Potential Energy

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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 two or more
  interacting objects or particles that exert
  forces on each other
• The forces are internal to the system

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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

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System Example

• This system consists of the Earth and a book
• Do work on the system by lifting the book through
  Dy
• The work done by an external force on the book is
  mgyb - mgya

Fig 7.1
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Potential Energy

• Similar as the work-kinetic energy theorem, the
  work equals to the difference between the final
  and initial values of some quantity of the
  system.
• The quantity is called potential energy of the
  system.
• Through the work, energy is transferred into the
  system in a form different from kinetic energy.
• The transferred energy is stored in the system.
• The quantity Ugmgy is called the gravitational
  potential energy of the book-Earth system.

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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
  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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Gravitational Potential Energy, final

• The gravitational potential energy depends only
  on the vertical height of the object above
  Earths surface
• A reference configuration of the system must be
  chosen so that the gravitational potential energy
  at the reference configuration is set equal to
  zero
• The choice is arbitrary because the difference in
  potential energy is independent of the choice of
  reference configuration

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7.2 Conservation of Mechanical Energy for
isolated systems

• 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

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Conservation of Mechanical Energy, example

• Look at the work done on the book by the
  gravitational force as it falls from some height
  yb to a lower height ya
• Won book DKbook
• Also, W mgyb mgya
• So, DK -Dug
• Kf Ugf Ki Ugi
• The isolated system is the book and the Earth.

Fig 7.2
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Fig 7.4
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Fig 7.5
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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 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

• 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

• 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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Conservation of Energy for isolated systems

• 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
• K would include all objects
• U would be all types of potential energy
• The internal energy is the energy stored in a
  system besides the kinetic and potential
  energies.

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7.3 Conservative Forces

• A conservative force is a force between members
  of a system that causes no transformation of
  mechanical energy to internal 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

• 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

• 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

Fig 7.7
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Mechanical Energy and Nonconservative Forces

• In general, if friction is acting in a system
• DEmech DK DU -kd
• 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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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 -kd

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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 -kd

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