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Chapter 6
Work and Energy
0) Conservation laws
• Newton’s laws (with kinematic equations) allow a complete and
continuous description of motion, but requires
– knowledge of the force at all times
– detailed calculations which may be prohibitive
• Conservation laws allow intermediate details to be ignored by
realizing some quantities are conserved
• Conservation of momentum, which follows directly from Newton’s
laws and will be considered later, was discovered first (Descartes).
Momentum (mv) was considered an essential quality of motion.
0) Conservation laws
• The idea that we should “pay” for producing motion is absent in
Newton’s laws and momentum.
• Huygens never cared for the idea that momentum was an essential
quality of motion: A bomb could explode and yet the total
momentum remains zero. What kind of essential quality of motion
remains zero when bodies are being hurtled through the air?
• Huygens found the quantity mv2 was conserved in billiard ball
collisions
• In 1807 Thomas Young associated the term energy with mv2
1) Energy
• It is important to realize that in physics today, we have no
knowledge of what energy is. Richard Feynman
• As we cannot give a general definition of energy, the
principle of the conservation of energy simply signifies
that there is something which remains constant. Well,
whatever new notions of the world future experiments
may give us, we know beforehand that there will be
something which remains constant and which we will be
able to call energy. Henri Poincaré (1854 - 1912)
1) Energy
• Energy is some quantity, which takes many forms,
associated with a system which we know intuitively is
conserved. Or else we could make a perpetual motion
machine. Feynman writes that the conservation of energy
is the statement that perpetual motion is not possible.
• State of a system in relation to fundamental forces
• Force is the agent of change; energy is measure of change
2) Work
Work is the change in the energy of a system resulting from the
application of a force acting over a distance.
a) Work along a straight line: F || s
F
m
s
Work done by force F, is
W  Fs
Units: 1 N m = 1 joule = 1 J
If the displacement s is zero, no work is done.
2) Work
(b) Force and displacement not parallel
F
m

s
The change in the system is the same as if a parallel force
Fs = F cos  acted over a distance s, so the work done by F is
W  Fs s  Fs cos 
2) Work
(c) Negative work
If a force acts opposite to the direction of motion
(like friction), it does negative work. It takes energy out
of the system.
f
v

v=0
m
s
W  Fs cos   Fs cos180º  Fs
3) Kinetic Energy
• Work produces a change in energy.
• Work on an otherwise free object produces motion.
• The energy associated with motion is called kinetic
energy, KE.
• Work - energy theorem: If as a result of work on an object
(system) the only change is its resulting motion:
W  KE = KE f  KE i
3) Kinetic Energy
vi
vf
F
m
s
Using
2nd law: F  ma
Define Kinetic Energy:
KE  mv
kinematics: v f 2  vi 2  2as
1
2
gives for the work done by F:
W  Fs  mas  m
(v f 2  vi 2 )
W  12 mv f 2  12 mvi 2
2
,
2
so that
or
W  KE
3) Kinetic energy
• Work-energy theorem qualifications:
– Force is the net force on the object (system)
– No internal changes to object (system)
e. g. Lifting a rock produces no kinetic energy because there is no
net force on the rock (on average).
e.g. Compressing a spring produces no kinetic energy because
there are internal changes to the system
4) Conservative forces
• Work done by a conservative force is independent of path
• Work done by a conservative force in a closed loop is zero
– e.g. gravity, spring, electromagnetic
• all fundamental forces
• any force determined by the configuration of the system
e.g. Work done by gravity
m
m
mg
W  mgh
m
mg
W  mgh
m
mg
W  mg cos 90º  0
mg
W  mgh  0  mgh  0  0
5) Potential Energy
• Work produces a change in energy
• Work against a conservative force (lifting a rock,
compressing a spring), can be converted to motion
(kinetic energy) by releasing the system; the system has a
potential to do work.
• Work on a system of interacting particles can produce a
change in the configuration (compressed spring).
• Energy associated with the configuration of a system is
called potential energy, PE.
• If, as a result of work (W) on a system (against
conservative forces), the only change is its configuration:
W  PE  PE f - PE i
5) Potential Energy
• If, as a result of work (W) on a system (against
conservative forces), the only change is its configuration:
W  PE  PE f - PE i
Work done by conservative forces in a system produces a negative
change in the potential energy.
W (by cons forces)  PE
6) Gravitational potential energy (near earth)
Consider an object lifted by an external force to a height h. If it is
at rest before and after, then the only change is the configuration
of the earth-object system
The change in the potential energy
is the work done:
PE  W  mgh
The position of zero PE is arbitrary;
only changes are interesting. Define
PE = 0 for h = 0. Then,
PE  mgh
h
F = mg
m
(Work done by gravity is -mgh)
7) Conservation of Mechanical Energy
Mechanical energy: E = PE + KE
If all forces in an isolated system are conservative, mechanical
energy is constant
Work-energy: W = KE
If all work is done by conservative forces: W = -PE
Therefore, -PE = KE, or
E  PE + KE = constant