Title: Chapter 9. Center of Mass and Linear Momentum
1Chapter 9. Center of Mass and Linear Momentum
- 9.1. What is Physics?    Â
- 9.2. The Center of Mass     Â
- 9.3. Newton's Second Law for a System of
Particles      - 9.4. Linear Momentum    Â
- 9.5. The Linear Momentum of a System of
Particles     - 9.6. Collision and Impulse   Â
- 9.7. Conservation of Linear Momentum   Â
- 9.8. Momentum and Kinetic Energy in
Collisions    - 9.9. Inelastic Collisions in One Dimension   Â
- 9.10. Elastic Collisions in One Dimension    Â
- 9.11. Collisions in Two Dimensions    Â
2What is physics?
central axis.)
3Defining 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
5Locating a System's Center of Mass
The components of the center of mass of a system
of particles are
6Velocity of center of mass
7Acceleration of center of mass
8 EXAMPLE 1Â Three Masses
- Three particles of masses mAÂ Â 1.2 kg, mBÂ Â 2.5
kg, and mCÂ Â 3.4 kg form an equilateral triangle
of edge length a  140 cm. Where is the center of
mass of this three-particle system?
9Solid 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.
10Question
- Where would you expect the center of mass of a
doughnut to be located? Why?
11Checkpoint 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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12EXAMPLE 2Â 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.
13Problem 3 Build your skill
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- 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.
14Newton'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.
15EXAMPLE 4Â 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?
16Collisions 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.
17What 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
18Linear Momentum of a particle
- Â Â m is the mass of the particle
- is its instantaneous velocity
19Newtons 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.
20The Linear Momentum of a System of Particles
M is the mass of the system
21Newton'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
22Collision and Impulse
The average impulse ltJgt
- Impulse is a vector quantity
- It has the same direction as the force
23Linear Momentum-Impulse Theorem
24Check 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. - Â Â a.In bringing you to a halt, the sand exerts a
greater impulse on you than does the water. Â Â - 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. Â - Â 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.
25Example 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.
26Example 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.
28Conceptual 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.
29Example 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.
30Example 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.
31Sample 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?
32Momentum 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.
33Velocity 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.
34Example 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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35Example 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)?
36Example 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.
37Example of completely inelastic collision
- Â 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.)Â