chapter 8 potential energy and conservation of...
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Units of Chapter 8
• Conservative and NonconservativeForces
• Potential Energy and the Work Done byConservative Forces
• Conservation of Mechanical Energy
• Work Done by Nonconservative Forces
• Potential Energy Curves andEquipotentials
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8-1 Conservative and NonconservativeForces
Conservative force: the work it does is stored inthe form of energy that can be released at a latertime
Example of a conservative force: gravity
Example of a nonconservative force: friction
Also: the work done by a conservative forcemoving an object around a closed path is zero;this is not true for a nonconservative force
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8-1 Conservative and NonconservativeForces
Work done by gravity on a closed path is zero:
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8-1 Conservative and NonconservativeForces
Work done by friction on a closed path is notzero:
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8-1 Conservative and NonconservativeForces
The work done by a conservative force is zeroon any closed path:
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8-2 The Work Done by ConservativeForces
If we pick up a ball and put it on the shelf, wehave done work on the ball. We can get thatenergy back if the ball falls back off the shelf; inthe meantime, we say the energy is stored aspotential energy.
(8-1)
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8-2 The Work Done by ConservativeForces
Gravitational potential energy:
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Example: What is the change in gravitational potentialenergy of the box if it is placed on the table? The table is1.0 m tall and the mass of the box is 1.0 kg.
( )( )( )( ) J 8.9m 0m 0.1m/s 8.9kg 0.1 2 +=!=
!="=" ifg yymgymgU
First: Choose the reference level at the floor. U = 0 here.
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Example continued:
( )( )( ) ( )( ) J 8.9m 0.1 m0.0m/s 8.9kg 1 2 +=!!=
!="=" ifg yymgymgU
Now take the reference level (U = 0) to be on top of thetable so that yi = −1.0 m and yf = 0.0 m.
The results do not depend onthe location of U = 0.
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8-3 Conservation of Mechanical EnergyDefinition of mechanical energy:
(8-6)
Using this definition and considering onlyconservative forces, we find:
Or equivalently:
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8-3 Conservation of Mechanical EnergyEnergy conservation can make kinematicsproblems much easier to solve:
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Example (text problem 6.28): A cart starts from position 4with v = 15.0 m/s to the left. Find the speed of the cart atpositions 1, 2, and 3. Ignore friction.
( ) m/s 5.202
2
1
2
1
34
2
43
2
33
2
44
3344
34
=!+=
+=+
+=+
=
yygvv
mvmgymvmgy
KUKU
EE
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( ) m/s 0.182
2
1
2
1
24
2
42
2
22
2
44
2244
24
=!+=
+=+
+=+
=
yygvv
mvmgymvmgy
KUKU
EE
( ) m/s 8.242
2
1
2
1
14
2
41
2
11
2
44
1144
14
=!+=
+=+
+=+
=
yygvv
mvmgymvmgy
KUKU
EE
Or useE3=E2
Or useE3=E1
E2=E1
Example continued:
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Example (text problem 6.84): A roller coaster car is about toroll down a track. Ignore friction and air resistance.
40 m
20 m
y=0
m = 988 kg
(a) At what speed does the car reach the top of the loop?
( ) m/s 8.192
2
10
2
=!=
+=+
+=+
=
fif
ffi
ffii
fi
yygv
mvmgymgy
KUKU
EE
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(b) What is the force exerted on the car by the track at the topof the loop?
N w
y
x
N 109.24
2
2
2
!="=
=+
"="=""=#
mgr
vmN
r
vmwN
r
vmmawNF ry
Example continued:
FBD for the car:
Apply Newton’s Second Law:
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(c) From what minimum height above the bottom of the trackcan the car be released so that it does not lose contact withthe track at the top of the loop?
Example continued:
2
min
2
10 mvmgymgy
KUKU
EE
fi
ffii
fi
+=+
+=+
=
Using conservation of mechanical energy:
g
vyy fi
2
2
min+=Solve for the starting height
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Example continued:
v = vmin when N = 0. This means that
grv
r
vmmgw
r
vmwN
=
==
=+
min
2
min
2
What is vmin?
m 0.255.22
22
2
min ==+=+= rg
grr
g
vyy fi
The initial height must be
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8-4 Work Done by Nonconservative Forces
In the presence of nonconservative forces, thetotal mechanical energy is not conserved:
Solving,
(8-9)
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8-4 Work Done by Nonconservative Forces
In this example, thenonconservative forceis water resistance:
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8-5 Potential Energy Curves andEquipotentials
The curve of a hill or a roller coaster is itselfessentially a plot of the gravitationalpotential energy:
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8-5 Potential Energy Curves andEquipotentials
The potential energy curve for a spring:
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8-5 Potential Energy Curves andEquipotentials
Contour maps are also a form of potentialenergy curve:
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Summary of Chapter 8
• Conservative forces conserve mechanicalenergy
• Nonconservative forces convert mechanicalenergy into other forms
• Conservative force does zero work on anyclosed path
• Work done by a conservative force isindependent of path
• Conservative forces: gravity, spring
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Summary of Chapter 8• Work done by nonconservative force on closedpath is not zero, and depends on the path
• Nonconservative forces: friction, airresistance, tension
• Energy in the form of potential energy can beconverted to kinetic or other forms
• Work done by a conservative force is thenegative of the change in the potential energy
• Gravity: U = mgy
• Spring: U = ½ kx2
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Summary of Chapter 8
• Mechanical energy is the sum of the kinetic andpotential energies; it is conserved only insystems with purely conservative forces
• Nonconservative forces change a system’smechanical energy
• Work done by nonconservative forces equalschange in a system’s mechanical energy
• Potential energy curve: U vs. position