Transcript The work done by a conservative force is

Lecture Outline
Chapter 8
Physics, 4th Edition
James S. Walker
Copyright © 2010 Pearson Education, Inc.
Chapter 8
Potential Energy and
Conservation of Energy
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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
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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
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8-1 Conservative and Nonconservative
Forces
Work done by gravity on a closed path is zero:
Wtotal = 0 + (-mgh) + 0 + mgh = 0
Work done by gravity on a closed path is zero
Gravity does no work on the two horizontal segments of the path. On the two
vertical segments of the amounts of work done are equal in magnitude but
opposite in sign. There fore, the total work done by gravity on this- or any – closed
path is zero.
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8-1 Conservative and Nonconservative
Forces
Work done by friction on a closed path is not
zero:
Work done by friction on a closed path is nonzero
The work done by friction when an object moves through a distance d is -µkmgd.
Thus, the total work done by friction on a closed path is nonzero. In this case, it is
equal to - 4µkmgd.
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8-1 Conservative and Nonconservative
Forces
The work done by a conservative force is zero
on any closed path:
The work done by a conservative force is independent of path
Considering paths 1 and 2, we see that W1 + W2 = 0, or W2 = W1. From paths 1 and 3,
however, we see that W1 + W3 = 0, or W3 = -W1. It follows, then, that W3 = W2, since they are
both equal to –W1; hence the work done in going from A to B in independent of the path.
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8-2 The Work Done by Conservative Forces
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; in the
meantime, we say the energy is stored as potential energy.
Potential Energy, U
When a conservative force does an amount of work Wc, the
corresponding potential energy U is changed according to the
following definition:
(8-1)
The work done by a conservative force is equal to the negative of the change in potential
energy. For example, when an object falls, gravity does positive work on it and its potential
energy decreases. Similarly, when an object is lifted, gravity does negative work and the
potential energy increases.
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8-2 The Work Done by Conservative Forces
Gravitational potential energy:
Gravitational Potential energy
A person drops from a diving board into a swimming pool. The driving board is at the
height y, and the surface of the water is at y = 0. We choose the gravitational potential
energy to be zero at y = 0; hence, the potential energy is mgy at the diving board.
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8-2 The Work Done by Conservative Forces
Gravitational potential energy:
Gravitational Potential energy
A candy bar called the Mountain Bar has a calorie content of 212 Cal = 212 Kcal, which is
equivalent to an energy of 8.87 x 105 J. If an 81.0-kg mountain climber eats a Mountain
Bar and marginally converts it all to potential energy, what gain of altitude would be
possible?
U = mgh
h = U/mg = 8.87 x 105 J/ (81.0 kg) (9.81 m/s2) = 1120 m
h = 1120 m
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8-2 The Work Done by Conservative Forces
Consider a spring that is stretched from its equilibrium position a distance
x. The work required to cause this stretch is W = 1/2kx2. If the spring is
released- and allowed to move from the stretched position back to the
equilibrium position- it will do the same work, 1/2kx2. From our definition
of potential energy, then, we see that
Springs:
Note that in this case Uf is the
potential energy when the spring
is at x = 0 and Ui is the potential
energy when the spring is
stretched by the amount x.
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(8-4)
8-3 Conservation of Mechanical Energy
Definition of mechanical energy:
(8-6)
Using this definition and considering only
conservative forces, we find:
Or equivalently:
In systems with conservative forces only, the mechanical energy E
is conserved; that is , E = U + K = constant.
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8-3 Conservation of Mechanical Energy
Energy conservation can make kinematics
problems much easier to solve:
9a). A set of keys fall to the floor. Ignoring frictional forces, we know that the mechanical energy at
points I and f must be equal; Ei = Ef. Using this condition, we can find the speed of the keys just before
they land. (b). The same physical situation as in part (a), except this time we have chosen y = 0 to be at
the point where the keys are dropped. As before, we set Ei = Ef to find the speed of the keys just before
they land.
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8-4 Work Done by Nonconservative Forces
In the presence of nonconservative forces, the
total mechanical energy is not conserved:
Solving,
(8-9)
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8-4 Work Done by Nonconservative Forces
In this example, the
nonconservative force
is water resistance:
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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:
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8-5 Potential Energy Curves and
Equipotentials
The potential energy curve for a spring:
A mass on a spring
(a) A spring is stretched by an amount A, giving it a potential energy of U = 1/2kA2.
(b) The potential energy curve, U = 1/2kx2, for the spring in (a). Because the mass starts at
rest, its initial mechanical energy is Eo = 1/2kA2. The mass oscillates between x = A and x = - A
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8-5 Potential Energy Curves and
Equipotentials
Contour maps are also a form of potential
energy curve:
A contour map
A small mountain (top, in side view) is very steep on the left, more gently sloping
on the right. A contour map of this mountain (bottom) shows a series of equal
altitude contour lines from 50 ft. to 450 ft. Notice that the contour lines are packed
close together where the terrain is steep, but are widely spaces where it is more level.
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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
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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
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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
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