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Lecture 18

Lecture 18. Agenda: Review for exam Assignment: For Monday, read chapter 14 (not 13). I. T 2. T 1. m 2. m 1. Newton’s Laws and the Atwood’s Machine.

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Lecture 18

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  1. Lecture 18 Agenda: • Review for exam • Assignment: For Monday, read chapter 14 (not 13)

  2. I T2 T1 m2 m1 Newton’s Laws and the Atwood’s Machine Two blocks are connected using a massless rope as shown. The 2.00 kg pulley has a moment of inertia about its pivot of 2.00 kg m2. The masses are m1 = 3.00 kg, m2 = 1.00kg. Initially they are at rest. a) What is the acceleration of the 1st mass? b) What is the difference in the tensions? c) How fast is the 1st mass going after it falls 2.00 m? 0 0 1: S Fy = m1 a1 = -m1 g + T1 2: S Fy = m2 a2 = -m2 g + T2 Notice a = a1 = -a2 I = mPulley R2 ( R=1.00 m) 3: S tPulley = I a = I (-a/R) = R T1 – R T2 (if a > 0, up, then a < 0) mPulley R2 (-a/R) = R T1 – R T2 – mPulley a = T1 – T2 1-2 gives m1a1 – m2a2 = (m1 + m2) a= (m2 - m1) g + (T1 - T2 )

  3. Continued…. 1-2+3 gives m1a1 – m2a2 = (m1 + m2) a= (m2 - m1) g – mPulley a or (m1 + m2 +mPulley) a= (m2 - m1) g a = (m2 - m1) g / (m1 + m2 +mPulley) a = (1.00 – 3.00) 10.0 / ( 3.00+1.00+2.00) = - 10.0/3.00 m/s2 b) – mPulley a = T1 – T2 = - 2.00 (-10.0/3.00) N = 20.0/3.00 N c) Use conservation of energy! Ki = 0 (at rest) Ui = 0 (by design) EMech = Ki + Ui = 0 = Kf + Uf = ½ m1v2+½ m2v2+½ Iw2-m1gh+m2gh ½ (m1+m2)v2+½ mPulley v2 = m1gh- m2gh v2 = 2(m1- m2)gh/(m1+m2 +mPulley) = 4.00x10.x2.0/(6.00) m2/s2 v2 = (80.0/6.00) m2/s2

  4. h = 1 m sin 30° = 0.5 m 1 meter 30° Exercise Work/Energy for Non-Conservative Forces • An air track is at an angle of 30° with respect to horizontal. The cart (with mass 1.0 kg) is released 1.0 meter from the bottom and hits the bumper at a speed, v1. This time the vacuum/ air generator breaks half-way through and the air stops. The cart only bounces up half as high as where it started. • How much work did friction do on the cart ? (g=10 m/s2) Notice the cart only bounces to a height of 0.25 m • 2.5 J • 5.0 J • 10. J • -2.5 J • -5.0 J • -10. J

  5. Exercise Work/Energy for Non-Conservative Forces • How much work did friction do on the cart ? (g=10 m/s2) W = F Dx = m mg cos q d is not easy to do, esp. if m not given • Work done (W) is equal to the change in the mech. energy of the “system” (U+K). Efinal - Einitial and is < 0. (E = U+K) • Here Ugravityis “in” the system and Kfinal = Kinitial = 0 Use W = Ufinal - Uinit = mg ( hf - hi ) = - mg sin 30° 0.5 m W = -2.5 N m = -2.5 J or (D) hi hf 1 meter 30° (A) 2.5 J (B) 5 J (C) 10 J (D) –2.5 J (E) –5 J (F) –10 J

  6. Exercise Work/Energy for Non-Conservative Forces • Alternatively we could look at Wnet=DK • Again Kfinal = Kinitial = 0 • Wnet=DK = 0 = Wgravity + Wfriction = (mg sin q ) (0.5 meter) + Wfriction Wfriction = -2.5 N m = -2.5 J or (D) And the result is the same hi hf 1 meter 30° (A) 2.5 J (B) 5 J (C) 10 J (D) –2.5 J (E) –5 J (F) –10 J

  7. Springs • A Hooke’s Law spring with a spring constant of 200 N/m is first stretched 3.0 m past its equilibrium distance and then is stretched 9.0 m. • How much work must be done to go from 3.0 m to 9.0 m? • W = Ufinal-Uinitial= ½ k (x-xeq)final2 -½ k (x-xeq)init2 = 100 [(9)2 –(3)2] J= 100(72) J = 7200 J

  8. Chapter 7 (Newton’s 3rd Law) & Chapter 8

  9. Chapter 8

  10. Chapter 8

  11. Chapter 9

  12. Chapter 9

  13. Chapter 10

  14. Chapter 10

  15. Chapter 10

  16. Chapter 11

  17. Chapter 11

  18. Chapter 12 and Center of Mass

  19. Chapter 12

  20. Important Concepts

  21. Chapter 12

  22. Example problem: Going in circles • A 2.0 kg disk tied to a 0.50 m string undergoes circular motion on a rough but horizontal table top. The kinetic coefficient of friction is 0.25. If the disk starts out at 5.0 rev/sec how many revolutions does it make before it comes to rest? • Work-energy theorem • W = F d = 0 – ½ mv2 • F = -mmg d = - ½ mv2 • d= v2 /(2mg)=(5.0 x 2p x 0.50)2/ (0.50 x 10) m = 5 p2 m • Rev = d / 2pr = 16 revolutions • What if the disk were tilted by 60° ? Top view

  23. Work and Energy • A block of mass m is connected by a spring to the ceiling. The block is held at a position where the spring is unstretched and then released. When released, the block (a) remains at rest. (b) oscillates about the unstretched position (c) oscillates about a position that is lower than the unstretched position (d) oscillates about a position that is higher than the unstretched position

  24. Momentum & Impulse • 0 • ½ v • 2 v • 4 v • A rubber ball collides head on (i.e., velocities are opposite) with a clay ball of the same mass. The balls have the same speed, v, before the collision, and stick together after the collision. What is their speed immediately after the collision?

  25. Momentum & Impulse • A rubber ball collides head on with a clay ball of the same mass. The balls have the same speed, v, before the collision, and stick together after the collision. What is their speed after the collision? (a) 0 (b) ½ v (c) 2 v (d) 4 v

  26. Momentum, Work and Energy • A 0.40 kg block is pushed up against a spring (with spring constant 270 N/m ) on a frictionless surface so that the spring is compressed 0.20 m. When the block is released, it slides across the surface and collides with the 0.60 kg bob of a pendulum. The bob is made of clay and the block sticks to it. The length of the pendulum is 0.80 m. (See the diagram.) • To what maximum height above the surface will the ball/block assembly rise after the collision? (g=9.8 m/s2) A. 2.2 cm B. 4.4 cm C. 11. cm D. 22 cm E. 44 cm F. 55 cm

  27. Momentum, Work and Energy • A 0.40 kg block is pushed up against a spring (with spring constant 270 N/m ) on a frictionless surface so that the spring is compressed 0.20 m. When the block is released, it slides across the surface and collides with the 0.60 kg bob of a pendulum. The bob is made of clay and the block sticks to it. The length of the pendulum is .80 m. (See the diagram.) • To what maximum height above the surface will the ball/block assembly rise after the collision? A. 2.2 cm B. 4.4 cm C. 11. cm D. 22 cm E. 44 cm F. 55 cm

  28. Momentum, Work and Energy ( Now with friction) • A 0.40 kg block is pushed up against a spring (with spring constant 270 N/m ) on a surface. • If mstatic = 0.54, how far can the spring be compressed and the block remain stationary (i.e., maximum static friction)? • S F = 0 = k u - f = k u - m N u = m mg/k = 0.54 (0.40x10 N) / 270 N/m= 0.0080 m

  29. Momentum, Work and Energy ( Now with friction) • A 0.40 kg block is pushed up against a spring (with spring constant 270 N/m ) on a surface. The spring is now compressed 2.0 m. • If mkinetic = 0.50 and the block is 8.0 m away from the unstretched spring, how high with the clay/block pair rise? • Emech (at collision) = Uspring + Wfriction = ½ k u2 - m mg d • 1/2 m v2 = 135(4.0)-0.50(0.40x10.)10.=(540-20) J=520 J • v2 = 1040/0.40 m2/s2 • Now the collision (cons. of momentum) and the swing.

  30. Work and Energy • A mass is attached to a Hooke’s law spring on a horizontal surface as shown in the diagram below. When the spring is at its natural length, the block is at position Y. • When released from position X, how will the spring potential energy vary as the block moves from X to Y to Z ? (a) It will steadily increase from X to Z. (b) It will steadily decrease from X to Z. (c) It will increase from X to Y and decrease from Y to Z. (d) It will decrease from X to Y and increase from Y to Z.

  31. Work and Energy • A mass is attached to a Hooke’s law spring on a horizontal surface as shown in the diagram below. When the spring is at its natural length, the block is at position Y. • When released from position X, how will the spring potential energy vary as the block moves from X to Y to Z ? (a) It will steadily increase from X to Z. (b) It will steadily decrease from X to Z. (c) It will increase from X to Y and decrease from Y to Z. (d) It will decrease from X to Y and increase from Y to Z.

  32. Work and Energy • An object moves along a line under the influence of a single force. The area under the force vs. position graph represents (a) the impulse delivered to the object (b) the work done on the object. (c) the change in the velocity of the object. (d) the momentum of the object.

  33. Work and Energy • An object moves along a line under the influence of a single force. The area under the force vs. position graph represents (a) the impulse delivered to the object (b) the work done on the object. (c) the change in the velocity of the object. (d) the momentum of the object.

  34. Momentum and Impulse • Henri Lamothe holds the world record for the highest shallow dive. He belly-flopped from a platform 12.0 m high into a tank of water just 30.0 cm deep! Assuming that he had a mass of 50.0 kg and that he stopped just as he reached the bottom of the tank, what is the magnitude of the impulse imparted to him while in the tank of water (in units of kg m/s)? (a) 121 (b) 286 (c) 490 (d) 623 (e) 767

  35. Momentum and Impulse • Henri Lamothe holds the world record for the highest shallow dive. He belly-flopped from a platform 12.0 m high into a tank of water just 30.0 cm deep! Assuming that he had a mass of 50.0 kg and that he stopped just as he reached the bottom of the tank, what is the magnitude of the impulse imparted to him while in the tank of water (in units of kg m/s)? (a) 121 (b) 286 (c) 490 (d) 623 (e) 767 Dp = sqrt(2x9.8x12.3)x50

  36. Work and Energy • Two particles, one positively charged and one negatively charged, are held apart. Since oppositely charged objects attract one another, the particles will accelerate towards each other when released. Let W+ be the work done on the positive charge by the negative charge. Let W– be the work done on the negative charge by the positive charge. While the charges are moving towards each other, which of the following statements is correct? (a) W+ is positive and W– is negative. (b) W+ is negative and W– is positive. (c) Both W+ and W– are positive. (d) Both W+ and W– are negative. (e) Without knowing the coordinate system, the sign of the work can not be determined.

  37. Work and Energy • Two particles, one positively charged and one negatively charged, are held apart. Since oppositely charged objects attract one another, the particles will accelerate towards each other when released. Let W+ be the work done on the positive charge by the negative charge. Let W– be the work done on the negative charge by the positive charge. While the charges are moving towards each other, which of the following statements is correct? (a) W+ is positive and W– is negative. (b) W+ is negative and W– is positive. (c) Both W+ and W– are positive. (d) Both W+ and W– are negative. (e) Without knowing the coordinate system, the sign of the work can not be determined.

  38. Momentum & Impulse • Suppose that in the previous problem, the positively charged particle is a proton and the negatively charged particle, an electron. (The mass of a proton is approximately 1,840 times the mass of an electron.) Suppose that they are released from rest simultaneously. If, after a certain time, the change in momentum of the proton is Dp, what is the magnitude of the change in momentum of the electron? (a) Dp / 1840 (b) Dp (c) 1840 Dp

  39. Momentum & Impulse • Suppose that in the previous problem, the positively charged particle is a proton and the negatively charged particle, an electron. (The mass of a proton is approximately 1,840 times the mass of an electron.) Suppose that they are released from rest simultaneously. If, after a certain time, the change in momentum of the proton is Dp, what is the magnitude of the change in momentum of the electron? (a) Dp / 1840 (b) Dp (c) 1840 Dp

  40. Work and Energy • A block slides along a frictionless surface before colliding with a spring. The block is brought momentarily to rest by the spring after traveling some distance. The four scenarios shown in the diagrams below are labeled with the mass of the block, the initial speed of the block, and the spring constant. • Rank the scenarios in order of the distance the block travels, listing the largest distance first. (a) B , A , C = D (b) B , C , A , D (c) B , C = D , A (d) C = B, A , D (e) C = B = D , A

  41. Work and Energy • A block slides along a frictionless surface before colliding with a spring. The block is brought momentarily to rest by the spring after traveling some distance. The four scenarios shown in the diagrams below are labeled with the mass of the block, the initial speed of the block, and the spring constant. • Rank the scenarios in order of the distance the block travels, listing the largest distance first. (a) B , A , C = D (b) B , C , A , D (c) B , C = D , A (d) C = B, A , D (e) C = B = D , A

  42. Newton’s Laws • Two boxes are connected to each other as shown. The system is released from rest and the 1.00 kg box falls through a distance of 1.00 m. The surface of the table is frictionless. What is the kinetic energy of box B just before it reaches the floor? (g=9.81 m/s2 ) (a) 2.45 J (b) 4.90 J (c) 9.80 J (d) 9.24 J (e) 9.32 J

  43. Work and Energy • If it takes 5.35 J of work to stretch a Hooke’s law spring 12.2 cm from its un-stretched length, how much work is required to stretch an identical spring by 17.2 cm from its un-stretched length? (a) 0.90 J (b) 5.3 J (c) 7.2 J (d) 10.6 J (e) 11.0 J

  44. Work and Energy • If it takes 5.35 J of work to stretch a Hooke’s law spring 12.2 cm from its un-stretched length, how much work is required to stretch an identical spring by 17.2 cm from its un-stretched length? (a) 0.90 J (b) 5.3 J (c) 7.2 J (d) 10.6 J (e) 11.0 J

  45. Work and Forces • A 25.0 kg chair is pushed 2.00 m at constant speed along a horizontal surface with a constant force acting at 30.0 degrees below the horizontal. If the friction force between the chair and the surface is 55.4 N, what is the work done by the pushing force? (a) 85 J (b) 98 J (c) 111 J (d) 113 J (e) 128 J

  46. Work and Forces • A 25.0 kg chair is pushed 2.00 m at constant speed along a horizontal surface with a constant force acting at 30.0 degrees below the horizontal. If the friction force between the chair and the surface is 55.4 N, what is the work done by the pushing force? (a) 85 J (b) 98 J (c) 111 J (d) 113 J (e) 128 J

  47. Work and Power • A 100 kg elevator is carrying 6 people, each weighing 70 kg. They all want to travel to the top floor, 75 m from the floor they entered at. How much power will the elevator motor supply to lift this in 45 seconds at constant speed? (a) 1.2 · 102 W (b) 7.0 · 102 W (c) 8.7 · 102 W (d) 6.9 · 103 W (e) 8.5 · 103 W

  48. Work and Power • A 100 kg elevator is carrying 6 people, each weighing 70 kg. They all want to travel to the top floor, 75 m from the floor they entered at. How much power will the elevator motor supply to lift this in 45 seconds at constant speed? (a) 1.2 · 102 W (b) 7.0 · 102 W (c) 8.7 · 102 W (d) 6.9 · 103 W (e) 8.5 · 103 W

  49. Conservation of Momentum • A woman is skating to the right with a speed of 12.0 m/s when she is hit in the stomach by a giant snowball moving to the left. The mass of the snowball is 2.00 kg, its speed is 25.0 m/s and it sticks to the woman's stomach. If the mass of the woman is 60.0 kg, what is her speed after the collision? (a) 10.8 m/s (b) 11.2 m/s (c) 12.4 m/s (d) 12.8 m/s

  50. Conservation of Momentum • A woman is skating to the right with a speed of 12.0 m/s when she is hit in the stomach by a giant snowball moving to the left. The mass of the snowball is 2.00 kg, its speed is 25.0 m/s and it sticks to the woman's stomach. If the mass of the woman is 60.0 kg, what is her speed after the collision? (a) 10.8 m/s (b) 11.2 m/s (c) 12.4 m/s (d) 12.8 m/s

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