Download Chapter 9 (Cont.) Linear Momentum and Collisions

Chapter 9 (Cont.)
Linear Momentum and
Collisions
Copyright © 2010 Pearson Education, Inc.
Units of Chapter 9
• Linear Momentum
• Momentum and Newton’s Second Law
• Impulse
• Conservation of Linear Momentum
• Inelastic Collisions
• Elastic Collisions
• Center of Mass
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Review: 9-1 Linear Momentum
Momentum is a vector; its direction is the
same as the direction of the velocity.
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Review: 9-2 Momentum and Newton’s
Second Law
Newton’s second law, Form I:
Form II:
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Review: 9-3 Impulse
Impulse is a vector, in the same direction
as the average force.
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Review: 9-3 Impulse
We can rewrite
(from Newton’s 2nd law)
as
So we see that
The impulse is equal to the change in
momentum.
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Review: 9-4 Conservation of Linear
Momentum
Internal Versus External Forces:
Internal forces act between objects within the system.
As with all forces, they occur in action-reaction pairs. As
all pairs act between objects in the system, the internal
forces always sum to zero:
Therefore, the net force acting on a system is
the sum of the external forces acting on it.
net external force=0Æ conservation of momentum
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Review: 9-5 Inelastic Collisions
Time of collision is short enough that external
forces may be ignored
Inelastic collision: momentum is conserved but
kinetic energy is not
Completely inelastic collision: objects stick
together afterwards
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9-6 Elastic Collisions
In elastic collisions, both kinetic energy and
momentum are conserved.
One-dimensional elastic collision:
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9-6 Elastic Collisions
We have two equations (conservation of
momentum and conservation of kinetic energy)
and two unknowns (the final speeds). Solving
for the final speeds:
Q: what if m1>>m2? Or m2>>m1?
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Question 9.10a Elastic Collisions I
Consider two elastic collisions:
1) a golf ball with speed v hits a
stationary bowling ball head-on.
2) a bowling ball with speed v
hits a stationary golf ball head-on. In
which case does the golf ball have the
greater speed after the collision?
v
a) situation 1
b) situation 2
c) both the same
at rest
at rest
1
v
2
Question 9.10a Elastic Collisions I
Consider two elastic collisions:
1) a golf ball with speed v hits a
stationary bowling ball head-on.
2) a bowling ball with speed v
hits a stationary golf ball head-on. In
which case does the golf ball have the
greater speed after the collision?
Remember that the magnitude of the
relative velocity has to be equal before
and after the collision!
a) situation 1
b) situation 2
c) both the same
v
1
In case 1 the bowling ball will almost
remain at rest, and the golf ball will
bounce back with speed close to v.
In case 2 the bowling ball will keep going
with speed close to v, hence the golf ball
will rebound with speed close to 2v.
v
2v
2
9-6 Elastic Collisions
Two-dimensional collisions can only be solved if
some of the final information is known, such as
the final velocity of one object:
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(Ki=Kf)
(Py=0)
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9-7 Center of Mass
Q: Where do you draw the gravitational force for an object in a free-body diagram?
The center of mass of a system is the point where
the system can be balanced in a uniform
gravitational field.
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9-7 Center of Mass
For two objects:
The center of mass is closer to the more
massive object.
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Question 9.20 Center of Mass
The disk shown below in (1) clearly has
its center of mass at the center.
a) higher
b) lower
Suppose the disk is cut in half and the
pieces arranged as shown in (2).
Where is the center of mass of (2) as
compared to (1) ?
c) at the same place
d) there is no definable
CM in this case
(1)
X
CM
(2)
Question 9.20 Center of Mass
The disk shown below in (1) clearly has
its center of mass at the center.
a) higher
b) lower
Suppose the disk is cut in half and the
pieces arranged as shown in (2).
c) at the same place
Where is the center of mass of (2) as
compared to (1) ?
The CM of each half is closer
to the top of the semicircle
than the bottom. The CM of
the whole system is located
at the midpoint of the two
semicircle CMs, which is
higher than the yellow line.
d) there is no definable
CM in this case
(1)
X
CM
(2)
CM
9-7 Center of Mass
The center of mass need not be within the object:
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Center of mass for a system of many objects:
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(a)
(b)
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9-7 Center of Mass
Motion of the center of mass:
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9-7 Center of Mass
The total mass multiplied by the acceleration of
the center of mass is equal to the net external
force:
The center of mass
accelerates just as
though it were a point
particle of mass M
acted on by
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Question 9.19 Motion of CM
Two equal-mass particles
(A and B) are located at
some distance from each
other. Particle A is held
stationary while B is
moved away at speed v.
What happens to the
center of mass of the
two-particle system?
a) it does not move
b) it moves away from A with speed v
c) it moves toward A with speed v
d) it moves away from A with speed ½v
e) it moves toward A with speed ½v
Question 9.19 Motion of CM
Two equal-mass particles
(A and B) are located at
some distance from each
other. Particle A is held
stationary while B is
moved away at speed v.
What happens to the
center of mass of the
two-particle system?
a) it does not move
b) it moves away from A with speed v
c) it moves toward A with speed v
d) it moves away from A with speed ½v
e) it moves toward A with speed ½v
Let’s say that A is at the origin (x = 0) and B is at
some position x. Then the center of mass is at x/2
because A and B have the same mass. If v = Δx/Δt
tells us how fast the position of B is changing,
1 2
then the position of the center of mass must be
changing like Δ(x/2)/Δt, which is simply v.
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(a)
(b)
(conservation of momentum)
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Summary of Chapter 9
•Elastic collision: kinetic energy is conserved
• Center of mass:
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Summary of Chapter 9
• Center of mass:
• Motion of center of mass:
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