Fundamentals of Physics

1. Fundamentals of Physics Chapter 12 Rolling, Torque & Angular Momentum • Rolling • The Kinetic Energy of Rolling • Rolling as Pure Rotation • The Forces of Rolling • Friction & Rolling • Rolling Down a Ramp • The Yoyo • Torque Revisited • Angular Momentum • Newton’s 2nd Law in Angular Momentum Form • Angular Momentum of a System of Particles • Angular Momentum of a Rigid Body Rotating About a Fixed Axis • Conservation of Angular Momentum Review & Summary Questions Exercises & Problems Physics 2111 Fundamentals of Physics

2. Rolling without Slipping Units: (m) (radians) (m) Units: (m/s) (rad/s) (m) Physics 2111 Fundamentals of Physics

3. Rolling Physics 2111 Fundamentals of Physics

4. Example: Value of x for rolling without slipping? x = 0  no initial spin Rolling without slipping: Independent of m and F! x > 2/5 R  topspin Physics 2111 Fundamentals of Physics

5. The Forces of Rolling Rolling with slipping initially No Initial Rotation: Topspin: Frictional force accelerates the ball in the direction of motion: increasing v decreasing w Frictional force reduces the speed & increases the angular speed until: Rolling without slipping Physics 2111 Fundamentals of Physics

6. Translation & Rotation pure rotation + pure translation = rolling motion Add: Linear velocity due to rotation Linear velocity of the center-of-mass To get: Linear velocity of each point on a rolling wheel. See p247 in HRW. Physics 2111 Fundamentals of Physics

7. The Kinetic Energy of Rolling A rolling object has two types of kinetic energy: a rotational KE due to its rotation about its center of mass and a translational KE due to the translation of its center of mass. The instantaneous axis of rotation is at Point P: Physics 2111 Fundamentals of Physics

8. The Forces of Rolling fs acts on the wheel at P, opposing its tending to slide. Wheel is rotating about point P. No work is done by fs! The wheel does not slow down. Friction & Rolling: friction is required! No slipping (aka “smooth rolling”) Physics 2111 Fundamentals of Physics

9. Rolling down a Ramp a> 0 counter clockwise a< 0 down the ramp Any body rolling down a ramp! Physics 2111 Fundamentals of Physics

10. Uniform Ball Rolling Down M = 6.00 kg Radius R q = 300 h = 1.20 m v = ? fs = ? Conservation of energy: Physics 2111 Fundamentals of Physics

11. Hoop, Disk and Sphere rolling down a ramp: Physics 2111 Fundamentals of Physics

12. Wheel has constant acceleration & no slipping fS = ? I = ? Physics 2111 Fundamentals of Physics

13. Ball rolls down the hill x = ? x H = 6.0 m h = 2.0 m Energy: Projectile: Physics 2111 Fundamentals of Physics

14. Loop the Loop Energy: Newton: Physics 2111 Fundamentals of Physics

15. Torque “Revisited” Physics 2111 Fundamentals of Physics

16. Angular Momentum Physics 2111 Fundamentals of Physics

17. All of the particles have the same momentum. Largest Magnitude of Angular Momentum about O ??? 1 and 3 2 and 3 Negative Angular Momentum??? 5 Zero Angular Momentum??? Physics 2111 Fundamentals of Physics

18. Angular Momentum • Angular momentum = ? m = 2.0 kg r = 3.0 m v = 4.0 m/s F = 2.0 N out of the page b) Torque = ? out of the page Physics 2111 Fundamentals of Physics

19. Newton’s 2nd Law in Angular Momentum Form Time derivative of the angular momentum: The vector sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle. Physics 2111 Fundamentals of Physics

20. Checkpoint Largest Torque about Point O??? Positive Torque about Point O??? Zero Torque about Point O??? Physics 2111 Fundamentals of Physics

21. Angular Momentum of a System of Particles Internal torques sum to zero: The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum. Physics 2111 Fundamentals of Physics

22. Angular Momentum About a Fixed Axis Physics 2111 Fundamentals of Physics

23. A Rigid Body Rotating About a Fixed Axis w Angular momentum of a mass element wrt the origin: Component of angular momentum in z-direction: Physics 2111 Fundamentals of Physics

24. A Rigid Body Rotating About a Fixed Axis w Total angular momentum in z-direction: Physics 2111 Fundamentals of Physics

25. Checkpoint : same radius same mass same force Largest Angular Momentum about Center??? Newton’s 2nd Law: All the angular momenta are the same. Largest Angular Speed after a time interval??? Physics 2111 Fundamentals of Physics

26. Conservation of Angular Momentum If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system. The Spinning Volunteer: Spins faster! Physics 2111 Fundamentals of Physics

27. Conservation of Angular Momentum The Diver: If the component of the net external torque on a system along a certain axis is zero, then the component of angular momentum of the system along that axis remains constant, no matter what changes take place within the system. Physics 2111 Fundamentals of Physics

28. Conservation of Angular Momentum The system = stool + person + spinning wheel Physics 2111 Fundamentals of Physics

29. A train starts running on a wheel; w = ? Physics 2111 Fundamentals of Physics

30. The sun goes out and then shrinks! Sun goes out Shrinks size Earth Period ? The sun will become a white dwarf. Physics 2111 Fundamentals of Physics