# Conservation of Energy

```Conservation of Energy
Forces
• Conservative Force - Gives back work done
against it. (Gravity &amp; springs)
• Non-conservative Force - Does not give
back work done against it. (Friction)
• Criteria for a conservative force
 
F

d
r

0

Work on a
Closed Path
• What total work is done by gravity going
from A to B to C and back to A again?
1m
B
A
-1m
0m
1m
C
Potential Energy
• If a force is conservative, it can be related to
potential energy.
U AB  
B
A
 
F  dr
• For gravity near the earth’s surface
y1
U AB     mg ˆj  dy ˆj
y0
U AB  mgy
Falling Brick
• A 5.0 kg brick is tossed from a 14 m tall
building. If the initial velocity were 20 m/s
upward, what is the change in potential
energy when the brick hits the ground?
h
Millenium Force
• Are the predictions of the
Millenium Force speed
accurate?
EK  EP
1
2
mv  mgh
2
v  2 gh
300ft Drop
&gt;92mph
Mechanical Energy
• From the work energy theorem
K  Wnet
K  Wcons  Wnoncons
K  U  Wnoncons
 K  U  Wnoncons
• If only conservative forces working, then
K  U  0
K 0  U 0  K1  U1
Springs
• The potential energy in a spring is
x1
U AB     kx iˆ  dx iˆ
x0
U AB  kx  kx
1
2
2
1
1
2
2
0
• Ex. As a compound bow is drawn, it exerts
a force of F=( 300x+150x3)N. What is the
change in potential as it is drawn from
x=0m to x=0.75m?
Arrow Speed
• What is the velocity of a 0.15 kg arrow fired
by a composite bow with a force of
F=( 300x+150x3)N and drawn to 0.75m?
Force and Potential
• If the potential is known, the force can be
determined.
dU
Fx  
dx
• Ex. The potential energy for a spring is
given by the function U=5(2 x)2 J. What
is the spring constant for this spring?
Energy Sources
Source: (2013)
https://www.eia.gov/
Oil Natural Gas
36%
27%
Coal
18%
Nuclear
8%
Renewable Sources
Hydroelectric
31%
Annual Energy Review: 2010
Solar
1%
Others
Geothermal, 3%
Biomass, 53%
Wind
11%
Energy &amp; Well Being
• Basis of HDI
– Longevity
– Education
– Standard of
Living
Chemical Fuels
• Gas
– Hydrogen (143 kJ/g)
– Natural, Syngas (56 kJ/g)
• Liquid
– Gasoline, Kerosene,
heating oil (44 - 48 kJ/g)
• Solid
– Coal (32 kJ/g)
– Wood (10 – 19 kJ/g)
– Stearic acid (40 kJ/g)
Einstein and Energy
• Relativity - Speed of
Light is the same for
everybody.
• Consequences
– Light bent by mass
– Energy related to mass
E  mc
2
•Ex. A nuclear power plant
generates 300 GW of power.
In one day how much mass is
converted to energy?
Nuclear Power
Nuclear Fusion
Nova
Tokomak
1 gallon sea water
= 300 gallon gasoline
```