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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Momentum and Impulse
Obeys a conservation law Simplifies complicated
motions Describes collisions Basis of rocket
propulsion space travel
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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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Newtons Second Law

• 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,

When then
then
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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 Non-identical Carts
Kinetic Energy

• Consider two carts, of masses m and 2m, at
  rest on an air track. If you push one cart for 3
  s 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 Same Momentum, Different Masses

• 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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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 the velocity is

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

• Total mass for discrete or continuous body (mass
  density ?)
• 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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Internal Force on a System of N Particles

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

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Total Force on a System of N Particles is the
External Force

• The total force on a system of particles is the
  sum of the total external and total internal
  forces. Since the total internal force is zero

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External Force and Momentum Change

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

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System of Particles Newtons Second and Third
Laws
The total momentum of a system remains constant
unless the system is acted on by an external force
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System of Particles Translational Motion of the
Center of Mass
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Translational Motion of the Center of Mass

• Momentum of system
• System can be treated as point mass located at
  center of mass. External force accelerates center
  of mass
• Impulse changes center of mass momentum

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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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Concept Question Pushing a Baseball Bat Recall
Issue with Phrasing
1
3
2

• The greatest 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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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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System of ParticlesConservation of Momentum
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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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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.

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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?