Transcript Lecture-10-10

Chapter 7
Work and
Kinetic Energy
http://people.virginia.edu/~kdp2c/downloads/WorkEnergySelections.html
Conservative and Nonconservative Forces
Conservative force:
- the work it does is stored in the form of energy that
can be released at a later time
- the work done by a conservative force moving an
object around a closed path is zero
- Force depends upon position only
Example of a conservative force: gravity
Example of a nonconservative force: friction
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Work done by gravity on a closed path is zero
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Work done by friction on a closed path is not zero
4
The work done by a conservative force is zero on
any closed path
Go A-B on path 1, the back B-A.
Wt = W1 + -W1
Go A-B on path 1, the B-A on
path 2.
Wt = W1 + -W2
So the work must be reversible (opposite when taking the
same path) AND path independent (same amount of work
for any two different paths connecting two points)
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Potential Energy
If we pick up a ball and put it on the shelf, we have
done work on the ball. We can get that energy back if
the ball falls back off the shelf (gravity does positive
work on the ball, “releasing” the work that we put in
before).
Until that happens, we say the energy is stored as
potential energy.
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Potential Energy
Consider the process
in which the book goes
from h=0 to h=0.50 m
Work done by gravity:
W = - (mg)h = -13.5 J
For the book to go up against gravity, another force
must be applied to overcome the weight. This other
force did a (minimum) work of 13.5 J
If I lft the book steadily, the “external force” is provided by
my hand with F~mg, work done by me: W=(mg)h = 13.5 J
The book’s potential energy changed by: 13.5 J
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Potential Energy
The work done against a conservative force is stored
in the form of (potential) energy that can be released
at a later time.
Note the minus sign:
•positive Wc (work by the conservative force) is negative
potential energy (energy is released)
•negative Wc is positive potential energy (another force as
done work against the conservative force)
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Gravitational Potential Energy
Q: What does “UG = 0” mean?
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Work Done by a Variable Force
on the spring
The force needed to stretch a
spring an amount x is F = kx.
Therefore, the work done in
stretching (or compressing)
the spring is
with positive work applied
leading to a positive change
in potential: W = Uf - Ui
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Potential energy in a spring
The corresponding conservative force is the force of the spring
acting on the hand: positive work by the spring releases
potential energy Wc = - ΔU
So, taking U=0 at x=0:
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Up the Hill
Two paths lead to the top of a big
hill. One is steep and direct, while
the other is twice as long but less
steep. How much more potential
energy would you gain if you take
the longer path?
a) the same
b) twice as much
c) four times as much
d) half as much
e) you gain no PE in either
case
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Up the Hill
Two paths lead to the top of a big
hill. One is steep and direct, while
the other is twice as long but less
steep. How much more potential
energy would you gain if you take
the longer path?
a) the same
b) twice as much
c) four times as much
d) half as much
e) you gain no PE in either
case
Because your vertical position (height) changes by
the same amount in each case, the gain in potential
energy is the same.
Follow-up: How much more work do you do in taking the steeper path?
Follow-up: Which path would you rather take? Why?
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Sign of the Energy
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
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Sign of the Energy
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
Gravitational PE is mgh, where height h is measured relative to some
arbitrary reference level where PE = 0. For example, a book on a table
has positive PE if the zero reference level is chosen to be the floor.
However, if the ceiling is the zero level, then the book has negative PE
on the table. Only differences (or changes) in PE have any physical
meaning.
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KE and PE
You and your friend both solve a
problem involving a skier going down a
slope, starting from rest. The two of
you have chosen different levels for y
= 0 in this problem. Which of the
following quantities will you and your
friend agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
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KE and PE
You and your friend both solve a
problem involving a skier going down a
slope, starting from rest. The two of
you have chosen different levels for y
= 0 in this problem. Which of the
following quantities will you and your
friend agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
The gravitational PE depends upon the reference level, but the
difference  PE does not! The work done by gravity must be
the same in the two solutions, so PE and KE should be the
same.
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Mechanical Energy
Consider the total amount of work done on a body by
the conservative and the non-conservative forces. This
is the change in kinetic energy (work-energy theorem)
It is useful to define the mechanical energy:
Then:
The work done by all non-conservative forces is the
change in the mechanical energy of a body
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Conservation of Mechanical Energy
The work done by all non-conservative forces is the
change in the mechanical energy of a body
If there are only conservative forces doing work during
a process, we find:
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Work-Energy Theorem vs. Conservation of
Energy?
Work-Energy Theorem
total work done (by both conservative and nonconservative forces) = change in kinetic energy
Conservation of mechanical energy
total work done by non-conservative forces
= change in mechanical energy
These two are completely equivalent. The
difference is only how to treat conservative forces.
Do NOT use both potential energy AND work by
the conservative force... that’s double-counting!
In general, energy conservation makes
kinematics problems much easier to solve...
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Runaway Truck
A truck, initially at rest, rolls
a) half the height
down a frictionless hill and
attains a speed of 20 m/s at the
bottom. To achieve a speed of
40 m/s at the bottom, how many
times higher must the hill be?
b) the same height
c) < 2 times the height
d) twice the height
e) four times the height
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Runaway Truck
A truck, initially at
rest, rolls down a
frictionless hill and
attains a speed of 20
m/s at the bottom.
UseTo
energy
conservation:
achieve
a speed
 initial energy: Ei = PEg = mgH
of 40 m/s at the 2
 final energy: Ef = KE = mv
bottom,
how many
Conservation
of Energy:
2
E
=
mgH
=
E
=
mv
i
f
times
higher
must
therefore: gH = v2
the
hill
be?
So if v doubles, H quadruples!
a) half the height
b) the same height
c) < 2 times the height
d) twice the height
e) four times the height
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Cart on a Hill
A cart starting from rest rolls down a hill and
at the bottom has a speed of 4 m/s. If the
cart were given an initial push, so its initial
speed at the top of the hill was 3 m/s, what
would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
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Cart on a Hill
A cart starting from rest rolls down a hill and
at the bottom has a speed of 4 m/s. If the
cart were given an initial push, so its initial
speed at the top of the hill was 3 m/s, what
would be its speed at the bottom?
a) 4 m/s
b) 5 m/s
c) 6 m/s
d) 7 m/s
e) 25 m/s
When starting from rest, the
cart’s PE is changed into KE:
 PE =  KE = m(4)2
When starting from 3 m/s, the
final KE is:
KEf = KEi +  KE
= m(3)2 + m(4)2
=
m(25)
= m(5)2
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Potential Energy Curves
The curve of a hill or a roller coaster is itself
essentially a plot of the gravitational potential
energy:
Q: at what point is speed maximized?
Q: where might apparent weight be minimized?
Potential Energy for a Spring
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Potential Energy Curves and
Equipotentials
Contour maps are also a form of potential energy curve:
Each contour is an equal
height, and so an
“equipotential” for
gravitational potential
energy
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Question 8.5 Springs and Gravity
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
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Question 8.5 Springs and Gravity
A mass attached to a vertical
spring causes the spring to
stretch and the mass to move
downwards. What can you
say about the spring’s
potential energy (PEs) and the
gravitational potential energy
(PEg) of the mass?
a) both PEs and PEg decrease
b) PEs increases and PEg decreases
c) both PEs and PEg increase
d) PEs decreases and PEg increases
e) PEs increases and PEg is constant
The spring is stretched, so its elastic PE increases,
because PEs =
kx2. The mass moves down to a
lower position, so its gravitational PE decreases,
because PEg = mgh.
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8-4 Work Done by Nonconservative Forces
In this example, the
nonconservative force
is water resistance:
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Chapter 9
Linear Momentum
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Linear Momentum
Momentum is a vector; its direction is the same
as the direction of the velocity.
Momentum is a vector
Change in momentum:
(a) mv
(b) 2mv
Going Bowling I
A bowling ball and a Ping-Pong ball are
rolling toward you with the same
momentum. Which one of the two has
the greater kinetic energy?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
p
p
Going Bowling I
A bowling ball and a Ping-Pong ball are
rolling toward you with the same
momentum. Which one of the two has
the greater kinetic energy?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
Momentum is p = mv
so the ping-pong ball must have a
much greater velocity
Kinetic Energy is KE = 1/2 mv2
so (for a single object): KE = p2 / 2m
p
p
Momentum and Newton’s Second Law
Newton’s second law, as we wrote it before:
is only valid for objects that have constant mass.
Here is a more general form (also useful when the
mass is changing):
Momentum and Force
A net force of 200 N acts on a 100-kg
boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to the
rate of change of the pebble’s
momentum?
a) greater than
b) less than
c) equal to
Momentum and Force
A net force of 200 N acts on a 100-kg
boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to the
a) greater than
b) less than
c) equal to
rate of change of the pebble’s
momentum?
The rate of change of momentum is, in fact, the force.
Remember that F = p/t. Because the force exerted on
the boulder and the pebble is the same, then the rate of
change of momentum is the same.
Impulse
The same change in momentum may be produced
by a large force acting for a short time, or by a
smaller force acting for a longer time.
Impulse quantifies the overall change in momentum
Impulse is a vector, in the same direction
as the average force.
Impulse
We can rewrite
as
So we see that
The impulse is equal to the change in momentum.
Why we don’t dive into concrete
The same change in momentum may be produced
by a large force acting for a short time, or by a
smaller force acting for a longer time.
Going Bowling II
A bowling ball and a Ping-Pong ball are
rolling toward you with the same
momentum. If you exert the same force
to stop each one, which takes a longer
time to bring to rest?
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
p
p
Going Bowling II
A bowling ball and a Ping-Pong ball are
rolling toward you with the same
momentum. If you exert the same force
to stop each one, which takes a longer
time to bring to rest?
We know:
p
Fav =
t
a) the bowling ball
b) same time for both
c) the Ping-Pong ball
d) impossible to say
so p = Fav t
Here, F and p are the same for both balls!
It will take the same amount of time to
stop them.
p
p
Going Bowling III
A bowling ball and a Ping-Pong ball
a) the bowling ball
are rolling toward you with the
b) same distance for both
same momentum. If you exert the
c) the Ping-Pong ball
same force to stop each one, for
d) impossible to say
which is the stopping distance
greater?
p
p
Going Bowling III
A bowling ball and a Ping-Pong ball
a) the bowling ball
are rolling toward you with the
b) same distance for both
same momentum. If you exert the
c) the Ping-Pong ball
same force to stop each one, for
d) impossible to say
which is the stopping distance
greater?
Use the work-energy theorem: W = KE.
The ball with less mass has the greater
speed, and thus the greater KE. In order to
remove that KE, work must be done, where W
= Fd. Because the force is the same in both
cases, the distance needed to stop the less
massive ball must be bigger.
p
p
Conservation of Linear Momentum
The net force acting on an object is the rate of
change of its momentum:
If the net force is zero, the momentum
does not change!
With no net force:
•A vector equation
•Works for each coordinate separately
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.
Momentum of components of a system
Internal forces cannot change the
momentum of a system.
However, the momenta of pieces of the
system may change.
With no net external force:
An example of internal forces moving
components of a system:
Kinetic Energy of a System
Another example of internal forces
moving components of a system:
The initial momentum
equals the final (total)
momentum.
But the final Kinetic Energy
is very large
Birth of the neutrino
Beta decay fails momentum
conservation?
First detection 1956
Pauli “fixes” it with a new ghost-like, undetectable
particle
Bohr scoffs
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