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Chapter 9 Center of Mass and Linear Momentum

Chapter 9 Center of Mass and Linear Momentum In this chapter we will introduce the following new concepts: -Center of mass (com) for a system of particles -The velocity and acceleration of the center of mass

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Chapter 9 Center of Mass and Linear Momentum

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  1. Chapter 9 Center of Mass and Linear Momentum In this chapter we will introduce the following new concepts: -Center of mass (com) for a system of particles -The velocity and acceleration of the center of mass -Linear momentum for a single particle and a system of particles We will derive the equation of motion for the center of mass, and discuss the principle of conservation of linear momentum. Finally, we will use the conservation of linear momentum to study collisions in one and two dimensions and derive the equation of motion for rockets. (9-1)

  2. (9-2)

  3. (9-3)

  4. Example:A 3.00 kg particle is located on the x axis at x = −5.00 m and a 4.00 kg particle is on the x axis at x = 3.00 m. Find the center of mass of this two–particle system.

  5. Example:A 3.00 kg particle is located on the x axis at x = −5.00 m and a 4.00 kg particle is on the x axis at x = 3.00 m. Find the center of mass of this two–particle system.

  6. . C C (9-4)

  7. z m1 F1 F2 m2 m3 F2 O y x (9-5)

  8. z m1 F1 F2 m2 m3 F2 O y x (9-6)

  9. (9-7)

  10. v m p (9-8)

  11. Example: A 3.00 kg particle has a velocity of (3.0 i − 4.0 j) m/s. Find its x and y components of momentum and the magnitude of its total momentum. Using the definition of momentum and the given values of m and v we have: p = mv = (3.00 kg)(3.0 i − 4.0 j) = (9.0 i − 12. j) So the particle has momentum components px = +9.0 and py = −12.

  12. z m1 p1 p3 m2 m3 p2 O y x (9-9)

  13. (9-10)

  14. (9-11)

  15. Example: A child bounces a ball on the sidewalk. The linear impulse delivered by the sidewalk is 2.00 N· s during the 1/800 s of contact. What is the magnitude of the average force exerted on the ball by the sidewalk?

  16. Example: A 3.0 kg steel ball strikes a wall with a speed of 10 m/s at an angle of 60o with the surface. It bounces off with the same speed and angle, as shown in the figure. If the ball is in contact with the wall for 0.20 s, what is the average force exerted on the wall by the ball?

  17. Since has no x component, the average force has magnitude 2.6 ×102 N and points in the y direction (away from the wall).

  18. Fave (9-12)

  19. z m1 p1 p3 m2 m3 p2 O y x (9-13)

  20. Example:A 4.5 kg gun fires a 0.1 kg bullet with a muzzle velocity of +150 m/s. What is the recoil velocity of the gun?

  21. (9-14)

  22. (9-15)

  23. Example: An object of 12.0 kg at rest explodes into two pieces of masses 4.00 kg and 8.00 kg. The velocity of the 8.00 kg mass is 6.00 m/s in the +ve x-direction. The change in the kinetic is:

  24. Example: An object of 12.0 kg at rest explodes into two pieces of masses 4.00 kg and 8.00 kg. The velocity of the 8.00 kg mass is 6.00 m/s in the +ve x-direction. The change in the kinetic is:

  25. Inelastic collisions in 1D

  26. Example: A 10.0 g bullet is stopped in a block of wood (m = 5.00 kg). The speed of the bullet–plus–wood combination immediately after the collision is 0.600 m/s. What was the original speed of the bullet?

  27. Elastic collisions in 1D

  28. (9-16)

  29. Example: A 10 kg ball (m1) with a velocity of +10 m/s collides head on in an elastic manner with a 5 kg ball (m2) at rest. What are the velocities after the collision?

  30. Example: A 10 kg ball (m1) with a velocity of +10 m/s collides head on in an elastic manner with a 5 kg ball (m2) at rest. What are the velocities after the collision?

  31. v2i = 0 v1i x m m v1f = 0 v2f x m m (9-17)

  32. v2i = 0 v1i x m1 m2 v2f x v1f m1 m2 (9-18)

  33. v1i v2i = 0 x m2 v1f m1 v2f x m2 m1 (9-19)

  34. Collisions in 2-D

  35. (9-20)

  36. Example:An unstable nucleus of mass 17 × 10**27 kg initially at rest disintegrates into three particles. One of the particles, of mass 5.0×10**27 kg, moves along the y axis with a speed of 6.0 × 10**6 m/s . Another particle of mass 8.4 × 10**27 kg, moves along the x axis with a speed of 4.0 × 10**6 m/s . Find (a) the velocity of the third particle and (b) the total energy given off in the process.

  37. The parent nucleus is at rest , so that the total momentum was (and remains) zero: Pi = 0.

  38. Example: A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of 30.0degree with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball’s velocity.

  39. Then the condition that the total x momentum be conserved gives us:

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