ACCOUNTING FOR LINEAR MOMENTUM

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ACCOUNTING FOR LINEAR MOMENTUM

• Class 28.1

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Rat 28.1
Turn off computer monitors. Take 3 minutes to
answer the following (true or false)

• 1. Mass is a state quantity.
• 2. Velocity is a state quantity.
• 3. Linear Momentum is a state quantity.
• 4. Linear Momentum is a vector quantity.
• 5. Linear Momentum is always conserved.

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Objectives

• Know that linear momentum is conserved
• Know that linear momentum can be added to a
  system by forces, or by adding mass
• Know the connection between Newton's Laws and
  accounting for linear momentum
• Be able to do calculations involving accounting
  for linear momentum

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Linear Momentum

• The concept of linear momentum was developed by
  Newton.
• Linear momentum, p, is defined as the product of
  the mass, m, and velocity, v
• pmv

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Linear Momentum

• Mass is a state quantity.
• Velocity is also a state quantity, but since it
  also has direction, it is also a vector quantity
  (thus the boldface notation).
• Since mass is a scalar and velocity is a vector,
  then momentum is also a vector.
• The algebraic combination of state quantities
  yields a state quantity, thus momentum is a state
  quantity (CLOSED SYSTEMS ONLY).

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Restatement of Newtons Laws

• Newtons laws can be re-stated in terms of linear
  momentum
• 1 Linear momentum is constant for a body in
  its natural state.
• 2 Linear momentum changes when net force is
  applied to a body.
• 3 Forces always exist by the interaction of
  bodies the force on one body is equal and
  opposite to the force on the other body.

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Revisiting the UAE

• Before applying the Universal Accounting Equation
  (UAE)
• Define a system.
• Determine what quantity will be counted.
• Define time interval for counting.

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Terms of the UAE

• FINAL AMOUNT in system at the END of time
  period
• INITIAL AMOUNT in system at the START of time
  period
• INPUT PASSING through boundary INTO system
  during time period
• OUTPUT PASSING through boundary OUT of system
  during time period
• GENERATION PRODUCED during time period within
  boundary
• CONSUMPTION DESTROYED during time period
  within boundary

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Revisiting the UAE

• The UAE equation states that the net difference
  in state quantities equals the net difference in
  path quantities, OR
• Accumulation Net Input Net Generation, OR,
• FINAL AMOUNT - INITIAL AMOUNT
• INPUT - OUTPUT GENERATION - CONSUMPTION

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Linear Momentum Is Conserved

• Linear momentum is a conserved quantity i.e. the
  total amount of linear momentum in the universe
  is constant and cannot be changed.
• Thus, generation consumption 0, so
• Final Amt - Initial Amt Input - Output,
• or
• Accumulation Net Input

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Conservation of Momentum
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Steps to Applying the UAE

• To use the UAE, define the system, then compute
• 1. INITIAL linear momentum (scalar) for the
  quantity(ies) of mass in the system,
• 2. FINAL linear momentum (scalar) for the
  quantity(ies) in the system,
• 3. Linear momentum (scalar) of quantity(ies)
  INPUT into the system,
• 4. Linear momentum (scalar) of quantity(ies)
  OUTPUT from the system.

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Linear Momentum Change

• Consider the accumulation term of the UAE
• There are two ways that the linear momentum of a
  system can change
• mass transfer
• unbalanced forces
• Otherwise, the accumulation of linear momentum is
  zero

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Individual Exercise (5 minutes)

• This is Problem 20.8
• A 5.0 g bullet fired horizontally at 300 m/s
  passes through a 500 g block of wood initially at
  rest on a frictionless surface. The bullet
  emerges with a speed of 100 m/s. What is the
  final speed of the block?

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Solution
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Linear Momentum Change by Mass Transfer

• Suppose you define a system that contains a
  single object of mass m1 moving at a velocity of
  v1 in the positive x direction
• As time passes, two more objects enter the system
  with masses m2 and m3 and velocities v2 and v3
  respectively, also in the x direction.

system boundary
m2v2
m2v2
Time Passes
Time Passes
m1v1
m1v1
m1v1
m3v3
m3v3
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Linear Momentum Change by Mass Transfer

• The initial linear momentum is
• pi m1v1
• The final linear momentum is
• pf (m1v1 m2v2 m3v3)
• Note that for this case we can write these as
  scalars if we assume that we are looking at the
  magnitude of the x component of the vectors.
• Then, accumulation pf - pi is not zero.

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Pairs Exercise (5 minutes)

• Problem 20.2
• A 1.0 ton (including ammo and pilot) military
  helicopter is flying at 40 mph. It has a machine
  gun that fires 60 bullets per second in the
  forward direction. Each 0.5 lbm bullet exits the
  gun at at speed of 1800 mph. After a 2 second
  burst of fire, what is the helicopter speed.

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Solution
Helicopter w/ ammo mi 1 ton vi 40 mph
Helicopter w/ less ammo mf ? vf ?
bullets
Initial
Final
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Pairs Exercise (2 minutes)

• For the previous exercise, if the final system
  had been defined to include the fired bullets,
  would the answer be different?

Helicopter w/ ammo mi 1 ton vi 40 mph
Helicopter w/ less ammo mf ? vf ?
bullets
Initial
Final
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Solution No, it would not change
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Linear Momentum Change by Unbalanced Forces
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Linear Momentum Change by Unbalanced Forces

• Momentum flow INTO or OUT OF a system can result
  from FORCES.
• Mathematically,
• SI Units of Force kgm/s2
• SI Units of Momentum kgm/s
• Thus, Units of Force Units of Momentum/s
  Units of Momentum Flow Rate across a boundary, or

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Relation to Newtons 2nd Law
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Linear Momentum and Newtons Laws

• Newtons 2nd Law, Fma, is equivalent to stating
  a net Force acting on a body changes the Linear
  Momentum of the body.
• Adding Newtons 3rd Law to this yields
  CONSERVATION OF LINEAR MOMENTUM (universe) since
  the linear momentum of one body changes the same
  as the other body, but in opposite directions.

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Designating the System

• When solving these types of problems, the system
  boundary must be defined.
• This boundary is at the discretion of the
  engineer.
• There is no requirement that the system contain
  all bodies involved in the process.
• Thus, a system can have unbalanced forces which
  will change the linear momentum of the system.

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Individual Exercise (7 minutes)

• This is Problem 20.5
• A 1.000 ton dragster has a jet engine that
  provides a thrust of 3000. lbf. The dragster
  starts from a dead stop at the start line and
  crosses the finish line 9 seconds later. Neglect
  mass loss.
• What is its final velocity?
• How long was the track?

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Solution (a)
Burning fuel neglected, so mass is constant
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Solution (b)
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Individual Exercise (7 minutes)

• This is Problem 20.19
• A 0.100 kg hockey puck is stationary on the ice.
  Then it is hit with a hockey stick and 0.020 s
  later, the puck is traveling at 100. km/h.
• What is the average force (N) on the puck?
• What is the average force (N) on the hockey stick?

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Solution (a)
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Solution (b)

• By Newtons 3rd Law of equal and opposite
  reactions, the force on the hockey stick is the
  same as that on the puck.

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Systems Without Net Linear Momentum Input

• No unbalanced forces no mass transfer into or
  out of the system.
• Thus, UAE simplifies to
  
• ACCUMULATION0, OR
• FINAL AMOUNT - INITIAL AMOUNT 0, OR
• FINAL AMOUNT INITIAL AMOUNT
• (CONSERVATIVE, STEADY STATE SYSTEM)
• Therefore, there is no change in linear momentum
  of the system.

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Individual Exercise (7 minutes)

• In Example 20.5, what is the change in linear
  momentum of the black ball? The white ball? What
  is the total kinetic energy of the two balls
  before the impact and after the impact? Is
  energy conserved in this impact?

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Solution

• White Ball
• Dpwhite mwhite (vwhite,final- vwhite,initial)
• 2 lbm (1.5 - 2) ft/s
• -1.0 lbmft/s -0.0311 lbf.s
• Black Ball
• Dpblack mblack (vblack,final- vblack,initial)
• 1 lbm (1 - 0) ft/s
• 1.0 lbmft/s 0.0311 lbf.s

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Assignment 12TEAM ASSIGNMENT

• Foundations 20.4, 20.5, 20.9, 20.11, 20.18,
  20.22 plus problem on next slide.
• DUE 4/22/03
• Include cover sheet with names of team members
  and participation
• FORMAT WILL BE GRADED AS WELL AS YOUR SOLUTION
  (i.e. Given, Required, etc.)

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Assignment 12 (cont.)TEAM ASSIGNMENT

• A 4.00 oz (0.250 lbm) baseball is thrown by a
  pitcher towards a batter at a velocity of 80.0
  ft/s.The batter impacts the ball with the bat for
  0.015 seconds and propels the baseball back at
  the pitcher at a velocity of 100. ft/s. Assuming
  the ball travels the same path as it was thrown
  (but in the opposite direction) compute the force
  exerted on the ball by the bat.