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Chapter 6
Work and Energy
Additional Concepts For Describing Motion
Define :
F  s  work
Units :
kg  m

Joule,
J
2
Work is done when a force acts
s
on an object AND the object moves
parallel to that force. Work is a scalar!
F  s is the dot (scalar) product:
F  s  Fs cos
  smaller angle between F and s
F  s  Fx s x  Fy s y
2
F

s
W  F  s  Fs cos  Fx s x  Fy s y
If the force and displacement (velocity) are in the same direction
 = 0°, cos = 1, and the work is positive…work is done ON the
object.
If the force and displacement (velocity) are in opposite directions
 = 180°, cos = -1, and the work is negative…work is done BY the
object.
If the force and displacement (velocity) are perpendicular
 = 90°, cos = 0, and no work is done.
Only the component of a force that is parallel to the displacement
(velocity) can do any work.
Examples of forces that NEVER do any work:
Normal Force
Centripetal Force
Tension Force in a Pendulum
Work Against Friction (Fluid Resistance)
Since friction always opposes motion the work associated with
friction is always negative…an object must do work against any
frictional force.
Also, the work against friction is dissipative and can never be
recovered. Therefore the work done against friction depends on the
distance an object travels not its displacement.
Wf   f  d
Define :
Kinetic energy is something an
Units :
object has because it is moving.
2
kg

m
2
= Joule,J
2
kinetic energy, KE  1
mv
s
2
KE is a scalar!
Change in kinetic energy,
2
2
1
1
KE = KEf - KEi  mv f  mvi
2
2
If KE is positive,
the kinetic energy has increased.
If KE is negative,
the kinetic energy has decreased.
Work-Energy Theorem
“When a net force causes an
object to accelerate, the work
done on or by the object equals
the change in the object’s kinetic
Wnet force  KE
energy.”
Another Type of Energy
Define:
Gravitational Potential Energy is something an object
has because of its position.
Gravitational Potential Energy, GPE  m g h
Units :
m
g  9.8
s
2
GPE is a Scalar!
kg  m 2
= Joule, J
2
s
The operational definition of GPE
requires the choice of a " reference level"
from which " h" can be measured.
The choice is arbitrary but is usually
chosen at the object' s initial position.
Let depends
the floor be
GPE
on the
where
reference
the objectlevel
is.
GPE can be positive or negative depending on
whether the object is above (h > 0) or below (h < 0)
the reference level.
h=0
h=1.1m h=2.2m
h=.6m
change in gravitational potential energy,
GPE = m g h f  m g hi
If GPE is positive, the gravitational Potential Energy
has increased.
If GPE is negative, the gravitational Potential Energy
has decreased.
“When an external force (equal to the object’s weight)
lifts an object at a constant velocity, the work done by
that force equals the change in the object’s Gravitational
Potential Energy.”
Wconstant velocity  GPE
Total Mechanical Energy, E
The Total Mechanical Energy of an object is defined as the sum of
its Kinetic Energy and Gravitational Potential Energy.
E = KE + GPE
Extended Work-Energy Theorem
“The work done by any force other
than the gravitational force equals
the change in Total Mechanical
Energy.” W
 E
FFG
What if the only force acting on an object is
the gravitational force?
Work done by the gravitational force does NOT change the total
mechanical energy it does cause a conversion between kinetic
energy and gravitational potential energy.
The gravitational force is called a Conservative Force.
If the work done by the gravitational force is positive…
the gravitational force is in the same direction as the displacement,
gravitational potential energy is converted into kinetic energy.
WFG ()
GPE  KE
If the work done by the gravitational force is negative…
the gravitational force is in the opposite direction as the
displacement, kinetic energy is converted into gravitational potential
energy.
W () KE  GPE
FG
The Law of Conservation of Energy
“If the only force acting on an object is
the gravitational force, or if there are
other forces acting on the object but
they do no work, the kinetic and
gravitational energies may change but
the total mechanical energy remains
constant.”
Wnonconservative  0  E  0
General Work-Energy Theorem
Work Done On Object
BY
Non-Conservative
Lifting Force
greater than
Forces
Net Force
weight
Total Energy Increases
Kinetic
Energy
e.g.,Friction
Work Done by
Total Energy Constant
Conservative Forces
e.g., gravity
Gravitational
Potential
Energy
Total Energy Decreases
Work Done BY Object
Against
Non-Conservative
Forces
Lifting Force
Less than
weight
Additional Concepts and
Relationships
Another Kind of Conservative Force and Potential Energy
Elastic Force
Felastic  kx
Is Felastic a conservative force?
Yes, therefore, there is a potential energy associated with the elastic
force.
What is the mathematical form of this potential energy?
Plotting a graph of Elastic Potential Energy versus displacement:
EPE versus x
500
k = 50 N/m
400
EPE, J
300
200
100
EP E, J
0
-5
-4
-3
-2
-1
0
x, m
1
2
3
4
5
Plotting a graph of Elastic Potential Energy versus displacement
squared:
EPE versus x^2
EPE = 25x^2
EPE, J
500
k = 50 N/m
EPE, J
400
300
200
100
0
0
5
10
15
x^2, m^2
20
25
EPE  25x
2
EPE(J)  25(?)x(m)
2
J  Nm
N  m  (?)m 2
? N 
m
units of force constant k
25 N  1 k
m 2
EPE  1 kx2
2
Alternate Form of Gravitational Potential Energy
The equation GPE = mgh assumes that g is constant over the range
of possible values of h.
If for example the motion of a satellite is described in terms of its
kinetic and potential energies an alternate form of GPE must be
used since over the range of possible altitude of the satellite the
value of g can not be taken as constant.
GPE satellite  m satellitegh
Gmearth r  distance from earth' s center
 e,s
g
2
re,s
Gm earth
GPEsatellite  msatellite
2 (re  h)
(re  h)
Gmsatellitemearth
GPEsatellite 
re  h
Is the altitude of a satellite changes from hi to hf the corresponding
change in its GPE will be given by:
Gm satellitemearth Gmsatellitemearth

GPEsatellite 
re  hi
re  h f
GPEsatellite  Gm satellitem earth( 1  1 )
re  h f re  hi
From this equation we see that as the altitude increases the GPE
decreases.
If h f  h i  r 1 h  r 1 h  r 1 h  r 1 h  0
e
f
e
i
e
f
e
i
GPE  0  GPE decreasing
as h  , GPE  0
Gmsatellitemearth
as h  0, GPE  
rearth
What minimum velocity must a rocket be given for it to escape
from the earth’s gravity?
v i  escape velocity, h i = 0
v f  0, hf  
E  0
KE i  GPE i  KE f  GPE f
0
0
Gmsatellitemearth 1
Gmsatellitemearth
2
2
1m
 msatellitev f 
2 satellitevi 
rearth  h i
2
rearth  h f
Gmsatellitemearth
2
1m
v

satellite
i
2
rearth
vi 
2Gmearth
rearth
2
24
Nm
2(6.67  10
2 )(6  10 kg)
kg
6.38  10 6 m
11
vi 
vi  1.12  10 m
s
4
1km  1mi  3600s
1.12  10 m

s 1000m 1.6km
1hr
4
vi  25, 200 mi
hr
Power
In the past we have defined average velocity as the rate of change of
position and average acceleration as the rate of change of velocity.
We now define the rate of change of energy as average power.
P  E
t
Units :
J  watt
s
Power may be expressed in two other ways:
W
P

1. Since W=E:
t
2. If the force is constant: P  Fv
Unit Conversion Factors:
1horsepower(hp)  746watts
1kilowatt hr(kW  hr)  3.6  10 6 J Note: kWhr is a unit of energy.
What horsepower engine would be required to accelerate a 1200kg
car from 0 to 60mph in 5s?
1.6km  1000m  3600s  26.7 m
60 mi

hr
1mi
1km
1hr
s
m 0
26.7
vf  vi
s
m
a


5.34
t
5s
s2
F  ma  1200kg  5.34 m2  6408N
s
m
0

26.7
v v
s  13.35 m
v i 2 f 
2
s
P  Fv  (6408N)(13.35 m )  8.55  10 4 watts
s
1hp
8.55  10 4 watts  746watts  115hp