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Chapter 7
Potential Energy
Potential Energy Potential energy is the energy
associated with the configuration of a system of objects that exert forces on each other The forces are internal to the system
Types of Potential Energy There are many forms of potential
energy, including: Gravitational Electromagnetic Chemical Nuclear
One form of energy in a system can be converted into another
System Example This system consists
of Earth and a book Do work on the
system by lifting the book through y
The work done is mgyb - mgya
Potential Energy 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
Gravitational Potential Energy 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
Gravitational Potential Energy, cont
The quantity mgy is identified as the gravitational potential energy, Ug
Ug = mgy Units are joules (J)
Gravitational Potential Energy, final 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 The choice is arbitrary because you normally need
the difference in potential energy, which is independent of the choice of reference configuration
Conservation of Mechanical Energy The mechanical energy of a system is the
algebraic sum of the kinetic and potential energies in the system Emech = K + Ug
The statement of Conservation of Mechanical Energy for an isolated system is Kf + Ugf = Ki+ Ugi
An isolated system is one for which there are no energy transfers across the boundary
Conservation of Mechanical Energy, example
Look at the work done by the book as it falls from some height to a lower height
Won book = Kbook
Also, W = mgyb – mgya
So, K = -Ug
Conservative Forces 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
Nonconservative Forces 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
Nonconservative Force, Example
Friction is an example of a nonconservative force The work done
depends on the path The red path will
take more work than the blue path
Elastic Potential Energy 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 W = 1/2 kxf
2 – 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
Elastic Potential Energy, cont 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
Elastic Potential Energy, final The elastic potential energy stored in a spring
is zero whenever the spring is not deformed (U = 0 when x = 0) 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 x2 will always be positive
Conservation of Energy, Extended Including all the types of energy
discussed so far, Conservation of Energy can be expressed as
K + U + Eint = E system = 0 or
K + U + E int = constant K would include all objects U would be all types of potential energy
Problem Solving Strategy – Conservation of Mechanical Energy
Conceptualize: Define the isolated system and the initial and final configuration of the system 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
Problem-Solving Strategy, 2 Categorize: Determine if any energy
transfers across the boundary If it does, use the nonisolated system
model, E system = T If not, use the isolated system model,
Esystem = 0
Determine if any nonconservative forces are involved
Problem-Solving Strategy, 3 Analyze: Identify the configuration for
zero potential energy 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
Problem-Solving Strategy, 4 If friction or air resistance is present,
mechanical energy of the system is not conserved
Use energy with non-conservative forces instead The difference between initial and final
energies equals the energy transformed to or from internal energy by the nonconservative forces
Problem-Solving Strategy, 5 If the mechanical energy of the system
is conserved, write the total energy as 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
Mechanical Energy and Nonconservative Forces In general, if friction is acting in a
system: Emech = K + U = -ƒkd U is the change in all forms of potential
energy If friction is zero, this equation becomes
the same as Conservation of Mechanical Energy
Nonconservative Forces, Example 1 (Slide)
Emech = K + U
Emech =(Kf – Ki) +
(Uf – Ui)
Emech = (Kf + Uf) –
(Ki + Ui)
Emech = 1/2 mvf2 –
mgh = -ƒkd
Nonconservative Forces, Example 2 (Spring-Mass)
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
Emech = -ƒkd
Conservative Forces and Potential Energy 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
U is negative when F and x are in the same direction
Conservative Forces and Potential Energy The conservative force is related to the
potential energy function through
The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system This can be extended to three dimensions
Conservative Forces and Potential Energy – Check Look at the case of an object located
some distance y above some reference point:
This is the expression for the vertical component of the gravitational force
Nonisolated System in Steady State A system can be nonisolated with 0 =
T 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
Nonisolated System in Steady State, House Example
Potential Energy for Gravitational Forces
Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation:
The potential energy then is
Potential Energy for Gravitational Forces, Final The result for the earth-object system
can be extended to any two objects:
For three or more particles, this becomes
Electric Potential Energy Coulomb’s Law gives the electrostatic
force between two particles
This gives an electric potential energy function of
Energy Diagrams and Stable Equilibrium 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
Energy Diagrams and Unstable Equilibrium 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
Neutral Equilibrium 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
Potential Energy in Fuels Fuel represents a storage mechanism
for potential energy Chemical reactions release energy that
is used to operate the automobile