Classical Mechanics Lecture 11

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1 Classical Mechanics Lecture 11 Today s Examples Center of Mass Today s Concept: Conservation of Momentum Inelastic Collisions Mechanics Lecture 11, Slide 1

2 Unit 10 Homework Problems Mechanics Lecture 10, Slide 2

3 Homework Problem i x N i i Total cm x v m M v, 1, 1 = = = = N i M Total m i 1 i y N i i Total cm y v m M v, 1, 1 = = Mechanics Lecture 10, Slide 3

4 m 5 CM moves toward m 5 cm Mechanics Lecture 10, Slide 4

5 Man on Boat F a net, external cm = 0 x cm = = 0 x cm,0 x cm = 1 M x = Total x N i= 1 cm, new cm,0 m i x i Let d be the distance the boat moves to the left. The man s new position is L-d. x j = M Total x cm m The boat s new position in L/2-d. ( m + m ) x = M x Total cm j N i j m x i i x = 1 x cm Mechanics Lecture 10, Slide 5

6 Let s try two men on a Boat x new = d 90, L xcm = ; xcm = xcm = L boat,70 0;,, 90 2 x cm,0 = m 70 x cm,70 + m m x + m cm, m + m boat boat x cm, boat 0 + (90)(20) + (400)(10) x cm, 0 = = m x cm, new d = = m ( m x x 70, new = 90, new = d L xboat, new = + d 2 L ( d) + m90( d) + mboat ( + d) 2 m70 + m90 + mboat x cm, new = xcm,0 L + m90 + mboat ) xcm,0 mboat 2 m + m + m boat (560)10.357m (400)(10) d = = 3. 2m (560) Mechanics Lecture 10, Slide 6

7 x cm = 1 M Total N i= 1 m i x i x = x cm, new cm,0 Mechanics Lecture 10, Slide 7

8 x x cm j = = 1 M M Total Total x N i= 1 cm m j m x i N i j i m x i i Mechanics Lecture 10, Slide 8

9 x x cm j = = 1 M M Total Total x N i= 1 cm m j m x i N i j i m x i i x x = x2 = x3 x4 x = M Total xcm m2x2 m3x3 m4x = m1 M Total xcm m2x1 m3x1 m4x1 = m m m + m + m 1 x = M x Total ( ) 1 cm 4 x = 1 x cm Mechanics Lecture 10, Slide 9

10 Classical Mechanics Lecture 11 Today s Concept: Conservation of Momentum Inelastic Collisions Mechanics Lecture 11, Slide 10

11 Unit 11 Lecture Thoughts Mechanics Lecture 11, Slide 11

12 Main Points Mechanics Lecture 11, Slide 12

13 Main Points Mechanics Lecture 11, Slide 13

14 Conservation of Total Momentum Mechanics Lecture 11, Slide 14

15 Conservation of Total Momentum Mechanics Lecture 11, Slide 15

16 Total Momentum & Center of Mass Velocity Mechanics Lecture 11, Slide 16

17 Conservation of Total Momentum Mechanics Lecture 11, Slide 17

18 Conservation of Momentum Mechanics Lecture 11, Slide 18

19 Example Mechanics Lecture 11, Slide 19

20 Example Mechanics Lecture 11, Slide 20

21 Example Mechanics Lecture 11, Slide 21

22 Momentum in 2-d More about this later Mechanics Lecture 11, Slide 22

23 Collisions: Conservation of Total Momentum Mechanics Lecture 11, Slide 23

24 Collision and Momentum Conservation Mechanics Lecture 11, Slide 24

25 Collision and Momentum Conservation What about internal forces? F 21 = F12 p 1 p F net, external = 0 P tot = 0 Mechanics Lecture 11, Slide 25

26 Checkpoint A. B. C. 0% 0% 0% Mechanics Lecture 11, Slide 26

27 Checkpoint Mechanics Lecture 11, Slide 27

28 Inelastic Collisions Internal forces do not affect conservation of momentum! Kinetic Energy of Center of Mass motion also unchanged However sum of Kinetic Energy of individual element may be reduced Mechanics Lecture 11, Slide 28

29 Collisions:Kinetic Energy Mechanics Lecture 11, Slide 29

30 Collisions: Kinetic Energy Total Kinetic Energy is not conserved Total Kinetic Energy is reduced in inelastic Collisions Mechanics Lecture 11, Slide 30

31 Inelastic Collision Mechanics Lecture 11, Slide 31

32 Which Kinetic Energy? Mechanics Lecture 11, Slide 32

33 Energy loss in collision Totally Inelastic Stick Together Mechanics Lecture 11, Slide 33

34 Mechanics Lecture 11, Slide 34

35 Center of Mass Frame Mechanics Lecture 11, Slide 35

36 Center of Mass Frame Mechanics Lecture 11, Slide 36

37 Center of Mass Frame Mechanics Lecture 11, Slide 37

38 Center of Mass Frame Mechanics Lecture 11, Slide 38

39 Checkpoint A. B. C. 0% 0% 0% Mechanics Lecture 11, Slide 39

40 Mechanics Lecture 11, Slide 40

41 What is the different between conservation of Momentum and conservation of energy? How do I know if the Momentum is conserved or energy is conserve? Momentum (Prelecture 11) Energy (Prelecture 8) Mechanics Lecture 11, Slide 41

42 Clicker Question A wood block rests at rest on a table. A bullet shot into the block stops inside, and the bullet plus block start sliding on the frictionless surface. The momentum of the bullet plus block remains constant A. B. C. D. E. Before After 20% 20% 20% 20% 20% A) Before the collision. B) During the collision C) After the collision D) All of the above E) Only A and C above As long as there are no external forces acting on the system Mechanics Lecture 11, Slide 42

43 Recap P tot = i m v i i Mechanics Lecture 11, Slide 43

44 CheckPoint Suppose you are on a cart initially at rest that rides on a frictionless track. If you throw a ball off the cart towards the left, will the cart be put into motion? Left Right A) Yes, and it moves to the right. B) Yes, and it moves to the left. Conservation of momentum means the cart must move to the right since the ball moves to the left. C) No, it remains in place. Mechanics Lecture 11, Slide 44

45 CheckPoint Suppose you are on a cart which is initially at rest that rides on a frictionless track. You throw a ball at a vertical surface that is firmly attached to the cart. If the ball bounces straight back as shown in the picture, will the cart be put into motion after the ball bounces back from the surface? Left Right A) Yes, and it moves to the right. B) Yes, and it moves to the left. C) No, it remains in place. Mechanics Lecture 11, Slide 45

46 A. Clicker Question B. C. Suppose you are on a cart that is moving at a constant speed v toward the left on a frictionless track. If you throw a massive ball straight up (relative to the cart), how will the speed of the cart change? Left Right v 33% 33% 33% A) Increase B) Decrease C) Will not change As long as there are no external forces acting on the system, P total is conserved Mechanics Lecture 11, Slide 46

47 CheckPoint Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. The ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck. Which box ends up moving faster? A) Box 1 B) Box 2 C) same 1 2 Mechanics Lecture 11, Slide 47

48 CheckPoint Which box ends up moving faster? A) Box 1 B) Box 2 C) same 1 2 A) As the ball bounces to the left, cart 1 moves faster to the right to conserve momentum. B) Some of the energy is not transferred to box 1, since the ball bounces back C) The starting velocity and mass is all the same so the momentum will be the same in both cases. Think of a 2-step bounce Mechanics Lecture 11, Slide 48

49 Clicker Question A. B. C. Two equal-mass balls swing down and hit identical bricks while traveling at identical speeds. Ball A bounces back, but ball B just stops when it hits the brick. Which ball is more likely to knock the brick over? 33% 33% 33% A B A) A B) B C) They both have the same chance. Mechanics Lecture 11, Slide 49

50 demo A B P A P B P A > P B The change in the momentum of the ball is bigger in A Mechanics Lecture 11, Slide 50

51 Ballistic Pendulum Mechanics Lecture 11, Slide 51

52 Ballistic Pendulum m v M H A projectile of mass m moving horizontally with speed v strikes a stationary mass M suspended by strings of length L. Subsequently, m + M rise to a height of H. Given H, what is the initial speed v of the projectile? Mechanics Lecture 11, Slide 52

53 Breaking it down into steps before during after m v M V H splat Which quantities are conserved before the collision? A) momentum B) mechanical energy C) both momentum and mechanical energy Mechanics Lecture 11, Slide 53

54 Breaking it down into steps before during after m v M V H splat Which quantities are conserved during the collision? A) momentum B) mechanical energy C) both momentum and mechanical energy mv = ( m + M ) V Mechanics Lecture 11, Slide 54

55 Breaking it down into steps before during after m v M V H splat Which quantities are conserved after the collision A) momentum B) mechanical energy C) both momentum and mechanical energy V 2 = 2gH Mechanics Lecture 11, Slide 55

56 Ballistic Pendulum m v M H mv = ( m + M ) V V 2 = 2gH V = m ( m + M ) v m + M v = 2gH v = 2gH 1 + m m M Mechanics Lecture 11, Slide 56

57 Ballistic Pendulum Mechanics Lecture 11, Slide 57

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