1. 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
-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
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
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.

8. . C
(9-4)

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

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

11. (9-7)

12. m v p
(9-8)

13. Example: A 3. 00 kg particle has a velocity of (3. 0 i − 4. 0 j) m/s
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 = and py = −12.

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

15. (9-10)

16. (9-11)

19. Example: A child bounces a ball on the sidewalk
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?

20. 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?

21. Since has no x component, the average force has magnitude 2
Since has no x component, the average force has magnitude 2.6 ×102 N and points in the y direction (away from the wall).

22. Fave
(9-12)

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

24. Example: A 4. 5 kg gun fires a 0
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?

25. (9-14)

26. (9-15)

27. 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:

28. 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:

31. Inelastic collisions in 1D

34. Example: A 10. 0 g bullet is stopped in a block of wood (m = 5. 00 kg)
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 m/s. What was the original speed of the bullet?

37. Elastic collisions in 1D

38. (9-16)

39. 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?

40. 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?

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

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

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

44. Collisions in 2-D

45. (9-20)

46. Example: An unstable nucleus of mass 17 × 10
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.

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

49. Example: A billiard ball moving at 5
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.

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

51. We can similarly find vy by using the condition that the total y momentum be conserved
in the collision. This gives us:

60. A 10. 0 kg toy car is moving along the x axis
A 10.0 kg toy car is moving along the x axis. The only force Fx acting on the car is shown in Fig. 5 as a function of time (t). At time t = 0 s the car has a speed of 4.0 m/s. What is its speed at time t = 6.0 s? (Ans: 8.0 m/s )

61. A 10. 0 kg toy car is moving along the x axis
A 10.0 kg toy car is moving along the x axis. The only force Fx acting on the car is shown in Fig. 5 as a function of time (t). At time t = 0 s the car has a speed of 4.0 m/s. What is its speed at time t = 6.0 s? (Ans: 8.0 m/s )

62. An impulsive force Fx as a function of time (in ms) is shown in the figure as applied to an object (m = 5.0 kg) at rest. What will be its final speed?

63. An impulsive force Fx as a function of time (in ms) is shown in the figure as applied to an object (m = 5.0 kg) at rest. What will be its final speed?

64. If the masses of m1 and m3 in Fig. 5 are 1. 0 kg each and m2 is 2
If the masses of m1 and m3 in Fig. 5 are 1.0 kg each and m2 is 2.0 kg, what are the coordinates of the center of mass? (Ans: (1.00, 0.50) m )

65. If the masses of m1 and m3 in Fig. 5 are 1. 0 kg each and m2 is 2
If the masses of m1 and m3 in Fig. 5 are 1.0 kg each and m2 is 2.0 kg, what are the coordinates of the center of mass? (Ans: (1.00, 0.50) m )

66. A 1500 kg car traveling at 90.0 km/h east collides with a 3000 kg car traveling at 60.0 km/h south. The two cars stick together after the collision (see Fig 2). What is the speed of the cars after collision?

67. A 1500 kg car traveling at 90.0 km/h east collides with a 3000 kg car traveling at 60.0 km/h south. The two cars stick together after the collision (see Fig 2). What is the speed of the cars after collision?

68. A 2. 00 kg pistol is loaded with a bullet of mass 3. 00 g
A 2.00 kg pistol is loaded with a bullet of mass 3.00 g. The pistol fires the bullet at a speed of 400 m/s. The recoil speed of the pistol when the bullet was fired is: (Ans: m/s)

69. A 2. 00 kg pistol is loaded with a bullet of mass 3. 00 g
A 2.00 kg pistol is loaded with a bullet of mass 3.00 g. The pistol fires the bullet at a speed of 400 m/s. The recoil speed of the pistol when the bullet was fired is: (Ans: m/s)

70. A 1. 0 kg ball falling vertically hits a floor with a velocity of 3
A 1.0 kg ball falling vertically hits a floor with a velocity of 3.0 m/s and bounces vertically up with a velocity of 2.0 m/s . If the ball is in contact with the floor for 0.10 s, the average force on the floor by the ball is:

71. A 1. 0 kg ball falling vertically hits a floor with a velocity of 3
A 1.0 kg ball falling vertically hits a floor with a velocity of 3.0 m/s and bounces vertically up with a velocity of 2.0 m/s . If the ball is in contact with the floor for 0.10 s, the average force on the floor by the ball is:

72. A 20-g bullet is fired into a 100-g wooden block initially at rest on a horizontal frictionless surface. If the initial speed of the bullet is 10 m/s and it comes out of the block with a speed of 5.0 m/s, find the speed of the block immediately after the collision.

73. A 20-g bullet is fired into a 100-g wooden block initially at rest on a horizontal frictionless surface. If the initial speed of the bullet is 10 m/s and it comes out of the block with a speed of 5.0 m/s, find the speed of the block immediately after the collision.

74. A 1.0-kg block at rest on a horizontal frictionless surface is connected to a spring (k = 200 N/m) whose other end is fixed (see figure). A 2.0-kg block moving at 4.0 m/s collides with the 1.0-kg block. If the two blocks stick together after the one-dimensional collision, what maximum compression of the spring does occur when the blocks momentarily stop?

75. A 1.0-kg block at rest on a horizontal frictionless surface is connected to a spring (k = 200 N/m) whose other end is fixed (see figure). A 2.0-kg block moving at 4.0 m/s collides with the 1.0-kg block. If the two blocks stick together after the one-dimensional collision, what maximum compression of the spring does occur when the blocks momentarily stop?