Momentum and Impulse

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Momentum and Impulse

• 8.01
• W06D2
• Associated Reading Assignment
• Young and Freedman 8.1-8.5

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Announcements
No Math Review Night this Week Next Reading
Assignment W06D3 Young and Freedman 8.1-8.5
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Todays Reading Questions

• Explain the difference between the concepts of
  impulse and work.
• Explain why the total force on a system of
  particles is only due to the sum of external
  forces.
• Under what conditions does the center of mass of
  a system of particles move with constant
  velocity?

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Momentum and Impulse
Obeys a conservation law Simplifies complicated
motions Describes collisions Basis of rocket
propulsion space travel
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Newtons Definition Quantity of Motion
DEFINITION II Newtons Principia The quantity of
motion is the measure of the same, arising from
the velocity and quantity of matter
conjointly. The motion of the whole is the sum
of the motions of all the parts and therefore in
a body double in quantity, with equal velocity,
the motion is double with twice the velocity, it
is quadruple.
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Momentum and Impulse Single Particle

• Momentum
• SI units
• Change in momentum
• Impulse
• SI units

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Integral Version of Newtons Second Law

For an object with fixed mass
Then impulse produces a change in momentum
The change of motion is proportional to the
motive force impresses, and is made in the
direction of the right line in which that force
is impressed
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Concept Question Pushing Identical Carts

• Identical constant forces push two identical
  objects A and B continuously from a starting line
  to a finish line. If A is initially at rest and B
  is initially moving to the right,
• Object A has the larger change in momentum.
• Object B has the larger change in momentum.
• Both objects have the same change in momentum
• Not enough information is given to decide.

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Concept Question Pushing Identical Carts Answer

• Answer 1 Both objects have the same mass, are
  pushed the same distance, by the same constant
  force, so they have the same acceleration. Since
  object A started from rest, an object B has an
  initial non-zero speed, object A needs a larger
  time interval to reach the finish than the
  corresponding time interval for object B
  Therefore the impulse on object A is larger than
  the corresponding impulse on object B. Hence
  object A has a larger change in momentum.

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Momentum, Kinetic Energy, and Work Single
Particle

• Kinetic energy and momentum for a single
  particle are related by
• Change in kinetic energy and work

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Concept Question Pushing Carts

• Consider two carts, of masses m and 2m, at
  rest on an air track. If you push one cart for 3
  seconds and then the other for the same length of
  time, exerting equal force on each, the kinetic
  energy of the light cart is
• larger than
• equal to
• 3) smaller than
• the kinetic energy of the heavy car.

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Concept Question Pushing Carts

• Answer 1. The kinetic energy of an object can be
  written as
• Because the impulse is the same for the two
  carts, the change in momentum is the same. Both
  start from rest so they both have the same final
  momentum. Since the mass of the lighter cart is
  smaller than the mass of the heavier cart, the
  kinetic energy of the light cart is larger than
  the kinetic energy of the heavy cart.

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Concept Question Stopping Distances

• Suppose a ping-pong ball and a bowling ball are
  rolling toward you. Both have the same momentum,
  and you exert the same force to stop each. How do
  the distances needed to stop them compare?
• It takes a shorter distance to stop the ping-pong
  ball.
• Both take the same distance.
• It takes a longer distance to stop the ping-pong
  ball.

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Concept Question Stopping Distances
Answer 3. The kinetic energy of an object can be
written as K p2/2m. Because the ping pong ball
and the bowling ball have the same momentum, the
kinetic energy of the less massive ping pong ball
is greater than the kinetic energy of the more
massive bowling ball. You must do work on an
object to change its kinetic energy. If you exert
a constant force, then the work done is the
product of the force with the displacement of the
point of application of the force. Since the work
done on an object is equal to the change in
kinetic energy, the ping pong ball has a greater
change in kinetic energy in order to bring it to
a stop, so the you need a longer distance to stop
the ping pong ball.
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Demo Jumping Off the Floorwith a Non-Constant
Force
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Demo Jumping Non-Constant Force

• Plot of total external force vs. time for Andy
  jumping off the floor. Weight of Andy is 911 N.

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Demo Jumping Impulse

• Shaded area represents impulse of total force
  acting on Andy as he jumps off the floor

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Demo Jumping Height

• When Andy leaves the ground, the impulse is
• So the y-component of his velocity is
• Andy jumped

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System of Particles Center of Mass
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Position and Velocity of Center of Mass

• Mass for collection of discrete bodies (system)
• Momentum of system
• Position of center of mass
• Velocity of center of mass

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Table Problem Center of Mass of Rod and Particle

• A slender uniform rod of length d and mass m
  rests along the x-axis on a frictionless,
  horizontal table. A particle of equal mass m is
  moving along the x-axis at a speed v0. At t 0,
  the particle strikes the end of the rod and
  sticks to it. Find a vector expression for the
  position of the center of mass of the system at t
  0.

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System of Particles Internal and External
Forces, Center of Mass Motion
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System of Particles Newtons Second and Third
Laws
The momentum of a system remains constant unless
the system is acted on by an external force in
which case the acceleration of center of mass
satisfies
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Force on a System of N Particles is the External
Force

• The force on a system of particles is the
  external force because the internal force is zero
  

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Internal Force on a System of N Particles is Zero

• The internal force on the ith particle is sum of
  the interaction forces with all the other
  particles
• The internal force is the sum of the internal
  force on each particle
• Newtons Third Law internal forces cancel in
  pairs
• So the internal force is zero

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External Force and Momentum Change

• The momentum of a system of N particles is
  defined as the sum of the individual momenta of
  the particles
• Force changes the momentum of the system
• Force equals external force
• Newtons Second and Third Laws for a system of
  particles The external force is equal to the
  change in momentum of the system

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External Forces and Constancy of Momentum Vector

• The external force may be zero in one direction
  but not others
• The component of the system momentum is constant
  in the direction that the external force is zero
• The component of system momentum is not constant
  in a direction in which external force is not zero

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Table Problem Center of Mass of Rod and Particle
Post- Collision

• A slender uniform rod of length d and mass m
  rests along the x-axis on a frictionless,
  horizontal table. A particle of equal mass m is
  moving along the x-axis at a speed v0. At t 0,
  the particle strikes the end of the rod and
  sticks to it. Find a vector expression for the
  position of the center of mass of the system for
  t gt 0.

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Demo Center of Mass trajectory B78

• http//tsgphysics.mit.edu/front/index.php?pagedem
  o.php?letnumB2078show0
• Odd-shaped objects with their centers of mass
  marked are thrown. The centers of mass travel in
  a smooth parabola. The objects consist of a
  squash racket, a 16 diameter disk weighted at
  one point on its outer rim, and two balls
  connected with a rod. This demonstration is shown
  with UV light.

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CM moves as though all external forces on the
system act on the CM
so the jumpers cm follows a parabolic trajectory
of a point moving in a uniform gravitational field
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Center of mass passes under the bar
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Table Problem Exploding Projectile Center of
Mass Motion

• An instrument-carrying projectile of mass m1
  accidentally explodes at the top of its
  trajectory. The horizontal distance between
  launch point and the explosion is x0. The
  projectile breaks into two pieces which fly apart
  horizontally. The larger piece, m3, has three
  times the mass of the smaller piece, m2. To the
  surprise of the scientist in charge, the smaller
  piece returns to earth at the launching station.
• How far has the center of mass of the system
  traveled from the launch when the pieces hit the
  ground?
• How far from the launch point has the larger
  piece graveled when it first hits the ground?

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Concept Question Pushing a Baseball Bat
1
3
2

• The greatest instantaneous acceleration of the
  center of mass
• will be produced by pushing with a force F at
• Position 1
• 2. Position 2
• 3. Position 3
• 4. All the same

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Concept Question Pushing a Baseball Bat
1
3
2
Answer 4. The external force is equal to the
total mass times the instantaneous acceleration
of the center-of-mass. It doesnt matter where
the external force acts with regards to the
center-of-mass acceleration.
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Table Problem Landing Plane and Sandbag

• A light plane of mass 1000 kg makes an
  emergency landing on a short runway. With its
  engine off, it lands on the runway at a speed of
  40 ms-1. A hook on the plane snags a cable
  attached to a sandbag of mass 120 kg and drags
  the sandbag along. If the coefficient of friction
  between the sandbag and the runway is µ 0.4,
  and if the planes brakes give an additional
  retarding force of magnitude 1400 N, how far does
  the plane go before it comes to a stop?

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Strategy Momentum of a System

• 1. Choose system
• 2. Identify initial and final states
• 3. Identify any external forces in order to
  determine whether any component of the momentum
  of the system is constant or not
• i) If there is a non-zero total external force
• ii) If the total external force is zero then
  momentum is constant

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Modeling Instantaneous Interactions

• Decide whether or not an interaction is
  instantaneous.
• External impulse changes the momentum of the
  system.
• If the collision time is approximately zero,
• then the change in momentum is approximately
  zero.