Conservation of Energy

1
Conservation of Energy
2
Path Dependence

• What happens to work as a rollercoaster goes down
  hill then up again?
• What if the roller coaster took a less steep path?

3
Reversible Process

• If an object is acted on by a force has its path
  reversed the work done is the opposite sign.
• This represents a reversible process.

y2
F -mg
h
y1
4
Closed Path

• If the work done by a force doesnt depend on the
  path it is a conservative force.
• Conservative forces do no work on a closed path.

From 1 to 2, the path A or B doesnt matter
From 1 to 2 and back to 1, the path A then the
reverse path B gives no work
5
Nonconservative Force

• Not all forces are conservative.
• In particular, friction and drag are not
  conservative.

d
Negative work is done by friction to get here
F -mFN
-d
Negative work is also done returning the box
F mFN
6
Net Work

• The work-energy principle is DK Wnet.
• The work can be divided into parts due to
  conservative and non-conservative forces.
• Kinetic energy DK Wcon Wnon

Ff
d
Fg
7
Kinetic and Potential Energy

• Potential energy is the negative of the work done
  by conservative forces.
• Potential energy DU -Wcon
• The kinetic energy is related to the potential
  energy.
• Kinetic energy DK -DU Wnon
• The energy of velocity and position make up the
  mechanical energy.
• Mechanical energy Emech K U

8
Conservation of Energy

• Certain problems assume only conservative forces.
• No friction, no air resistance
• The change in energy, DE DK DU 0
• If the change is zero then the total is constant.
• Total energy, E K U constant
• Energy is not created or destroyed it is
  conserved.

9
Solving Problems

• There are some general techniques to solve energy
  conservation problems.
• Identify all the potential and kinetic energy at
  the beginning
• Identify all the potential and kinetic energy at
  the end
• Set the initial and final energy equal to one
  another
• Nonconservative forces reduce the final energy

10
Using Friction and Energy

• The hill is 2.5 km long with a drop of 800 m.
• The skier is 75 kg.
• The speed at the finish is 120 km/h.
• How much energy was dissipated by friction?

q
11
Friction and Height

• Find the total change in kinetic energy.
• Find the total change in potential energy.
• The difference is due to friction (air and
  sliding).

• DK ½ mv2 - 0
• ½(75 kg)(130 m/s)2
• 5.4 x 105 J
• DU mgh
• (75 kg)(9.8 m/s2)(-800 m)
• -5.9 x 105 J
• Wnon DK DU
• -0.5 x 105 J