Transcript Chapter 9. Center of Mass and Linear Momentum

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
What is physics?
Defining the Position of a Complex
Object
The effective “position” of the system is:
M sys  mA  mB
The effective “position” of a system of particles is the point
that moves as though
(1)all of the system’s mass were concentrated there and
(2) all external forces were applied there.
N particles system
R
eff
1

(m1r1  m2 r2    mN rN )
M sys
M sys  m1  m2    mN
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
Locating a System's Center of Mass
The components of the center of mass of a system of
particles are:
xCOM
1

(m1 x1  m2 x2    mN xN )
M sys
yCOM
1

(m1 y1  m2 y2    mN y N )
M sys
zCOM
1

(m1 z1  m2 z2    mN z N )
M sys
M sys  m1  m2    mN
Velocity of center of mass
vcom
m1v1  m2 v2    mN vN

m1  m2    mN
Acceleration of center of mass
acom
m1a1  m2 a2    mN aN

m1  m2    mN
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?
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.
Question:
Where would you expect the center of
mass of a doughnut to be located? Why?
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).
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.
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.
Newton's Laws for a System of Particles
net
sys
F
 M sys acom
net
F
• sys 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.
• acom 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.
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 FA=6 N ,
FB=12 N , and FC=14 N. What is
the magnitude of the
acceleration of the center of
mass of the system, and in what
direction does it move?
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.
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
Linear Momentum of a particle
•
•
m is the mass of the particle
is its instantaneous velocity
Newton’s 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.
The Linear Momentum of a System of Particles
M is the mass of the system
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:
Collision and Impulse
Impulse:
The average impulse <J> :
•Impulse is a vector quantity
•It has the same direction as the force
J  F t
Linear Momentum-Impulse Theorem

tf
ti
dP  Pf  Pi
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
impulse–momentum 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.
Example 1 A Well-Hit Ball
A baseball (m=0.14 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 vf=+58 m/s. (a) Determine the impulse
applied to the ball by the bat. (b) Assuming that
the time of contact is Δt=1.6 × 10–3 s, find the
average force exerted on the ball by the bat.
Example 2 A Rain Storm
During a storm, rain comes straight down with a velocity
of v0=–15 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 (vf=0 m/s), find the average
force exerted by the rain on the roof.
Conservation of Momentum
Fnet 
dPsys
dt
If no net external force acts on a system of
particles, the total translational momentum of
the system cannot change.
Psys (t1 )  Psys (t2 )
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.
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.
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.
Example 7
The drawing shows a collision
between two pucks on an airhockey 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.
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 vfc=5.0m/s. (a) What is
the speed of piece B, with mass 0.20M? (b) What is the
speed of piece A?
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.
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.
Example of elastic collision
• Two metal spheres, suspended by vertical cords,
initially just touch, as shown in Fig. 9-22. Sphere 1,
with mass m1=30 g, is pulled to the left to height
h1=8.0cm, and then released from rest. After swinging
down, it undergoes an elastic collision with sphere 2,
whose mass m2=75 g. What is the velocity v1f of
sphere 1 just after the collision?
Example of elastic collision
A small ball of mass m is aligned
above a larger ball of mass M=0.63 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
h=1.8m. (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. 970b)?
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 dA=8.2m and dB=6.1m .
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.
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.)