Classical Mechanics Lecture 16

1. Classical Mechanics Lecture 16 Today’s Concepts: a) Rolling Kinetic Energy b) Angular Acceleration

2. Schedule • One unit per lecture! • I will rely on you watching and understanding pre-lecture videos!!!! • Lectures will only contain summary, homework problems, clicker questions, Example exam problems…. Midterm 3 Wed Dec 11

3. Main Points

4. Main Points Rolling without slipping 

5. Rotational Kinetic Energy Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy Energy Conservation H Rolling without slipping 

6. Acceleration of Rolling Ball Newton’s Second Law Newton’s 2nd Law for rotations Rolling without slipping  a f a Mg

7. Rolling Motion Objects of differentIrolling down an inclined plane: v=0 =0 K=0 K= -U=Mgh R M h v=R

8. Rolling If there is no slipping: v  v Wherev = R v In the lab reference frame In the CM reference frame

9. Rolling c c c c Hoop: c= 1 Disk: c= 1/2 Sphere: c= 2/5 etc... Use v = R and I = cMR2. v So: Doesn’t depend on M or R, just on c (the shape)

10. Clicker Question A hula-hoop rolls along the floor without slipping. What is the ratio of its rotational kinetic energy to its translational kinetic energy? A) B) C) Recall that I = MR2for a hoop about an axis through its CM:

11. CheckPoint A) Block B) Ball C) Both reach the same height. v w v A block and a ball have the same mass and move with the same initial velocity across a floor and then encounter identical ramps. The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp?

12. CheckPoint The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Same • B) The ball has more total kinetic energy since it also has rotational kinetic energy. Therefore, it makes it higher up the ramp. v w v

13. Rolling vs Sliding Rolling Ball Sliding Block Rolling without slipping  Ball goes 40% higher!

14. CheckPoint A cylinder and a hoop have the same mass and radius. They are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Cylinder B) Hoop C) Both reach the bottom at the same time

15. Which one reaches the bottom first? A) Cylinder B) Hoop C) Both reach the bottom at the same time A) same PE but the hoop has a larger rotational inertia so more energy will turn into rotational kinetic energy, thus cylinder reaches it first.

16. CheckPoint A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time

17. A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? CheckPoint A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time C) The mass is canceled out in the velocity equation and they are the same shape so they move at the same speed. Therefore, they reach the bottom at the same time.

18. a v M R f

19. a v M R f

20. w v M a R v Oncev=wRit rollswithout slipping v0 wR= aRt v0 t w = at w t t

21. w v M a R Plug in a and t found in parts 2) & 3)

22. w v M a R Interesting aside: how v is related tov0 : Doesn’t depend on m We can try this…

23. a v M R f

24. Atwood's Machine with Massive Pulley: A pair of masses are hung over a massive disk-shaped pulley as shown. Find the acceleration of the blocks. y x For the pulley use =I m2g (Since for a disk) m1g M For the hanging masses useF = ma -m1g + T1=-m1a -m2g+T2=m2a  R T2 T1 a m2 m1 T1R - T2R a

25. We have three equations and three unknowns (T1, T2, a). Solve for a. -m1g + T1=-m1a (1) -m2g + T2= m2a (2) T1-T2(3) Atwood's Machine with Massive Pulley: y x m2g m1g M  R T2 T1 a m2 m1 a

26. Three Masses

27. Three Masses

28. Three Masses

29. Three Masses

30. Three Masses 2

31. Three Masses 2