Chapter 9. Center of Mass and Linear Momentum

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Chapter 9. Center of Mass and Linear Momentum

• 9.1. What is Physics?     
• 9.2. The Center of Mass      
• 9.3. Newton's Second Law for a System of
  Particles      
• 9.4. Linear Momentum     
• 9.5. The Linear Momentum of a System of
  Particles     
• 9.6. Collision and Impulse    
• 9.7. Conservation of Linear Momentum    
• 9.8. Momentum and Kinetic Energy in
  Collisions    
• 9.9. Inelastic Collisions in One Dimension    
• 9.10. Elastic Collisions in One Dimension     
• 9.11. Collisions in Two Dimensions     

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What is physics?
central axis.)
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Defining the Position of a Complex Object
The effective position of the system is

• The effective position of a system of particles
  is the point that moves as though
• all of the systems mass were concentrated there
  and
• all external forces were applied there.

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N particles system

• The effective position is also called as the
  center of mass of a system. It represents the
  average location for the total mass of a system

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Locating a System's Center of Mass
The components of the center of mass of a system
of particles are
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Velocity of center of mass
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Acceleration of center of mass
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EXAMPLE 1 Three Masses

• Three particles of masses mA  1.2 kg, mB  2.5
  kg, and mC  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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Solid Bodies


If objects have uniform density,

• For objects such as a golf club, the mass is
  distributed symmetrically and the center-of-mass
  point is located at the geometric center of the
  objects.

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Question

• Where would you expect the center of mass of a
  doughnut to be located? Why?

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Checkpoint 1

• The figure shows a uniform square plate from
  which four identical squares at the corners will
  be removed. (a) Where is the center of mass of
  the plate originally? Where is it after the
  removal of (b) square 1 (c) squares 1 and 2 (d)
  squares 1 and 3 (e) squares 1, 2, and 3 (f) all
  four squares? Answer in terms of quadrants, axes,
  or points (without calculation, of course).

    
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EXAMPLE 2 U-Shaped Object

• The U-shaped object pictured in Fig. has
  outside dimensions of 100 mm on each side, and
  each of its three sides is 20 mm wide. It was cut
  from a uniform sheet of plastic 6.0 mm thick.
  Locate the center of mass of this object.

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Problem 3 Build your skill
    

• Figure 9-4a shows a uniform metal plate P of
  radius 2R from which a disk of radius R has been
  stamped out (removed) in an assembly line. Using
  the x-y coordinate system shown, locate the
  center of mass comP of the plate.

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Newton's Laws for a System of Particles

• is the net force of all external
  forces that act on the system.
• Msys 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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EXAMPLE 4 Center-of-Mass Acceleration

• The three particles in Fig. a 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
  FA6 N , FB12 N , and FC14 N. What is the
  magnitude of the acceleration of the center of
  mass of the system, and in what direction does it
  move?

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Collisions and Explosions

• A COLLISION or EXPLOSION is an isolated event
  in which two or more bodies exert relatively
  strong forces on each other over a short time
  compared to the period over which their motions
  take place.

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What is Properties of Collision?

• When objects collide or a large object explodes
  into smaller fragments, the event can happen so
  rapidly that it is impossible to keep track of
  the interaction forces

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Linear Momentum of a particle

•   m is the mass of the particle
• is its instantaneous velocity

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Newtons second law

• The rate of change of the momentum of a
  particle is proportional to the net force acting
  on the particle and is in the direction of that
  force.

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The Linear Momentum of a System of Particles
M is the mass of the system
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Newton's Laws

• The sum of all external forces acting on all
  the particles in the system is equal to the time
  rate of change of the total momentum of the
  system. That leaves us with the general
  statement

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Collision and Impulse

• Impulse

The average impulse ltJgt

• Impulse is a vector quantity
• It has the same direction as the force

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Linear Momentum-Impulse Theorem
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Check Your Understanding 1

• Suppose you are standing on the edge of a dock
  and jump straight down. If you land on sand your
  stopping time is much shorter than if you land on
  water. Using the impulsemomentum theorem as a
  guide, determine which one of the following
  statements is correct.
•   a.In bringing you to a halt, the sand exerts a
  greater impulse on you than does the water.   
• b.In bringing you to a halt, the sand and the
  water exert the same impulse on you, but the sand
  exerts a greater average force.  
•  c.In bringing you to a halt, the sand and the
  water exert the same impulse on you, but the sand
  exerts a smaller average force.

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Example 1  A Well-Hit Ball

• A baseball (m0.14 kg) has an initial velocity
  of v0 38 m/s as it approaches a bat. We have
  chosen the direction of approach as the negative
  direction. The bat applies an average force that
  is much larger than the weight of the ball, and
  the ball departs from the bat with a final
  velocity of vf58 m/s. (a) Determine the impulse
  applied to the ball by the bat. (b) Assuming that
  the time of contact is ?t1.6 103 s, find the
  average force exerted on the ball by the bat.

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Example 2  A Rain Storm

• During a storm, rain comes straight down with
  a velocity of v015 m/s and hits the roof of a
  car perpendicularly (see Figure ). The mass of
  rain per second that strikes the car roof is
  0.060 kg/s. Assuming that the rain comes to rest
  upon striking the car (vf0 m/s), find the
  average force exerted by the rain on the roof.

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Conservation of Momentum

• If no net external force acts on a system of
  particles, the total translational momentum of
  the system cannot change.

Note 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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Conceptual Example 4  Is the Total Momentum
Conserved?

• Imagine two balls colliding on a billiard
  table that is friction-free. Use the momentum
  conservation principle in answering the following
  questions. (a) Is the total momentum of the
  two-ball system the same before and after the
  collision? (b) Answer part (a) for a system that
  contains only one of the two colliding balls.

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Example 5

• Bullet and Two Blocks In Fig. a, a 3.40 g
  bullet is fired horizontally at two blocks at
  rest on a frictionless tabletop. The bullet
  passes through the first block, with mass 1.20
  kg, and embeds itself in the second, with mass
  1.80 kg. Speeds of 0.630 m/s and 1.40 m/s,
  respectively, are thereby given to the blocks
  (Fig.b). Neglecting the mass removed from the
  first block by the bullet, find (a) the speed of
  the bullet immediately after it emerges from the
  first block and (b) the bullet's original speed.

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Example 7

• The drawing shows a collision between two
  pucks on an air-hockey table. Puck A has a mass
  of 0.025 kg and is moving along the x axis with a
  velocity of 5.5 m/s. It makes a collision with
  puck B, which has a mass of 0.050 kg and is
  initially at rest. The collision is not head-on.
  After the collision, the two pucks fly apart with
  the angles shown in the drawing. Find the final
  speed of (a) puck A and (b) puck B.

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Sample Problem 9

• Two-dimensional explosion 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. 9-14a. Piece C,
  with mass 0.30M, has final speed vfc5.0m/s. (a)
  What is the speed of piece B, with mass 0.20M?
  (b) What is the speed of piece A?

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Momentum and Kinetic Energy in Collisions

• If the collision occurs in a very short time
  or external forces can be ignored, the momentum
  of system is conserved.

• If the kinetic energy of the system is conserved,
  such a collision is called an elastic collision.
• If the kinetic energy of the system is not
  conserved, such a collision is called an
  inelastic collision.
• The inelastic collision of two bodies always
  involves a loss in the kinetic energy of the
  system. The greatest loss occurs if the bodies
  stick together, in which case the collision is
  called a completely inelastic collision.

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Velocity of the Center of Mass

• In a closed, isolated system, the velocity of
  the center of mass of the system cannot be
  changed by a collision because, with the system
  isolated, there is no net external force to
  change it.

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Example of elastic collision

• Two metal spheres, suspended by vertical cords,
  initially just touch, as shown in Fig. 9-22.
  Sphere 1, with mass m130 g, is pulled to the
  left to height h18.0cm, and then released from
  rest. After swinging down, it undergoes an
  elastic collision with sphere 2, whose mass m275
  g. What is the velocity v1f of sphere 1 just
  after the collision?

    


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Example of elastic collision

• A small ball of mass m is aligned above a
  larger ball of mass M0.63 kg (with a slight
  separation, as with the baseball and basketball
  of Fig. 9-70a), and the two are dropped
  simultaneously from a height of h1.8m. (Assume
  the radius of each ball is negligible relative to
  h.) (a) If the larger ball rebounds elastically
  from the floor and then the small ball rebounds
  elastically from the larger ball, what value of m
  results in the larger ball stopping when it
  collides with the small ball? (b) What height
  does the small ball then reach (Fig. 9-70b)?

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Example of inelastic collision

• In the before part of Fig. 9-60, car A (mass
  1100 kg) is stopped at a traffic light when it is
  rear-ended by car B (mass 1400 kg). Both cars
  then slide with locked wheels until the
  frictional force from the slick road (with a low
  µk of 0.13) stops them, at distances dA8.2m and
  dB6.1m . What are the speeds of (a) car A and
  (b) car B at the start of the sliding, just after
  the collision? (c) Assuming that linear momentum
  is conserved during the collision, find the speed
  of car B just before the collision. (d) Explain
  why this assumption may be invalid.

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Example of completely inelastic collision

•  A completely inelastic collision occurs
  between two balls of wet putty that move directly
  toward each other along a vertical axis. Just
  before the collision, one ball, of mass 3.0 kg,
  is moving upward at 20 m/s and the other ball, of
  mass 2.0 kg, is moving downward at 12 m/s. How
  high do the combined two balls of putty rise
  above the collision point? (Neglect air drag.)