Physics 1710 Chapter 8 Potential Energy and Conservation

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Physics 1710Chapter 8 Potential Energy and
Conservation

• Quiz
• Dr. M drags a chair across the floor for a
  distance of 3.00 m. He is pulling on it with a
  force of 60.0 N at an angle of 30 to the
  horizontal. A) How much work does he do? B) If
  he does it in 5.00 seconds what power is he
  delivering?

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• 1 Lesson
• Potential Energy U is the energy stored in a
  system and may later be used to produce work.
• The Potential Energy is equal to the negative
  of the work done on the system to put it in its
  present state.
• The sum of all energy, potential and kinetic,
  of a system is conserved.

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Potential Energy
• W ? Fd r
• U -W
• Potential Energy is the negative of the work
  required to put the system in the current state.

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Example Mass on a Spring
• F -k x
• Potential Energy
• U -?0xFdx -?0x(-k x) dx
• U k?0x x dx ½ k x 2
• Thus, the potential energy stored in a stretched
  spring is proportional to the square of the
  extension x and the spring constant k.

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Example Elevated Mass
• F -mg
• Potential Energy
• Ug -?0hFdy -?0h(- mg) dy
• Ug mg?0h dy mgh
• Thus, the potential energy stored in an elevated
  mass is proportional to the height h and the
  weight of the mass.

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Thought Experiment
• Consider water impounded behind a dam. Where
  does the energy come from to produce
  hydroelectricity?

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Relationship Between F and U
• U -? Fd r
• So
• U -? Fx dx Fy dy Fz dz
• Then
• Fx -dU/dx Fy -dU/dy Fz -dU/dz
• F -?U
• F -gradient of U

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• The Force is equal to the negative gradient of
  the potential energy
• F -?U
• Fx -?U/?x
• Fy -?U/?y
• Fz -?U/?z

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Example Ball on a slope
• h ax by
• U mgh
• Fx -?U/?x -?(mgh)/?x -mg?h/?x
• Similarly
• Fy -?U/?y -mg b
• Thus, F -mg( a i b j )

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Conservation of Energy
• The sum of all energy in a system is conserved,
  ie remains the same.
• E U K

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Example Pendulum
• U mg h
• h L(1- cos ? )
• U mg L(1- cos ? )
• K ½ m v 2
• ½ m (Ld ?/dt) 2
• E mg L(1- cos ? ) ½ m (Ld ?/dt) 2
• constant

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Pendulum of Doom

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Thought (Gedanken) Experiment
• Why does a pendulum stop moving?

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Dissipative (non-conservative) Forces
• W ? Fd r
• ? (C vx 2 )dx
• ? (C vx 2 )(dx /dt) dt
• ? (C vx 3 )dt
• E U K -W

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Physics 1710 Chapter 8 Potential Energy and
Conservation

• Summary
• The Potential Energy is equal to the negative of
  the work done on the system to put it in its
  present state.
• U -? Fd r
• The sum of all energy, potential and kinetic,
  of a system is conserved, in the absence of
  dissipation.
• E U K W
• F - ?U