyright © 2010 Pearson Education, Inc. 10-5 Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular objects can be calculated:

24. Copyright © 2010 Pearson Education, Inc. 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.

25. Copyright © 2010 Pearson Education, Inc. 10-6 Conservation of Energy If these two objects, of the same mass and radius, are released simultaneously, the disk will reach the bottom first – more of its gravitational potential energy becomes translational kinetic energy, and less rotational.

26. Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 Describing rotational motion requires analogs to position, velocity, and acceleration Average and instantaneous angular velocity: Average and instantaneous angular acceleration:

27. Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 Period: Counterclockwise rotations are positive, clockwise negative Linear and angular quantities:

28. Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 Linear and angular equations of motion: Tangential speed: Centripetal acceleration: Tangential acceleration:

29. Copyright © 2010 Pearson Education, Inc. 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.