Chapter 9 Systems of Particles

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Chapter 9 Systems of Particles Linear Momentum

• System of Particles
• Center of Mass for a System of Particles
• Center of Mass for a Solid Body
• Newtons 2nd Law for a System of Particles
• Linear Momentum
• Collision and Impulse
• Linear Momentum for a System of Particles
• Conservation of Linear Momentum
• Momentum and Kinetic Energy, Elastic and
  Inelastic Collsion

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• Center of Mass

The center of mass (com) of a body or a system
of bodies is the point that moves as though all
of the mass were concentrated there and all
external forces were applied there.
A Special Point
Center of mass for two particles
where M m1m2 is the total mass
In general (many particles) and for a 3-D
distribution
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• In the language of the vectors

• The three scalar equation can now be replaced by
  a single vector equation

• Solid Bodies
• A solid body can be thought as consisting of
  infinite number of small mass element dm. The
  particles then becomes the differential mass
  elements dm the sum becomes the integral

• For objects having uniform density r

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A method for finding the center of mass of any
object.

• Hang object from two or more points.
• Draw extension of suspension line.
• Center of mass is at intercept of these lines.

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Sample Problem 9-1
Three particles of masses m1 1.2 kg, m2 2.5
kg, and m3 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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5E
A uniform square plate 6 m on a side has had a
square piece 2 m on a side cut out of it. The
center of that piece is at x 2 m, y 0. The
center of the square plate is at x y 0. Find
the coordinates of the center of mass of the
remaining piece.
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• Newtons Second Law for a system of Particles

The center of mass of a system of particles
(combined mass M) moves like one equivalent
particle of mass M would move under the influence
of an external force.
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Sample Problem 9-3
The three particles in figure (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
F1 6.0 N, F2 12 N, and F3 14 N. What is the
acceleration of the center of mass of the system,
and in what direction does it move?
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• The Linear Momentum

The linear momentum of a particle of mass m and
velocity v is defined as
The linear momentum is a vector quantity. Its
direction is along v.
The components of the momentum of a particle
From Newtons second law
The time rate of change in linear momentum is
equal to the net forces acting on the particle.
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• The Linear Momentum of a System of Particles

For a system of particles
Taking the time derivative of total linear
momentum of the system of particles
For a system of particles
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• Collision and Impulse

What is a Collision?
A collision is an isolated event in which two or
more bodies (the colliding bodies) exert
relatively strong forces on each other for a
relatively short time.
Impulse and Linear Momentum
single collision
Two particle-like bodies L and R collide with
each other. During the collision, body L exert
force on body R, and body R exerts
force on body L a third-law force pair.
Their magnitude vary with time during the
collision.
Impulse is a vector quantity, the direction is
the same as the direction of the force
For a constant (average) force
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Impulse causes a change in momentum Linear
Momentum Impulse Theory
Think of hitting a soccer ball A force F acting
over a time Dt causes a change Dp in the momentum
(velocity) of the ball.
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Example Problem

• A soccer player hits a ball (mass m 440 g)
  coming at him with a velocity of 20 m/s. After it
  was hit, the ball travels in the opposite
  direction with a velocity of 30 m/s.
• What impulse acts on the ball while it is in
  contact with the foot?
• The impact time is 0.1s. What is force the acting
  on the ball?

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Example Impulse Applied to Auto Collisions

• The most important factor is the collision time
  or the time it takes the person to come to a rest
• This will reduce the chance of dying in a car
  crash
• Ways to increase the time
• Seat belts
• Air bags

• The air bag increases the time of the collision
  and absorbs some of the energy
  from the body

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ConcepTest 1
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 time intervals to stop them compare? 1. It
takes less time to stop the ping-pong ball. 2.
Both take the same time. 3. It takes more time
to stop the ping-pong ball.
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Impulse on target - impulse on n projectiles
where is the momentum change for each
projectile.

• If the projectiles stop after impact, then

• If the projectiles bounce backward, with
  , then

In the time interval , an amount of mass
collides with the target, therefore
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Conservation of linear momentum

• Conservation of Linear Momentum

If no external force (isolated) is acting on a
particle, its momentum is conserved. This is
also true for a system of particles If no
external forces (isolated) interact with a closed
system of particles the total momentum of the
system remains constant.
(For a closed, isolated system)
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Sample Problem 9-5
A ballot box with mass m 6.0 kg slides with
speed v 4.0 m/s across a frictionless floor in
the positive direction of an x axis. It suddenly
explodes into two pieces. One piece, with mass m1
2.0 kg, moves in the positive direction of the
x axis with speed v1 8.0 m/s. What is the
velocity of the second piece, with mass m2 ?
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Sample Problem 9-7
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.
9-10a. Piece C, with mass 0.30M, has final speed
vfC 5.0 m/s.
(a)  What is the speed of piece B, with mass
0.20M?
The net external force acting on our closed
system (coconut) is zero, the explosive forces
are internal meaning this is an isolated
system. Therefore the linear momentum of this
system is conserved.
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Sample Problem 9-7
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.
9-10a. Piece C, with mass 0.30M, has final speed
vfC 5.0 m/s.
(a)  What is the speed of piece A?
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• Momentum and Kinetic Energy in Collisions

• Closed System (no mass enters or leaves)
• Isolated System (no external net force)
• Elastic Collision (kinetic energy conserved)
• Inelastic Collision (kinetic energy not
  conserved)
• Completely Inelastic Collision (bodies always
  stick together)

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Types of Collisions

• Momentum is conserved in any collision
• what about kinetic energy?
• Inelastic collisions
• Kinetic energy is not conserved
• Some of the kinetic energy is converted into
  other types of energy such as heat, sound, work
  to permanently deform an object
• Completely inelastic collisions occur when the
  objects stick together
• Maximum KE is converted to other forms of energy.
• Elastic collision
• Kinetic energy is (also) conserved

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ConcepTest 2
Suppose a person jumps on the surface of Earth.
The Earth 1. will not move at all 2. will
recoil in the opposite direction with tiny
velocity 3. might recoil, but there is not
enough information provided to see if that
could happen
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ConcepTest 2
Suppose a person jumps on the surface of Earth.
The Earth 1. will not move at all 2. will
recoil in the opposite direction with tiny
velocity 3. might recoil, but there is not
enough information provided to see if that
could happen
Convince your neighbor!
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ConcepTest 2
Suppose a person jumps on the surface of Earth.
The Earth 1. will not move at all 2. will
recoil in the opposite direction with tiny
velocity 3. might recoil, but there is not
enough information provided to see if that
could happen

Note momentum is conserved. Lets estimate
Earths velocity after a jump by a 80-kg person.
Suppose that initial speed of the jump is 4 m/s,
then
tiny negligible velocity, in opposite direction
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• Inelastic Collision in One Dimension

Conservation of Linear Momentum
Total Linear Momentum is conserved even when the
collision is inelastic, in a closed, isolated
system
Completely Inelastic Collision
Special case in which v2i 0 and the two bodies
stick together after the collision. Note that the
velocity V after the collision depends on the
ratio of m1 and (m1 m2).
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• Inelastic Collision in One Dimension

Velocity of Center of Mass
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Sample Problem 9-8
The ballistic pendulum was used to measure the
speeds of bullets before electronic timing
devices were developed. The version shown in Fig.
10-11 consists of a large block of wood of mass M
5.4 kg, hanging from two long cords. A bullet
of mass m 9.5 g is fired into the block, coming
quickly to rest. The block bullet then swing
upward, their center of mass rising a vertical
distance h 6.3 cm before the pendulum comes
momentarily to rest at the end of its arc. What
is the speed of the bullet just prior to the
collision?
Energy is not conserved before and after the
collision. Only the momentum
Where V is the velocity of the bulletbock system
just after collision.
Mechanical Energy is conserved during the swing
of the pendulum after the collision
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• Elastic Collision in One Dimension

In an elastic collision, the kinetic energy of
each colliding body may change, but the total
kinetic energy of the system does not change. The
total KE is also conserved in addition to the
total momentum.
(v2i 0)
Moving Target
Stationary Target
In the special case that m1 is much, much larger
than m2
If m1m2
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• Collision in Two Dimension

Special case when target body is at rest before
the collision (v2i 0)
This is true only if the collision is ELASTIC
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P55
A cart with mass 340 g moving on a frictionless
linear air track at an initial speed of 1.2 m/s
undergoes an elastic collision with an initially
stationary cart of unknown mass. After the
collision, the first cart continues in its
original direction at 0.66 m/s. (a) What is the
mass of the second cart? (b) What is the speed of
the two-cart center of mass?
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Exam2 042 - P13
A 2.0 kg block with a speed of 4.0 m/s undergoes
a head on ELASTIC collision with a 4.0 kg block
initially at rest. After the collision, the 4.0
kg block has 14.2 J of kinetic energy . The speed
of the 2.0 kg block after the collision is (1.3
m/s)
Exam2 022 - P13
Sphere A of mass 200 g is moving with VAi 6.0
m/s. It makes a head-on collision with sphere B
of mass 400 g at rest. After collision sphere B
moves with VBf 3.0 m/s. What is the velocity
of sphere A after collision? (0)
Exam2 032 - P13
Body A with mass m moves along an x axis with
kinetic energy of 9.0 J before having an elastic
collision with body B with the same mass m ,
which is initially at rest. What is the final
kinetic energy of B? (9.0 J)