Chapter 7Conservation of Energy
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
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.
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
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
Conservative Forces
The work done ona particle bya conservative force is independent of the path takenbetween any two points
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
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
Potential-Energy Function of a Spring
210 2U U kx By convention,
one choosesU0 =U(0) = 0
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
7-1Conservation of Energy
Conservation of Energy
inE 0sysE
Energy can be neither created nor destroyed
outE
0sysE Closed System
Open System
Conservation of Mechanical Energy
constantmechE K U
If the forces acting are conservativethen the mechanical energy is conserved
Example 7-3 (1)
How high does the block go?
Initial mechanical energy of system
212iE kx
Final mechanical energy of system
fE mgh
Example 7-3 (2)
Forces are conservative, therefore,mechanical energy is conserved
212 kx mgh
Height reached2
2
kxh
mg
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
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
Example 7-4 (3)
2mgd
k
Solve for d
12( ) 0kd mg d
2mg
Since d ≠ 0
Example 7-4 (4)
gravE mgd
Note21
2springE kd
2mg
is equal to loss in gravitational potentialenergy
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.
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
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
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
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
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
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.
E = mc2
In a brief paper in 1905Albert Einstein wrotedown the most famousequation in science
E = mc2
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
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
Problems
To go…
Ch. 7, Problem 19
Ch. 7, Problem 29
Ch. 7, Problem 74