Download Ch.7 Energy Conservation

7 Conservation of Energy
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Potential Energy
The Conservation of Mechanical Energy
The Conservation of Energy
Mass and Energy
• Hk: 23, 27, 39, 47, 55, 65, 69, 71
Potential Energy
• Potential Energy is stored energy
• Potential Energy is position dependent
(KE is speed dependent)
• Ex. object at higher height has more PE
• Types of PE: gravitational, elastic, electric,
magnetic, chemical, nuclear.
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Conservative Forces
• When the work done by a force moving
from position 1 to 2 is independent of the
path, the force is Conservative.
• The work done by a Conservative Force is
zero for any closed path.
• Conservative Forces have associated
Potential Energies
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Non Conservative Forces
• Produce thermal energy, e.g. friction
• Work done by Non Conservative Forces is
path dependent, e.g. longer path, more
work required
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Potential Energy Functions
W 
2
1
 
F  d    U
PE must decrease to produce  work
 
U  U 2  U1   F  d 
2
1
Definition Potential Energy Function
Elastic Potential Energy
 
dU   F  d    Fdx  (kx)dx
dU  kxdx
U   kxdx  kx  U o
1
2
2
U  kx
1
2
2
Definition Elastic Potential Energy
Ex. Elastic Potential Energy
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100N/m spring is compressed 0.2m.
F = -kx = -(100N/m)(0.2m) = -20N
U = ½kx2 = ½(100N/m)(0.2m)2 = 2J
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Gravitational Potential Energy
 
dU   F  d    Fdx  (mg )dy
dU  mgdy
U   mgdy  mgy  U o
U  mgy
Definition Gravitatio nal Potential Energy
Ex. Gravitational Potential Energy
• Ex: A 2kg object experiences weight
(2kg)(9.8N/kg) = 19.6N.
• At 3m above the floor it has a stored
energy of mgy:
• (2kg)(9.8N/kg)(3m) = 48.8Nm = 48.8J.
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Conservation of Energy
• Individual energy levels change.
• Sum of all individual energies is constant.
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Conservation of Mechanical Energy
Wtotal  Wext  Wnc  Wc
Wext  Wnc  Wtotal  Wc
Wext  Wnc  KEsys  U sys  ( KEsys  U sys )
Emech  KEsys  U sys
Definition of Mechanical Energy
Conserved when Wext & Wnc  0
Ex. Conservation of Mechanical Energy:
Object dropped from height h above floor.
ME1  mgh  m(0)  mgh
2
1
2
ME2  mg (0)  mv  mv
1
2
2
1
2
ME2  ME1
1
2
mv  mgh
2
v  2 gh
2
Energy
E1
E2
E3
Kinetic
0
½mv22
0
PE-g
0
0
mgh
PEspring
½kx2
0
0
Totals

½kx2
½mv22
mgh
Energy
Kinetic
PE-g
Totals
E(h)
E(y)
0
½mv2
mgh
mgy
mgh
½mv2
+ mgy
Energies and speeds are same at
height y
Accelerations at y are not same
Work Energy with Friction
Wext  Wnc  Emech
Wext  (Etherm )  Emech
Wext  Emech  Etherm  Emech  f k srel
Work Energy wit h Friction
0  Emech  Etherm
E  Emech  Etherm  constant
Total Energy of an isolated system is conserved
Example: The smaller the
frictional force fk, the
larger the distance, s, it will
travel before stopping.
s
Energy
Ei
Ef
Kinetic
½mvi2
0
PE-g
0
0
Thermal
0
fks
Totals
½mvi2
fks
A 2.00kg ball is dropped from rest from a height of 1.0m above
the floor. The ball rebounds to a height of 0.500m. A movieframe type diagram of the motion is shown below.
Type
E1
E2
E3
E4
E5
gravitational
mg(1)
0
0
0
mg(1/2)
kinetic
0
½ m(v2)2
0
½ m(v4)2
0
elastic
0
0
PE-elastic
0
0
thermal
0
0
E-thermal
E-thermal
E-thermal
By energy conservation, the sum of all energies in each column
is the same, = E1 = mg(1) = 19.6J
Calculate v2: (use 1st and 2nd columns)
mg(1) = ½ m(v2)2.
g = ½ (v2)2.
v2 = 4.43m/s
Calculate PE-thermal: (use 1st and 5th columns)
mg(1) = mg(1/2) + PE-thermal
mg(1/2) = PE-thermal
PE-thermal = 9.8J
Calculate PE-elastic: (use 1st and 3rd columns)
PE-elastic + PE-thermal = mg(1)
PE-elastic + 9.8 = 19.6
PE-elastic = 9.8J
Calculate v4: (use 1st and 4th columns)
½ m(v4)2 + PE-thermal = mg(1)
½ m(v4)2 + 9.8 = 19.6
½ m(v4)2 = 9.8
(v4)2 = 2(9.8)/2
v4 = 3.13m/s
Potential Energy & Force
 
dU   F  d    Fx dx
dU
Fx  
dx
Ex. U  kx
1
2
2
dU
d 1 2
Fx  
  ( 2 kx )  kx
dx
dx
Equilibrium
• Stable: small displacement in any direction
results in a restoring force toward
Equilibrium Point
• Unstable: small displacement in any
direction results in a force away from
Equilibrium Point
• Neutral: small displacement in any
direction results in zero force
Mass and Energy
E  mc
2
c  3.0 10 m / s E  mc
8
Energy of 1 milligram
110 g 1kg
-6
 10 kg
1
1000 g
-3
E  mc  10 kg(3 10 m/s)
2
 9 10 J
10
-6
8
2
2
Efficiency & Thermodynamics
W  U
This equation implies 100% efficiency
Less efficient :
Wout  Qout  U sys
Q out  heat, e.g. gasoline engine
U sys  Wout  Qout
U sys  Won  Qin
First Law of Thermodyna mics
Summary
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Potential Energy function & force
The Conservation of Mechanical Energy
The Conservation of Energy
Mass and Energy
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