Transcript 8-1 Conservative and Nonconservative Forces The work done by a

Chapter 8
Potential Energy and
Conservation of Energy
Units of Chapter 8
• Conservative and Nonconservative
Forces
• Potential Energy and the Work Done by
Conservative Forces
• Conservation of Mechanical Energy
• Work Done by Nonconservative Forces
• Potential Energy Curves and
Equipotentials
8-1 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
Example of a conservative force: gravity
Example of a nonconservative force: friction
Also: the work done by a conservative force
moving an object around a closed path is zero;
this is not true for a nonconservative force
8-1 Conservative and Nonconservative
Forces
Work done by gravity on a closed path is zero:
8-1 Conservative and Nonconservative
Forces
Work done by friction on a closed path is not
zero:
8-1 Conservative and Nonconservative
Forces
The work done by a conservative force is zero
on any closed path:
Example
• Calculate the work
one by gravity as
a 3.2 kg object is
moved from point
A to point B on the
figure along paths
1, 2, and 3
8-2 The Work Done by Conservative Forces
(8-1)
8-2 The Work Done by Conservative Forces
Gravitational potential energy:
8-2 The Work Done by Conservative Forces
Springs:
(8-4)
ConcepTest 8.4 Elastic Potential Energy
How does the work required to
1) same amount of work
stretch a spring 2 cm compare
2) twice the work
with the work required to
3) 4 times the work
stretch it 1 cm?
4) 8 times the work
ConcepTest 8.4 Elastic Potential Energy
How does the work required to
1) same amount of work
stretch a spring 2 cm compare
2) twice the work
with the work required to
3) 4 times the work
stretch it 1 cm?
4) 8 times the work
The elastic potential energy is 1/2 kx2. So in the second case,
the elastic PE is 4 times greater than in the first case. Thus,
the work required to stretch the spring is also 4 times greater.
Pre- and Post- Tests
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Percent
40
Pre Test
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Post Test
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0
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Number
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8-3 Conservation of Mechanical Energy
Definition of mechanical energy:
(8-6)
Using this definition and considering only
conservative forces, we find:
Or equivalently:
8-3 Conservation of Mechanical Energy
Energy conservation can make kinematics
problems much easier to solve:
ConcepTest 8.2 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
1) only B
2) only C
3) A, B and C
4) only A and C
5) only B and C
C) skier’s final KE
ConcepTest 8.2 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
1) only B
2) only C
3) A, B and C
4) only A and C
5) only B and C
C) skier’s final KE
The gravitational PE depends upon the reference level, but
the difference DPE does not! The work done by gravity
must be the same in the two solutions, so DPE and DKE
should be the same.
Follow-up: Does anything change physically by the choice of y = 0?
ConcepTest 8.6 Down the Hill
Three balls of equal mass start from rest and roll down different
ramps. All ramps have the same height. Which ball has the
greater speed at the bottom of its ramp?
4) same speed
for all balls
1
2
3
ConcepTest 8.6 Down the Hill
Three balls of equal mass start from rest and roll down different
ramps. All ramps have the same height. Which ball has the
greater speed at the bottom of its ramp?
4) same speed
for all balls
1
2
3
All of the balls have the same initial gravitational PE,
since they are all at the same height (PE = mgh). Thus,
when they get to the bottom, they all have the same final
KE, and hence the same speed (KE = 1/2 mv2).
Follow-up: Which ball takes longer to get down the ramp?
ConcepTest 8.8a Water Slide I
Paul and Kathleen start from rest at
1) Paul
the same time on frictionless water
2) Kathleen
slides with different shapes. At the
bottom, whose velocity is greater?
3) both the same
ConcepTest 8.8a Water Slide I
Paul and Kathleen start from rest at
1) Paul
the same time on frictionless water
2) Kathleen
slides with different shapes. At the
bottom, whose velocity is greater?
Conservation of Energy:
Ei = mgH = Ef = 1/2 mv2
therefore: gH = 1/2 v2
Since they both start from the
same height, they have the
same velocity at the bottom.
3) both the same
Example
• In the figure below, the water slide ends at a height of
1.50 m above the pool. If the person starts from rest at A
and lands in the water at B, what is the height h of the
slide?
8-4 Work Done by Nonconservative Forces
In the presence of nonconservative forces, the
total mechanical energy is not conserved:
Solving,
(8-9)
8-4 Work Done by Nonconservative Forces
In this example, the
nonconservative force
is water resistance:
ConcepTest 8.10a Falling Leaves
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf until just before it hits the
ground, at which point it has final
total energy PEf + KEf. How do
these total energies compare?
1) PEi + KEi > PEf + KEf
2) PEi + KEi = PEf + KEf
3) PEi + KEi < PEf + KEf
4) impossible to tell from
the information provided
ConcepTest 8.10a Falling Leaves
You see a leaf falling to the ground
with constant speed. When you
first notice it, the leaf has initial
total energy PEi + KEi. You watch
the leaf until just before it hits the
ground, at which point it has final
total energy PEf + KEf. How do
these total energies compare?
1) PEi + KEi > PEf + KEf
2) PEi + KEi = PEf + KEf
3) PEi + KEi < PEf + KEf
4) impossible to tell from
the information provided
As the leaf falls, air resistance exerts a force on it opposite to
its direction of motion. This force does negative work, which
prevents the leaf from accelerating. This frictional force is a
non-conservative force, so the leaf loses energy as it falls,
and its final total energy is less than its initial total energy.
Follow-up: What happens to leaf’s KE as it falls? What net work is done?
8-5 Potential Energy Curves and
Equipotentials
The curve of a hill or a roller coaster is itself
essentially a plot of the gravitational
potential energy:
8-5 Potential Energy Curves and
Equipotentials
The potential energy curve for a spring:
Summary of Chapter 8
• Conservative forces conserve mechanical
energy
• Nonconservative forces convert mechanical
energy into other forms
• Conservative force does zero work on any
closed path
• Work done by a conservative force is
independent of path
• Conservative forces: gravity, spring
Summary of Chapter 8
• Work done by nonconservative force on closed
path is not zero, and depends on the path
• Nonconservative forces: friction, air
resistance, tension
• Energy in the form of potential energy can be
converted to kinetic or other forms
• Work done by a conservative force is the
negative of the change in the potential energy
• Gravity: U = mgy
• Spring: U = ½ kx2
Summary of Chapter 8
• Mechanical energy is the sum of the kinetic and
potential energies; it is conserved only in
systems with purely conservative forces
• Nonconservative forces change a system’s
mechanical energy
• Work done by nonconservative forces equals
change in a system’s mechanical energy
• Potential energy curve: U vs. position