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Chapter 7

Chapter 7. Linear Momentum. Linear Momentum. Definition of Momentum Impulse Conservation of Momentum Center of Mass Motion of the Center of Mass Collisions (1d, 2d; elastic, inelastic). F 21. F 12. m 1. m 2. Momentum From Newton’s Laws.

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Chapter 7

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  1. Chapter 7 Linear Momentum

  2. Linear Momentum • Definition of Momentum • Impulse • Conservation of Momentum • Center of Mass • Motion of the Center of Mass • Collisions (1d, 2d; elastic, inelastic) Chap07b- Linear Momentum: Revised 6/28/2010

  3. F21 F12 m1 m2 Momentum From Newton’s Laws Consider two interacting bodies with m2 > m1: If we know the net force on each body then The velocity change for each mass will be different if the masses are different. Chap07b- Linear Momentum: Revised 6/28/2010

  4. Momentum Rewrite the previous result for each body as: From the 3rd Law Combine the two results: Chap07b- Linear Momentum: Revised 6/28/2010

  5. Define Momentum The quantity (mv) is called momentum (p). p = mv and is a vector. The unit of momentum is kg m/s; there is no derived unit for momentum. Chap07b- Linear Momentum: Revised 6/28/2010

  6. Momentum From slide (4): The change in momentum of the two bodies is “equal and opposite”. Total momentum is conserved during the interaction; the momentum lost by one body is gained by the other. Δp1 +Δp2 = 0 Chap07b- Linear Momentum: Revised 6/28/2010

  7. Impulse Definition of impulse: We use impulses to change the momentum of the system. One can also define an average impulse when the force is variable. Chap07b- Linear Momentum: Revised 6/28/2010

  8. 30 N m1 or m2 Impulse Example Example (text conceptual question 7.2): A force of 30 N is applied for 5 sec to each of two bodies of different masses. Take m1 < m2 (a) Which mass has the greater momentum change? Since the same force is applied to each mass for the same interval, p is the same for both masses. Chap07b- Linear Momentum: Revised 6/28/2010

  9. Example continued: (b) Which mass has the greatest velocity change? Since both masses have the same p, the smaller mass (mass 1) will have the larger change in velocity. (c) Which mass has the greatest acceleration? Since av the mass with the greater velocity change will have the greatest acceleration (mass 1). Chap07b- Linear Momentum: Revised 6/28/2010

  10. Change in Momentum Example An object of mass 3.0 kg is allowed to fall from rest under the force of gravity for 3.4 seconds. Ignore air resistance. What is the change in momentum? Want p = mv. Chap07b- Linear Momentum: Revised 6/28/2010

  11. Change in Momentum Example What average force is necessary to bring a 50.0-kg sled from rest to 3.0 m/s in a period of 20.0 seconds? Assume frictionless ice. The force will be in the direction of motion. Chap07b- Linear Momentum: Revised 6/28/2010

  12. Impulse Since we are applying the change of momentum over a time interval consisting of the instant before the impact to the instant after the impact, we ignore the effect of gravity over that time period Chap07b- Linear Momentum: Revised 6/28/2010

  13. Impulse and Momentum The force in impulse problems changes rapidly so we usually deal with the average force Chap07b- Linear Momentum: Revised 6/28/2010

  14. Work and Impulse Comparison Work changes ET Work = 0 ET is conserved Work = Force x Distance Impulse changes Impulse = 0 is conserved Impulse = Force x Time Chap07b- Linear Momentum: Revised 6/28/2010

  15. v1i v2i m1 m2 m1 m2 Conservation of Momentum m1>m2 A short time later the masses collide. What happens? Chap07b- Linear Momentum: Revised 6/28/2010

  16. N2 N1 F12 F21 y w2 w1 x During the interaction: There is no net external force on either mass. Chap07b- Linear Momentum: Revised 6/28/2010

  17. The forces F12 and F21 are internal forces. This means that: In other words, pi = pf. That is, momentum is conserved. This statement is valid during the interaction (collision) only. Chap07b- Linear Momentum: Revised 6/28/2010

  18. Momentum Examples Chap07b- Linear Momentum: Revised 6/28/2010

  19. Generic Zero Initial Momentum Problem These problems occur whenever all the objects in the system are initially at rest and then separate. Chap07b- Linear Momentum: Revised 6/28/2010

  20. Changing the Mass While Conserving the Momentum Chap07b- Linear Momentum: Revised 6/28/2010

  21. Changing the Mass While Conserving the Momentum Chap07b- Linear Momentum: Revised 6/28/2010

  22. Changing the Mass While Conserving the Momentum Chap07b- Linear Momentum: Revised 6/28/2010

  23. Astronaut Tossing How Long Will the Game Last? Chap07b- Linear Momentum: Revised 6/28/2010

  24. How Long Will the Game Last? Chap07b- Linear Momentum: Revised 6/28/2010

  25. The Ballistic Pendulum Conservation of Energy Conservation of Momentum Chap07b- Linear Momentum: Revised 6/28/2010

  26. The High Velocity Ballistic Pendulum Chap07b- Linear Momentum: Revised 6/28/2010

  27. Relative Speed Relationship • The relative speed relationship applies to elastic collisions. • In an elastic collision the total kinetic energy is also conserved. • If a second equation is needed to solve a momentum problem, the relative speed relationship is easier to deal with than the KE equation. Chap07b- Linear Momentum: Revised 6/28/2010

  28. Relative Speed Relationship Chap07b- Linear Momentum: Revised 6/28/2010

  29. Relative Speed in Problem Solving Need a second equation when solving for two variables in an elastic collision problem? The relative speed relationship is preferred over the conservation ofkinetic energy relationship Chap07b- Linear Momentum: Revised 6/28/2010

  30. Rifle Example A rifle has a mass of 4.5 kg and it fires a bullet of 10.0 grams at a muzzle speed of 820 m/s. What is the recoil speed of the rifle as the bullet leaves the barrel? As long as the rifle is horizontal, there will be no net external force acting on the rifle-bullet system and momentum will be conserved. Chap07b- Linear Momentum: Revised 6/28/2010

  31. Center of Mass Chap07b- Linear Momentum: Revised 6/28/2010

  32. Center of Mass The center of mass (CM) is the point representing the mean (average) position of the matter in a body. This point need not be located within the body. Chap07b- Linear Momentum: Revised 6/28/2010

  33. Center of Mass The center of mass (of a two body system) is found from: This is a “weighted” average of the positions of the particles that compose a body. (A larger mass is more important.) Chap07b- Linear Momentum: Revised 6/28/2010

  34. Center of Mass In 3-dimensions write where The components of rcm are: Chap07b- Linear Momentum: Revised 6/28/2010

  35. y x x A Center of Mass Example Particle A is at the origin and has a mass of 30.0 grams. Particle B has a mass of 10.0 grams. Where must particle B be located so that the center of mass (marked with a red x) is located at the point (2.0 cm, 5.0 cm)? Chap07b- Linear Momentum: Revised 6/28/2010

  36. Center of Mass Example Chap07b- Linear Momentum: Revised 6/28/2010

  37. y 2 x 1 3 Find the Center of Mass The positions of three particles are (4.0 m, 0.0 m), (2.0 m, 4.0 m), and (1.0 m, 2.0 m). The masses are 4.0 kg, 6.0 kg, and 3.0 kg respectively. What is the location of the center of mass? Chap07b- Linear Momentum: Revised 6/28/2010

  38. Example continued: Chap07b- Linear Momentum: Revised 6/28/2010

  39. Motion of the Center of Mass For an extended body, it can be shown that p = mvcm. From this it follows that Fext = macm. Chap07b- Linear Momentum: Revised 6/28/2010

  40. Center of Mass Velocity For the center of mass: Solving for the velocity of the center of mass: Or in component form: Chap07b- Linear Momentum: Revised 6/28/2010

  41. Center of Mass Reference System Chap07b- Linear Momentum: Revised 6/28/2010

  42. Center of Mass Treatment Chap07b- Linear Momentum: Revised 6/28/2010

  43. Center of Mass Treatment Chap07b- Linear Momentum: Revised 6/28/2010

  44. Center of Mass Treatment Chap07b- Linear Momentum: Revised 6/28/2010

  45. Center of Mass Treatment Chap07b- Linear Momentum: Revised 6/28/2010

  46. Center of Mass Reference System Center of Mass System Laboratory System Chap07b- Linear Momentum: Revised 6/28/2010

  47. Collisions in One Dimension When there are no external forces present, the momentum of a system will remain unchanged. (pi = pf) If the kinetic energy before and after an interaction is the same, the “collision” is said to be perfectly elastic. If the kinetic energy changes, the collision is inelastic. Chap07b- Linear Momentum: Revised 6/28/2010

  48. Collisions in One Dimension If, after a collision, the bodies remain stuck together, the loss of kinetic energy is a maximum, but not necessarily a 100% loss of kinetic energy. This type of collision is called perfectly inelastic. Chap07b- Linear Momentum: Revised 6/28/2010

  49. Elastic Collision - Unequal Masses In the later slides, the initially moving mass, mn , will called be m1 while the initially stationary mass, mc , will be m2. Chap07b- Linear Momentum: Revised 6/28/2010

  50. Elastic Collision - KE Transfer Chap07b- Linear Momentum: Revised 6/28/2010

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