Solution:

1
Physics 1710 Chapter 78Power Energy
0

• Solution

K ½ mv 2 ½ (2.0 kg) (5.0 m/s) 2 25. kg
m2/s2 25. J
2
Physics 1710 Chapter 78Power Energy
0

• What is the minimum height from which a small
  rolling ball must be started from rest so that it
  will complete a loop-the-loop?

h
3
Physics 1710 Chapter 78Power Energy
0

• What is the minimum height from which a small
  rolling ball must be started from rest so that it
  will complete a loop-the-loop?

Peer Instruction Time
4
Physics 1710 Chapter 78Power Energy
0

• What is the minimum height from which a small
  rolling ball must be started from rest so that it
  will complete a loop-the-loop?

v2/R g K U ½ mv 2 mg(h-R) v 2 2 g
(h-R) g R 2g (h-R) h 3R
h
5
Physics 1710 Chapter 78Power Energy
0

• What is the minimum height from which a small
  rolling ball must be started from rest so that it
  will complete a loop-the-loop?

K ½ mv 2 ½ (2.0 kg) (5.0 m/s) 2 25. kg
m2/s2 25. J
6
Physics 1710 Chapter 78Power Energy
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• 1' Lecture
• Power is the time rate of change in energy.
  Power Watt Joule/second
• Potential Energy U is the energy stored in a
  system and may later 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,
  is conserved in an isolated system.

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Physics 1710 Chapter 78Power Energy
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• Power
• P dE/dt
• Power is the time rate of change in the energy
  of a system, the rate of work down on or by the
  system.
• Unit of power Joule/second Watt

8
Physics 1710 Chapter 78Power Energy
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• Unit of Work and Energy
• F ? d N?m Joule J

Joule
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Physics 1710 Chapter 78Power Energy
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• Power Watt Joule/second

Watts Steam Engine 1774
10
Physics 1710 Chapter 78Power Energy
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• Power
• P dE/dt
• P ?E/?t
• ?E P ?t
• ?E (100 W)(3600 s)
• ?E 360 000 J 360 kJ

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Physics 1710 Chapter 78Power Energy
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• 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 78Power Energy
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• Potential Energy

U - (- F h) m g h
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Physics 1710 Chapter 78Power Energy
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• 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 78Power Energy
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• Where does the energy come from to produce
  electrical power in a hydroelectric dam?

Peer Instruction Time
15
Physics 1710 Chapter 78Power Energy
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• Potential Energy

U m g h P dU/dt mg dh/dt mg (100.
kg)(9.8N/kg) 98.0 N dh/dt 10 m/10 s 1
m/s P 98. W
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Physics 1710 Chapter 78Power Energy
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• 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 78Power Energy
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• Example Mass on a Spring
• Potential Energy
• U ½ k x 2
• F dU/dx
• F -½ k dx2/dx
• F -k x
• Thus, the force is equal to the negative of the
  gradient of the potential energy.

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Physics 1710 Chapter 78Power Energy
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• The Force is equal to the negative gradient of
  the potential energy
• F -?U
• Fx -?U/?x
• Fy -?U/?y
• Fz -?U/?z

19
Physics 1710 Chapter 8 Potential Energy and
Conservation
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• 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 78Power Energy
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• Conservation of Energy
• The sum of all energy in a system is conserved,
  i.e. remains the same.
• E U K

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Physics 1710 Chapter 78Power Energy
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• 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 78Power Energy
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• Thought (Gedanken) Experiment
• Why does a pendulum stop moving?

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Physics 1710 Chapter 78Power Energy
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• 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 78Power Energy
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• 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
• P dE/dt