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Chapter 7Conservation of Energy
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Recap – Work & Energy
2 21 12 2f iW mv mv
The total work done on a particle is equal to the change in its kinetic energy
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Potential Energy
The total work done on an object equals the change in its kinetic energy
But the total work done on a system of objects may or may not change its total kinetic energy. The energy may be stored as potential energy.
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Potential Energy – A Spring
Both forces do work on the spring. Butthe kinetic energy of the spring is unchanged. The energy is stored as
potential energy
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Conservative Forces
If the ski lift takes youup a displacement h, thework done on you, bygravity, is –mgh.
But when you ski downhill the work done by gravity is +mgh, independent of the path you take
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Conservative Forces
The work done ona particle bya conservative force is independent of the path takenbetween any two points
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Potential-Energy Function
2
12 1
s
sU U U F ds
If a force is conservative, then we candefine a potential-energy function as thenegative of the work done on the particle
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Potential-Energy Function
0
0
0
0
0
( ˆˆ ˆ) ( )
( )
ˆ
s
s
s
s
y
y
i
U U U F ds
mg dx dy dz
mg dy
m
k
g y y
j j
potential-energy function associatedwith gravity (taking +y to be up)
0 0( )U U mg y y
The valueof U0 = U(y0)can be set to any convenientvalue
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Potential-Energy Function of a Spring
210 2U U kx By convention,
one choosesU0 =U(0) = 0
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Force & Potential-Energy Function
dUF
dx
In 1-D, given the potential energy function associated with a force one can compute the latter using:
Example:
212
dUU kx F kx
dx
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7-1Conservation of Energy
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Conservation of Energy
inE 0sysE
Energy can be neither created nor destroyed
outE
0sysE Closed System
Open System
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Conservation of Mechanical Energy
constantmechE K U
If the forces acting are conservativethen the mechanical energy is conserved
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Example 7-3 (1)
How high does the block go?
Initial mechanical energy of system
212iE kx
Final mechanical energy of system
fE mgh
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Example 7-3 (2)
Forces are conservative, therefore,mechanical energy is conserved
212 kx mgh
Height reached2
2
kxh
mg
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Example 7-4 (1)
How far does the mass drop?
2 21 12 2i i i iE mgy ky mv
Final mech. energy2 21 1
2 2f f ff mgy ky vE m
Initial mech. energy
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Example 7-4 (2)
2 21 12 2
2 21 12 2
( ) ( ) (0
(0)
)
(0) (0)mg m
mg d d m
Final mech. energy = Initial mech. energy
2 21 12
2 2
2
2 21 1f f f
i i imgy ky mv
mgy ky mv
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Example 7-4 (3)
2mgd
k
Solve for d
12( ) 0kd mg d
2mg
Since d ≠ 0
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Example 7-4 (4)
gravE mgd
Note21
2springE kd
2mg
is equal to loss in gravitational potentialenergy
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Conservation of Energy & Kinetic Friction
Non-conservative forces, such as kinetic friction, cause mechanical energy to be transformed into other forms of energy, such as thermal energy.
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Work-Energy Theorem
Work done, on a system, by external forces is equal to the change in energyof the system
ext sysW EThe energy in a system can be distributed in many different ways
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Example 7-11 (1)
ext sysW E
Find speed of blocks after spring isreleased. Consider spring & blocks as system. Write down initial energy.Write down final energy.Subtract initialfrom final
ext sysW E
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Example 7-11 (2)
212i iE kx
Initial Energy
ext sysW E
Take potentialenergy of system to be zero initially
Kinetic energy of system is zeroinitially
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Example 7-11 (3)
21
2112 2 1
212 20
othermf s m
k
E
m
E
m v m g
E E
gv
E
ms s
Final Energy
ext sysW E
Kinetic and potential energies of system have changed
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Example 7-11 (4)
ext f iW E E Subtract initial energy from final energy
ext sysW E
But since noexternal forcesact, Wext = 0, soEf = Ei
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Example 7-11 (5)
22 1
1 2
2 2i kkx m g s m g sv
m m
And the answer is…
ext sysW ETry to derivethis.
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E = mc2
In a brief paper in 1905Albert Einstein wrotedown the most famousequation in science
E = mc2
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Sun’s Power Output
Power1 Watt = 1 Joule/second100 Watt light bulb = 100 Joules/second
Sun’s power output3.826 x 103.826 x 102626 Watts Watts
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Sun’s Power Output
Mass to Energy Kg/s = 3.826 x 10 3.826 x 102626 Watts Watts / (3 x 108 m/s)2
The Sun destroys mass at~ 4 billion kg / s
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Problems
To go…
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Ch. 7, Problem 19
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Ch. 7, Problem 29
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Ch. 7, Problem 74