Kinetic Energy and Work;
Potential Energy;Conservation
of Energy.
Lecture 07
Thursday: 5 February 2004
WORK
•Work provides a means of determining the
motion of an object when the force applied to it
is known as a function of position.
•For example, the force exerted by a spring
varies with position:
F=-kx
where k is the spring constant and x is the
displacement from equilibrium.
WORK (Constant Force)
W  Fd
W  Fd cos
WORK (Variable Force)
W   F ( x )dx
xf
xi
Work Energy Theorem
• Wnet is the work done by
• Fnet the net force acting on a body.
Wnet   Fnet ( x )dx
xf
xi
Work Energy Theorem
(continued)
Wnet   Fnet dx
xf
xi
dv
  madx  m
dx
dt
v dx
v
 mv
dv  mv vdv
dt
xf
xi
xf
xi
f
i
f
i
Work Energy Theorem
(continued)
Wnet  m vdv
vf
vi
2 vf
v 
2
2
1
 m   2 m(v f  vi )
 2 v
i
Wnet  mv  mv
1
2
2
f
1
2
2
i
Work Energy Theorem
(concluded)
• Define Kinetic Energy
K  mv
1
2
2
• Then,
• Wnet = Kf - Ki
• Wnet = DK
Recall Our Discussion of the
Concept of Work
W  F d
W  F d cos
•Work has no direction associated with it (it is a scalar).
•However, work can still be positive or negative.
•Work done by a force is positive if the force has a component (or
is totally) in the direction of the displacement.
CONSERVATIVE FORCES
•A force is conservative if the work it does on a
particle that moves through a closed path is zero.
Otherwise, the force is nonconservative.
 F  dr  0
•Conservative forces include: gravitational force
and restoring force of spring.
Fg
• Nonconservative forces include: friction,
pushes and pulls by a person .
d
CONSERVATIVE FORCES
If a force is conservative, then the work it does on
a particle that moves between two points is the
same for all paths connecting those points.
This is handy to know because it means
that we can indirectly calculate the work
done along a complicated path by calculating
the work done along a simple (for example, linear) path.
Work Done by Conservative
Forces is of Special Interest
• The work “done” in the course of a motion, is
“undone” in if you move back.
Fg
d
This encourages us to define another kind of energy (as opposed to kinetic energy)
- a “stored” energy associated with conservative forces.
• We call this new type of energy potential energy
and define it as follows:
DU = – Wc
Potential Energy Associated
with the Gravitational Force
DU  W   
rf
ri
DU   
yf
Fds
Fy dy
yi
Fy   mg
DU   
yf
(  mg )dy
yi
 mg 
yf
yi
dy
DU  mg ( y f  yi )  mgDy
Potential Energy Associated
with the Spring Force
We know (or should know) from our homework,
Wspring force  12 kxi2  12 kx2f .
So, we can deduce that for a spring force,
DU  12 kx2f  12 kxi2
Tying Together What We
Know about Work and Energy
 DU = – Wc
• Wnet = DK
So, under the condition that there are
only conservative forces present :
Wnet = Wc
In that case, DK = – DU
DK + DU = 0
The “Bottom Line”
• Ei = Ef
• Ki + Ui = Kf + Uf
• The “Total Mechanical Energy” of a System is the
sum of Kinetic and Potential energies. This is
what is “conserved” or constant.
 Gravitational
force: U= mgh
 Restoring force of a spring: U =1/2kx2
 (KE=1/2mv2)
An Example
A 70 kg skate boarder is moving at 8 m/s on
flat stretch of road. If the skate boarder now
encounters a hill which makes an angle of
10o with the horizontal, how much further up
the road will the he be able to go without
additional pushing? Ignore Friction.
d
10o
h
KEi+Ui=KEf +Uf (only conservative forces)
so
KEi + 0 = 0+Uf (Ui=0 and KEf=0)
1/2mv2 = mgh
1/2v2 = gh
h = v2/(2g) = 82/(2*9.8) = 3.26 m
h/d = Sin 10o
d = 18.8 m