Chapter 9 Systems of Particles The Center of Mass

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Chapter 9 Systems of ParticlesThe Center of
Mass

• The center of mass of a body or a system of
  bodies is the point that moves as though all of
  the mass were concentrated there and all external
  forces were applied there.

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Solid Bodies
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• The center of mass of an object needs not lie
  within the object. There is no dough at the
  center of mass of a doughnut, and no iron at the
  center of mass of a horseshoe.

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Sample Problem 9-1

• Three particles of masses m1 1.2 kg, m2 2.5
  kg, and m3 3.4 kg form an equilateral triangle
  of edge length a 140 cm. Where is the center of
  mass of this three-particle system?

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SOLUTION 
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Newtons Second Law for a System of Particles

• is the net force of all external forces
  that act on the system. Forces on one part of the
  system from another (internal forces) are not
  included.
• M is the total mass of the system. We assume
  that no mass enters or leaves the system as it
  moves, so that M remains constant. The system is
  said to be closed.
• is the acceleration of the center of
  mass of the system. Equation 9-14 gives no
  information about the acceleration of any other
  point of the system.

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• This means that the center of mass of the
  fragments follows the same parabolic trajectory
  that the rocket would have followed had it not
  exploded.

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Sample Problem 9-3

• The three particles in Fig. 9-7a are initially at
  rest. Each experiences an external force due to
  bodies outside the three-particle system. The
  directions are indicated, and the magnitudes are
  F1 6.0 N, F2 12 N, and F3 14 N. What is the
  acceleration of the center of mass of the system,
  and in what direction does it move?

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SOLUTION We can apply Newtons Second Law to the
center of mass.
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Linear Momentum

• Linear momentum is a vector and the SI unit is
  kg-m / s.
• For constant mass,

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Linear Momentum for a System of Particles

• The linear momentum of a system of particles is
  equal to the product of the total mass M of the
  system and the velocity of the center of mass.
• For constant mass,

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Sample Problem 9-4

• Figure 9-8a shows a 2.0 kg toy race car before
  and after taking a turn on a track. Its speed is
  0.50 m/s before the turn and 0.40 m/s after the
  turn. What is the change D in the linear
  momentum of the car due to the turn?

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SOLUTION 
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Conservation of Linear Momentum

• If no net external force acts on a system of
  particles, the total linear momentum of the
  system cannot change.
• If the component of the net external force on a
  closed system is zero along an axis, then the
  component of the linear momentum of the system
  along that axis cannot change.

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Sample Problem 9-5

• A ballot box with mass m 6.0 kg slides with
  speed v 4.0 m/s across a frictionless floor in
  the positive direction of an x axis. It suddenly
  explodes into two pieces. One piece, with mass m1
  2.0 kg, moves in the positive direction of the
  x axis with speed v1 8.0 m/s. What is the
  velocity of the second piece, with mass m2 ?

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SOLUTION 
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Sample Problem 9-6

• Figure 9-9a shows a space hauler and cargo
  module, of total mass M traveling along an x axis
  in deep space. They have an initial velocity
  of magnitude 2100 km/h relative to the Sun. With
  a small explosion, the hauler ejects the cargo
  module, of mass 0.20M (Fig. 9-9b). The hauler
  then travels 500 km/h faster than the module
  along the x axis that is, the relative speed
  vrel between the hauler and the module is 500
  km/h. What then is the velocity of the
  hauler relative to the Sun?

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SOLUTION 
Note that Vi, VMS and VHS are all relative to the
sun
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Sample Problem 9-7

• A firecracker placed inside a coconut of mass M,
  initially at rest on a frictionless floor, blows
  the coconut into three pieces that slide across
  the floor. An overhead view is shown in Fig.
  9-10a. Piece C, with mass 0.30M, has final speed
  vfC 5.0 m/s.
• (a)  What is the speed of piece B, with mass
  0.20M?

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SOLUTION 
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(b)  What is the speed of piece A?
SOLUTION 
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• Skip sections 9-7 and 9-8.

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Homework (due Nov 1)

• 3E
• 7P
• 11E
• 13P
• 17P
• 21E

• 23P
• 27E
• 29E
• 37P
• 39P