Solution:

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Physics 1710 Chapter 8Potential Energy
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• Solution

Power dW/dt (Fdx)/dt F dx/dt F v (for F
constant) (20.0 N )(36 x 103 m/ 3600 sec)
200. N m/s 200. W (4)
2
Physics 1710 Chapter 8Potential Energy
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• 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
Review
3
Physics 1710 Chapter 8Potential Energy
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• 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 - W ½ mv 2 ½ (2/5 mv 2) mgh
v 2 Rg 10/7 hg h 0.7 R h 7/10(22.0 cm)
15.4 cmd 2Rh 69.4 cm
Physics Works! (When you include all relevant
effects)
h
4
Physics 1710 Chapter 8Potential Energy
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• 1' Lecture
• Potential Energy is 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 negative gradient of U.

5
Physics 1710 Chapter 8Potential Energy
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• Potential Energy
• W ? Fd r
• U -W -? Fd r
• Potential Energy is the negative of the work
  required to put the system in the current state.

6
Physics 1710 Chapter 8Potential Energy
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• What is the potential energy of a 0.100 kg ball
  placed up a 45 o ramp 0.50 above the table?

7
Physics 1710 Chapter 8Potential Energy
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• What is the potential energy of a 0.100 kg ball
  placed up a 45 o ramp 0.50 above the table?

U - F?x - (-mg)h mg h
8
Physics 1710 Chapter 8Potential 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 8Potential 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

10
Physics 1710 Chapter 8Potential 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

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Physics 1710 Chapter 8Potential Energy
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• Example Pendulum
• U mg h
• h L(1- cos ? )
• U mg L(1- cos ? )
• s L ?
• FS - (1/L)dU/d ?
• - mg sin ?

?
L
s
12
Physics 1710 Chapter 8Potential Energy
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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 8Potential 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 8Potential Energy
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• Force
• z ar2
• U mgz
• Fr -?U/?r -?(mgz)/?r -mg?z/?r
• - 2amgr - k r
• Like a mass on a spring!

15
Physics 1710 Chapter 8Potential 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

16
Physics 1710 Chapter 8Potential 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 8Potential 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
• F - ?U
• The sum of all energy, potential and kinetic,
  of a system is conserved, in the absence of
  dissipation.
• E U K W

18
Physics 1710 Chapter 8Potential Energy

• 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