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Chapter 9. Center of Mass and Linear Momentum

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Title: Chapter 9. Center of Mass and Linear Momentum


1
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     

2
What is physics?
central axis.)
3
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.

4
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

5
Locating a System's Center of Mass
The components of the center of mass of a system
of particles are
6
Velocity of center of mass
7
Acceleration of center of mass
8
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?

9
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.

10
Question
  • Where would you expect the center of mass of a
    doughnut to be located? Why?

11
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).

    
12
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.

13
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.

14
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.

15
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?

16
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.

17
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

18
Linear Momentum of a particle
  •   m is the mass of the particle
  • is its instantaneous velocity

19
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.

20
The Linear Momentum of a System of Particles
M is the mass of the system
21
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

22
Collision and Impulse
  • Impulse

The average impulse ltJgt
  • Impulse is a vector quantity
  • It has the same direction as the force

23
Linear Momentum-Impulse Theorem
24
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.

25
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.

26
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.

27
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.
28
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.

29
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.

30
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.

31
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?

32
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.

33
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.

34
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?

    


35
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)?







36
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.





37
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.) 
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