Potential Energy

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Potential Energy

• March 1, 2006

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Coming Up

• Exam on Friday
• Material through today.
• Look for current WA
• Next Week
• Important Topic Conservation of Momentum and
  collisions between objects.

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Potential Energy

• Potential energy is the energy associated with
  the configuration of a system of objects that
  exert forces on each other
• This can be used only with conservative forces
• Conservative forces are NOT Republicans
• When conservative forces act within an isolated
  system, the kinetic energy gained (or lost) by
  the system as its members change their relative
  positions is balanced by an equal loss (or gain)
  in potential energy.
• This is Conservation of Mechanical Energy

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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
• Nuclear ?heat easy
• Heat? Nuclear probably impossible!
• Conversion from one type to another type of
  energy is not always reversible.

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Systems with Multiple Particles

• We can extend our definition of a system to
  include multiple objects
• The force can be internal to the system
• The kinetic energy of the system is the algebraic
  sum of the kinetic energies of the individual
  objects
• Sometimes, the kinetic energy of one of the
  objects may be negligible

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System Example

• This system consists of Earth and a book
• Do work on the system by lifting the book through
  Dy
• The work done by you is mg(yb ya)
• At the top it is at rest.
• The amount of work that you did is called the
  potential energy of the system with respect to
  the ground.
• The PEs initial value is mgya
• The FINAL value is mgyb
• The difference is the work done.

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Lets drop the book from yb and see what it is
doing at ya.
Multiply by m and divide by 2
The decrease in Potential Energy equals the
increase in Kinetic Energy
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Potential Energy

• The energy storage mechanism is called potential
  energy
• A potential energy can only be associated with
  specific types of forces (conservative)
• Potential energy is always associated with a
  system of two or more interacting objects

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Gravitational Potential Energy

• Gravitational Potential Energy is associated with
  an object at a given distance above Earths
  surface
• Assume the object is in equilibrium and moving at
  constant velocity
• The work done on the object is done by Fapp and
  the upward displacement is

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Gravitational Potential Energy, cont

• The quantity mgy is identified as the
  gravitational potential energy, Ug
• Ug mgy
• Units are joules (J)

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Energy Problems

• Draw a diagram of the situation.
• ESTABLISH AN ORIGIN.
• THE POTENTIAL ENERGY OF A PARTICAL OF MASS M IS
  ALWAYS MEASURED WITH RESPECT TO THIS ORIGIN.
• The potential energy of a particle is defined as
  being ZERO when it is at the origin.
• At some height above the origin, the value of the
  PE is mgh.

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Gravitational Potential Energy, final

• The gravitational potential energy depends only
  on the vertical height of the object above
  Earths 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

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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 Uf Ki Ui
• An isolated system is one for which there are no
  energy transfers across the boundary

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Lets look at the more general case.
yh
W0
d
d
D
D
WmgD
WmgD
We do work to move mass to yh. Wmgh
F
y0 (origin)
Add'em Up - Same result ... mgh
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ANY PATH

• Can be broken up into a series of very small
  vertical moves and horizontal moves.
• The horizontal moves require no work.
• The force is at right angles to the motion. Dot
  product is zero.
• The vertical moves are

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The work done on a mass in raising it a distance
h is "mgh" no matter what the path!
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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 DKbook
• Also, W mgyb mgya
• So, DK -DUg

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Elastic Potential Energy

• Elastic Potential Energy is associated with a
  spring
• The force the spring exerts (on a block, for
  example) is Fs - kx
• The work done by an external applied force on a
  spring-block system is
• W ½ kxf2 ½ 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

• This expression is the elastic potential energy
  Us ½ 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

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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

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Problem Solving Strategy Conservation of
Mechanical Energy

• 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

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Problem-Solving Strategy, 2

• 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

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Problem-Solving Strategy, 3

• 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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Problem-Solving Strategy, 4

• 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

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Conservation of Energy, Example 1 (Drop a Ball)

• Initial conditions
• Ei Ki Ui mgh
• The ball is dropped, so Ki 0
• The configuration for zero potential energy is
  the ground
• Conservation rules applied at some point y above
  the ground gives
• ½ mvf2 mgy mgh

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Conservation of Energy, Example 2 (Pendulum)

• As the pendulum swings, there is a continuous
  change between potential and kinetic energies
• At A, the energy is potential
• At B, all of the potential energy at A is
  transformed into kinetic energy
• Let zero potential energy be at B
• At C, the kinetic energy has been transformed
  back into potential energy

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Conservation of Energy, Example 3 (Spring Gun)

• Choose point A as the initial point and C as the
  final point
• EA EC
• KA UgA UsA KA UgA UsA
• ½ kx2 mgh

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Conservative Forces

• 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
• A closed path is one in which the beginning and
  ending points are the same

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Conservative Forces, cont

• Examples of conservative forces
• Gravity
• Spring force
• We can associate a potential energy for a system
  with any conservative force acting between
  members of the system
• This can be done only for conservative forces
• In general WC - DU

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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

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Mechanical Energy and Nonconservative Forces

• In general, if friction is acting in a system
• 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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Nonconservative Forces, cont

• The work done against friction is greater along
  the red path than along the blue path
• Because the work done depends on the path,
  friction is a nonconservative force

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Problem Solving Strategies Nonconservative
Forces

• Define the isolated system and the initial and
  final configuration of the system
• Identify the configuration for zero potential
  energy
• These are the same as for Conservation of Energy
• The difference between the final and initial
  energies is the change in mechanical energy due
  to friction

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Nonconservative Forces, Example 1 (Slide)

• DEmech DK DU
• DEmech (Kf Ki)
• (Uf Ui)
• DEmech (Kf Uf)
• (Ki Ui)
• DEmech ½ mvf2 mgh -ƒkd

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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
• DEmech -ƒkd

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Nonconservative Forces, Example 3 (Connected
Blocks)

• The system consists of the two blocks, the
  spring, and Earth
• Gravitational and potential energies are involved
• The kinetic energy is zero if our initial and
  final configurations are at rest

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Connected Blocks, cont

• Block 2 undergoes a change in gravitational
  potential energy
• The spring undergoes a change in elastic
  potential energy
• The coefficient of kinetic energy can be measured

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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
• DU is negative when F and x are in the same
  direction

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Conservative Forces and Potential Energy

• The conservative force is related to the
  potential energy function through
• The x component of a conservative force acting on
  an object within a system equals the negative of
  the potential energy of the system with respect
  to x

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Conservative Forces and Potential Energy Check

• Look at the case of a deformed spring
• This is Hookes Law

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Energy Diagrams and Equilibrium

• Motion in a system can be observed in terms of a
  graph of its position and energy
• In a spring-mass system example, the block
  oscillates between the turning points, x xmax
• The block will always accelerate back toward x
  0

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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
• xxmax and x-xmax are called the turning points

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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

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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 either restoring or
  disrupting forces

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Potential Energy in Molecules

• There is potential energy associated with the
  force between two neutral atoms in a molecule
  which can be modeled by the Lennard-Jones
  function

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Potential Energy Curve of a Molecule

• Find the minimum of the function (take the
  derivative and set it equal to 0) to find the
  separation for stable equilibrium
• The graph of the Lennard-Jones function shows the
  most likely separation between the atoms in the
  molecule (at minimum energy)

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Force Acting in a Molecule

• The force is repulsive (positive) at small
  separations
• The force is zero at the point of stable
  equilibrium
• The force is attractive (negative) when the
  separation increases
• At great distances, the force approaches zero