Classical Mechanics Lecture 16

Transcription

1 Classical Mechanics Lecture 16 Today s Concepts: a) Rolling Kine6c Energy b) Angular Accelera6on Physics 211 Lecture 16, Slide 1

2 Thoughts from the Present & Past Can we do that cookie 6n demo again, but with math this 6me? Despite me taking notes on the pre-lecture, I do not feel I understand anything that was taught within it. The checkpoint ques6ons were simple enough, but most of my reasoning had to do with moment of iner6a rather than any knowledge of what was taught in the pre-lecture. It seems weird that the ball would roll slower than a block sliding down a ramp. My intuition says the ball would go faster, but alas... What unmerciful doom doth rot upon the error of my soul? What does the center of mass have to do with this sec6on? A liple brush up on last lectures material would help, just to reinforce what has been taught moments of iner6a s6ll throw me it takes me a couple tries to think through them and with really rota6on period I am finding it more challenging despite the fact I had trouble learning it last year as well just stop teaching these things plz - It seems weird that the ball would roll slower than a block sliding down a ramp. My intui6on says the ball would go faster, but alas... Started off terrible, but now that we've got the ball rolling I'm star6ng to get it. Mechanics Lecture 16, Slide 2

3 I felt like every slide just had a ton of equa6ons just being used to find new equa6ons. Other than that the actual use of the equa6ons doesn't seem so bad, but I s6ll don't really know what to do for the CheckPoints. Energy Conserva6on H α f a F = ma Mg Mechanics Lecture 16, Slide 4

4 Let s work through this again. Mechanics Lecture 16, Slide 5

5 Rolling Motion Objects of different I rolling down an inclined plane: R M v = 0 ω = 0 K = 0 Δ K = ΔU = M g h h v = ωr Mechanics Lecture 16, Slide 6

6 Rolling If there is no slipping: v = ωr v ω v Where v = ωr In the lab reference frame In the CM reference frame Mechanics Lecture 16, Slide 7

7 Rolling v = ωr c Use v = ωr and I = cmr 2. c Hoop: c = 1 Disk: c = 1/2 Sphere: c = 2/5 etc... So: c c Doesn t depend on M or R, just on c (the shape) Ramp demo Mechanics Lecture 16, Slide 8

8 Clicker Question A hula-hoop rolls along the floor without slipping. What is the ra6o of its rota6onal kine6c energy to its transla6onal kine6c energy? A) B) C) Recall that I = MR 2 for a hoop about an axis through its CM. Mechanics Lecture 16, Slide 9

9 CheckPoint A block and a ball have the same mass and move with the same ini6al velocity across a floor and then encounter iden6cal ramps. The block slides without fric6on and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Both reach the same height. v ω v Mechanics Lecture 16, Slide 10

10 CheckPoint The block slides without fric6on and the ball rolls without slipping. Which one makes it furthest up the ramp? v A) Block B) Ball C) Same ω v A) The ball losses energy to rota6on B) The ball has more total kine6c energy since it also has rota6onal kine6c energy. Therefore, it makes it higher up the ramp. C) If they have the same velocity then they should go the same height. The rota6onal energy should not affect the ball. Mechanics Lecture 16, Slide 11

11 CheckPoint A cylinder and a hoop have the same mass and radius. They are released at the same 6me and roll down a ramp without slipping. Which one reaches the bopom first? A) Cylinder B) Hoop C) Both reach the bopom at the same 6me Mechanics Lecture 16, Slide 12

12 Which one reaches the bottom first? A) Cylinder B) Hoop C) Both reach the bopom at the same 6me A) same PE but the hoop has a larger rota6onal iner6a so more energy will turn into rota6onal kine6c energy, thus cylinder reaches it first. B) It has more kine6c energy because it has a bigger moment of iner6a. C) Both have the same PE so they will have the same KE and reach the ground at the same 6me. Mechanics Lecture 16, Slide 13

13 CheckPoint A small light cylinder and a large heavy cylinder are released at the same 6me and roll down a ramp without slipping. Which one reaches the bopom first? A) Small cylinder B) Large cylinder C) Both reach the bopom at the same 6me Mechanics Lecture 16, Slide 14

14 CheckPoint A small light cylinder and a large heavy cylinder are released at the same 6me and roll down a ramp without slipping. Which one reaches the bopom first? A) Small cylinder B) Large cylinder C) Both reach the bopom at the same 6me A) Because the smaller one has a smaller moment of iner6a. B) The large cylinder has a larger moment of iner6a, and therefore, ends up with more energy. C) The mass is canceled out in the velocity equa6on and they are the same shape so they move at the same speed. Therefore, they reach the bopom at the same 6me. Mechanics Lecture 16, Slide 15

15 What you saw in your Prelecture: Accelera6on depends only on the shape, not on mass or radius. Mechanics Lecture 16, Slide 16

16 Clicker Question A small light cylinder and a large heavy cylinder are released at the same 6me and slide down the ramp without fric6on. Which one reaches the bopom first? A) Small cylinder B) Large cylinder C) Both reach the bopom at the same 6me Mechanics Lecture 16, Slide 17

17 f M R α v Mechanics Lecture 16, Slide 18

18 a f M R v Mechanics Lecture 16, Slide 19

19 a M R ω v v v 0 Once v=ωr it rolls without slipping ω t ω = αt v 0 ωr = α Rt t t Mechanics Lecture 16, Slide 20

20 a M R ω v Plug in a and t found in parts 2) & 3) Mechanics Lecture 16, Slide 21

21 ω a M R v Interesting aside: how v is related to v 0 : v = v 0 µg 2v! 0 7µg Doesn t depend on µ We can try this Mechanics Lecture 16, Slide 22

22 Position vs time for Bowling Ball Quadra6c curve fit to region 8.66 s to 9.05 s: v 0 = 2.67 m/s Linear fit to region 9.1 s to 10.3 s v = 1.78 m/s

23 Linear velocity vs time throwing slipping rolling the velocity vs 6me graph shows that clearly v is nearly (5/7) v 0

24 f M R α v = 1 2! 2 v 2 5 MR2 R Mechanics Lecture 16, Slide 23

25 Clicker Question Suppose a cylinder (radius R, mass M) is used as a pulley. Two masses (m 1 > m 2 ) are apached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. Compare the magnitudes of the accelera6on of the two masses: α M R A) a 1 > a 2 B) a 1 = a 2 C) a 1 < a 2 T 1 T 2 m 1 m 2 a 2 a 1 Mechanics Lecture 16, Slide 24

26 Clicker Question Suppose a cylinder (radius R, mass M) is used as a pulley. Two masses (m 1 > m 2 ) are apached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. How is the angular accelera6on of the wheel related to the linear accelera6on of the masses? A) α = Ra B) α = a/r C) α = R/a M α R T 1 T 2 m 1 m 2 a 2 a 1 Mechanics Lecture 16, Slide 25

27 Clicker Question Suppose a cylinder (radius R, mass M) is used as a pulley. Two masses (m 1 > m 2 ) are apached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. Compare the tension in the string on either side of the pulley: α M R A) T 1 > T 2 B) T 1 = T 2 C) T 1 < T 2 T 1 T 2 a 2 m 1 m 2 a 1 Mechanics Lecture 16, Slide 26

28 Atwood's Machine with Massive Pulley: A pair of masses are hung over a massive disk-shaped pulley as shown.! Find the accelera6on of the blocks. y x For the hanging masses use F = ma m 1 g + T 1 = m 1 a α M R m 2 g + T 2 = m 2 a For the pulley use τ = Iα T 1 T 2 a T 1 R T 2 R (Since for a disk) m m 2 1 a m 2 g m 1 g Mechanics Lecture 16, Slide 27

29 Atwood's Machine with Massive Pulley: We have three equa6ons and three unknowns (T 1, T 2, a). Solve for a. y m 1 g + T 1 = m 1 a (1) m 2 g + T 2 = m 2 a (2) α M R x T 1 T 2 (3) T 1 T 2 a a =! m 1 m 2 g m 1 + m 2 + M/2 m m 2 1 a m 2 g m 1 g Mechanics Lecture 16, Slide 28

30 Solu6on coming later... Yo-Yo

31 direction at a distance of 53 cm from the pivot. What is the magnitude of the force he must exert? UNIT 12 HOMEWORK AFTER SESSION TWO Id = SmartPhysics: View the Prelecture 16 Rotational Dynamics before the class of next session and answer the checkpoint questions. (The online Homework is due the next day.) For the hole in the centr rh = rubber washer maybe not Ih = needed. Work supplemental problem SP12-4 (Problem SP12-4 will help you prepare for the experiment to be done during the next class session.) metal washer bolt SP12-4) (20 pts) A small spool of radius rs and a large Lucite disk of radius rd are connected by an axle that is free to rotate in an almost frictionless manner inside of a bearing as shown in the diagram below. CD stack rubber washer use 12 LP rotational motion! sensor ICD = Id Ih = (b) Calculate the theor the spool using basic m Note: You'll have to do state units. rs = Is = This problem is similar to the previous but m2 is zero and the radii of the spool and disk are different: Rd and Rs. (c) Calculate the theoret the whole system. Don't noting how small the rot to that of the disk, you s you can neglect the rotat calculations. Assembled Apparatus I= (c) In preparation for cal summarize the measure radius of the spool in the Dept. of Physics and Astronomy, Dickinson Co and NSF. Modified for SFU by N. Alberding, 2005.