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Chapter 7
Potential Energy
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7.1 Potential Energy

Potential energy is the energy
associated with the configuration of a
system of objects that exert forces on
each other
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The forces are internal to the system
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Types of Potential Energy
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There are many forms of potential
energy, including:
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Gravitational
Electromagnetic
Chemical
Nuclear
One form of energy in a system can be
converted into another
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System Example
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This system consists
of Earth and a book
Do work on the
system by lifting the
book through Dy
The work done is
mgyb - mgya
Fig 7.1
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Potential Energy
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The energy storage mechanism is
called potential energy
A potential energy can only be
associated with specific types of forces
Potential energy is always associated
with a system of two or more interacting
objects
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Gravitational Potential Energy
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Gravitational Potential Energy is
associated with an object at a given
distance above Earth’s surface
Assume the object is in equilibrium and
moving at constant velocity
The work done on the object is done by
and the upward displacement is
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Gravitational Potential Energy,
cont
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The quantity mgy is identified as the
gravitational potential energy, Ug
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Ug = mgy
Units are joules (J)
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Gravitational Potential Energy,
final
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The gravitational potential energy depends
only on the vertical height of the object above
Earth’s surface
In solving problems, you must choose a
reference configuration for which the
gravitational potential energy is set equal to
some reference value, normally zero
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The choice is arbitrary because you normally need
the difference in potential energy, which is
independent of the choice of reference
configuration
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7.2 Conservation of
Mechanical Energy
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The mechanical energy of a system is the
algebraic sum of the kinetic and potential
energies in the system
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Emech = K + Ug
The statement of Conservation of Mechanical
Energy for an isolated system is
Kf + Ugf = Ki+ Ugi
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An isolated system is one for which there are no
energy transfers across the boundary
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Conservation of Mechanical
Energy, example
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Look at the work done
by the book as it falls
from some height to a
lower height
Won book = DKbook
Also, W = mgyb – mgya
So, DK = -DUg
Fig 7.2
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Active Figure
7.3
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A ball of mass m is dropped from rest at a
height h above the ground as in Figure 7.4.
Ignore air resistance.
Fig 7.4
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You are designing apparatus to support an actor ofmass 65 kg
who is to “fly” down to the stage during the performance of a
play. You attach the actor’s harness to a 130-kg sandbag by
means of a lightweight steel cable running smoothly over two
frictionless pulleys as in Figure 7.5a. You need 3.0 m of cable
between the harness and the nearest pulley so that the pulley can
be hidden behind a curtain. For the apparatus to work
successfully, the sandbag must never lift above the floor as the
actor swings from above the stage to the floor. Let us identify the
angle  as the angle that the actor’s cable makes with the vertical
when he begins his motion from rest. What is the maximum
value  can have such that the sandbag does not lift off the floor
during the actor’s swing?
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Fig 7.5
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Since T = mbagg
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7.3 Conservative Forces
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A conservative force is a force between
members of a system that causes no
transformation of mechanical energy within
the system
The work done by a conservative force on a
particle moving between any two points is
independent of the path taken by the particle
The work done by a conservative force on a
particle moving through any closed path is
zero
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Nonconservative Forces
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A nonconservative force does not
satisfy the conditions of conservative
forces
Nonconservative forces acting in a
system cause a change in the
mechanical energy of the system
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Nonconservative Force,
Example
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Friction is an
example of a
nonconservative
force
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The work done
depends on the path
The red path will
take more work than
the blue path
Fig 7.7
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Elastic Potential Energy
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Elastic Potential Energy is associated
with a spring, Us = 1/2 k x2
The work done by an external applied
force on a spring-block system is
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W = 1/2 kxf2 – 1/2 kxi2
The work is equal to the difference
between the initial and final values of an
expression related to the configuration of
the system
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Elastic Potential Energy, cont
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This expression is the
elastic potential energy:
Us = 1/2 kx2
The elastic potential
energy can be thought
of as the energy stored
in the deformed spring
The stored potential
energy can be
converted into kinetic
energy
Fig 7.6
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Elastic Potential Energy, final
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The elastic potential energy stored in a spring
is zero whenever the spring is not deformed
(U = 0 when x = 0)
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The energy is stored in the spring only when the
spring is stretched or compressed
The elastic potential energy is a maximum
when the spring has reached its maximum
extension or compression
The elastic potential energy is always positive
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x2 will always be positive
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Active Figure
7.6
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Conservation of Energy,
Extended
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Including all the types of energy
discussed so far, Conservation of
Energy can be expressed as
DK + DU + DEint = DE system = 0 or
K + U + E int = constant
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K would include all objects
U would be all types of potential energy
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Problem Solving Strategy –
Conservation of Mechanical Energy
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Conceptualize: Define the isolated
system and the initial and final
configuration of the system
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The system may include two or more
interacting particles
The system may also include springs or
other structures in which elastic potential
energy can be stored
Also include all components of the system
that exert forces on each other
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Problem-Solving Strategy, 2
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Categorize: Determine if any energy
transfers across the boundary
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If it does, use the nonisolated system
model, DE system = ST
If not, use the isolated system model,
DEsystem = 0
Determine if any nonconservative forces
are involved
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Problem-Solving Strategy, 3
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Analyze: Identify the configuration for
zero potential energy
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Include both gravitational and elastic
potential energies
If more than one force is acting within the
system, write an expression for the
potential energy associated with each force
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Problem-Solving Strategy, 4
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If friction or air resistance is present,
mechanical energy of the system is not
conserved
Use energy with non-conservative
forces instead
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The difference between initial and final
energies equals the energy transformed to
or from internal energy by the
nonconservative forces
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Problem-Solving Strategy, 5
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If the mechanical energy of the system
is conserved, write the total energy as
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Ei = Ki + Ui for the initial configuration
Ef = Kf + Uf for the final configuration
Since mechanical energy is conserved,
Ei = Ef and you can solve for the
unknown quantity
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Mechanical Energy and
Nonconservative Forces
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In general, if friction is acting in a
system:
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DEmech = DK + DU = -ƒkd
DU is the change in all forms of potential
energy
If friction is zero, this equation becomes
the same as Conservation of Mechanical
Energy
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A 3.00-kg crate slides down a ramp
at a loading dock. The ramp is 1.00
m in length and is inclined at an
angle of 30.0° as shown in Figure 7.8.
The crate starts from rest at the top
and experiences a constant friction
force of magnitude 5.00 N. Use
energy methods to determine the
speed of the crate when it reaches
the bottom of the ramp.
Fig 7.8
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Nonconservative Forces,
Example 1 (Slide)
DEmech = DK + DU
DEmech =(Kf – Ki) +
(Uf – Ui)
DEmech = (Kf + Uf) –
(Ki + Ui)
DEmech = 1/2 mvf2 –
mgh = -ƒkd
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Nonconservative Forces,
Example 2 (Spring-Mass)
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Without friction, the
energy continues to be
transformed between
kinetic and elastic
potential energies and
the total energy remains
the same
If friction is present, the
energy decreases
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DEmech = -ƒkd
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7.4 Conservative Forces and
Potential Energy
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Define a potential energy function, U,
such that the work done by a
conservative force equals the decrease
in the potential energy of the system
The work done by such a force, F, is
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DU is negative when F and x are in the
same direction
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Conservative Forces and
Potential Energy
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The conservative force is related to the
potential energy function through
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The conservative force acting between parts
of a system equals the negative of the
derivative of the potential energy associated
with that system
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This can be extended to three dimensions
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Conservative Forces and
Potential Energy – Check
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Look at the case of an object located
some distance y above some reference
point:
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This is the expression for the vertical
component of the gravitational force
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7.5 Nonisolated System in
Steady State
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A system can be nonisolated with 0 = DT
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This occurs if the rate at which energy is
entering the system is equal to the rate at
which it is leaving
There can be multiple competing transfers
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Nonisolated System in Steady
State, House Example
Fig 7.11
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7.6 Potential Energy for
Gravitational Forces
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Generalizing
gravitational potential
energy uses Newton’s
Law of Universal
Gravitation:
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The potential energy
then is
Fig 7.12
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Potential Energy for
Gravitational Forces, Final
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The result for the earth-object system
can be extended to any two objects:
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For three or more particles, this
becomes
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Fig 7.13
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Fig 7.14
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Electric Potential Energy
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Coulomb’s Law gives the electrostatic
force between two particles
This gives an electric potential energy
function of
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7.7 Energy Diagrams and
Stable Equilibrium
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The x = 0 position is
one of stable
equilibrium
Configurations of stable
equilibrium correspond
to those for which U(x)
is a minimum
x=xmax and x=-xmax are
called the turning points
Fig 7.15
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Energy Diagrams and
Unstable Equilibrium
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Fx = 0 at x = 0, so the
particle is in equilibrium
For any other value of x,
the particle moves away
from the equilibrium
position
This is an example of
unstable equilibrium
Configurations of
unstable equilibrium
correspond to those for
which U(x) is a
maximum
Fig 7.16
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Active Figure
7.15
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Active Figure
7.15b
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Neutral Equilibrium
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Neutral equilibrium occurs in a
configuration when U is constant over
some region
A small displacement from a position in
this region will produce neither restoring
nor disrupting forces
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7.8 Potential Energy in Fuels
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Fuel represents a storage mechanism
for potential energy
Chemical reactions release energy that
is used to operate the automobile
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