Center of Mass and Linear Momentum

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Chapter 9 Center of Mass and Linear Momentum
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• Linear momentum
• Linear momentum (or, simply momentum) of a
  point-like object (particle) is
• SI unit of linear momentum is kgm/s
• Momentum is a vector, its direction coincides
  with the direction of velocity

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• Newtons Second Law revisited
• Originally, Newton formulated his Second Law in
  a more general form
• The rate of change of the momentum of an object
  is equal to the net force acting on the object
• For a constant mass

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• Center of mass
• In a certain reference frame we consider a
  system of particles, each of which can be
  described by a mass and a position vector
• For this system we can define a center of mass

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• Center of mass of two particles
• A system consists of two particles on the x axis
• Then the center of mass is
• Changing the reference frame

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• Newtons Second Law for a system of particles
• For a system of particles, the center of mass is
• Then

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• Newtons Second Law for a system of particles
• From the previous slide
• Here is a resultant force on particle i
• According to the Newtons Third Law, the forces
  that particles of the system exert on each other
  (internal forces) should cancel
• Here is the net force of all external
  forces that act on the system (assuming the mass
  of the system does not change)

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Newtons Second Law for a system of particles
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• Linear momentum for a system of particles
• We define a total momentum of a system as
• Using the definition of the center of mass
• The linear momentum of a system of particles is
  equal to the product of the total mass of the
  system and the velocity of the center of mass

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• Linear momentum for a system of particles
• Total momentum of a system
• Taking a time derivative
• Alternative form of the Newtons Second Law for
  a system of particles

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• Conservation of linear momentum
• From the Newtons Second Law
• If the net force acting on a system is zero,
  then
• If no net external force acts on a system of
  particles, the total linear momentum of the
  system is conserved (constant)
• This rule applies independently to all components

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• Center of mass of a rigid body
• For a system of individual particles we have
• For a rigid body (continuous assembly of matter)
  with volume V and density ?(V) we generalize a
  definition of a center of mass

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Chapter 9 Problem 82
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• Impulse
• During a collision, an object is acted upon by a
  force exerted on it by other objects
  participating in the collision
• We define impulse as
• Then (momentum-impulse theorem)

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• Elastic and inelastic collisions
• During a collision, the total linear momentum is
  always conserved if the system is isolated (no
  external force)
• It may not necessarily apply to the total
  kinetic energy
• If the total kinetic energy is conserved during
  the collision, then such a collision is called
  elastic
• If the total kinetic energy is not conserved
  during the collision, then such a collision is
  called inelastic
• If the total kinetic energy loss during the
  collision is a maximum (the objects stick
  together), then such a collision is called
  completely inelastic

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Elastic collision in 1D
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• Elastic collision in 1D stationary target
• Stationary target v2i 0
• Then

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Chapter 9 Problem 58
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Completely inelastic collision in 1D
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Chapter 9 Problem 62
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• Answers to the even-numbered problems
• Chapter 9
• Problem 2
• -1.50 m
• (b) -1.43 m

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Answers to the even-numbered problems Chapter 9
Problem 10 6.2 m
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Answers to the even-numbered problems Chapter 9
Problem 18 4.9 kg m/s
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• Answers to the even-numbered problems
• Chapter 9
• Problem 26
• 42 N s
• (b) 2.1 kN

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Answers to the even-numbered problems Chapter 9
Problem 46 3.1 102 m/s
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• Answers to the even-numbered problems
• Chapter 9
• Problem 66
• (10 m/s)i (15 m/s)j
• (b) -500 J

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• Answers to the even-numbered problems
• Chapter 9
• Problem 70
• 2.7
• (b) 7.4