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Chapter 4
Work and Energy
Additional Concepts For Describing Motion
PSC 150
Exercise
"Conservation of Energy"
Results and Conclusions
h, m
20.86
64.63
81.48
73.53
39.16
1.5
Free-fall
mass = 10kg
v, m/s KE, J GPE, J
34.51 5955 2044
18.23 1662 6334
1.36
9
7985
-12.56 789
7206
-28.84 4159 3838
-39.63 7853
147
E, J
8000
8000
7990
8000
8000
8000
Questions:
1.) Based on your table, as the object moves UPWARD its kinetic energy:
Decreases
2.) Based on your table, as the object moves UPWARD its gravitational potential
energy: Increases
3.) Based on your table, as the object moves DOWNWARD its kinetic energy:
Increases
4.) Based on your table, as the object moves DOWNWARD its gravitational
potential energy: Decreases
5.) Based on your table, as the object moves UPWARD its total mechanical energy:
Remains Constant
6.) Based on your table, as the object moves DOWNWARD its total mechanical
energy: Remains Constant
7.) As the object moves the only force acting on it is: Gravitational Force
Conclusion
If the only force acting on an object
is the gravitational force the kinetic
and gravitational potential energies
may change but the total mechanical
energy remains constant.
h, m
4.64
1.73
0.15
1.52
3.02
5.91
Pendulum
mass = 5kg
v, m/s KE, J GPE, J
82
227
5.71
210
85
9.16
287
7
10.71
220
74
9.38
146
148
7.65
5
290
1.36
E, J
309
295
294
294
294
295
Questions
8.) Based on your table, as the pendulum swings DOWNWARD its kinetic energy:
Increases
9.) Based on your table, as the pendulum swings DOWNWARD its gravitational
potential energy: Decreases
10.) Based on your table, as the pendulum swings UPWARD its kinetic energy:
Decreases
11.) Based on your table, as the pendulum swings UPWARD its gravitational
potential energy: Increases
12.) Based on your table, as the pendulum swings UPWARD or DOWNWARD its
total mechanical energy:
Remains Constant
13.) As the pendulum swings the two forces acting on it are:
Gravitational Force and Tension
Tension is always perpendicular to the direction of motion.
Conclusion
If the only force acting on an object
is the gravitational force or if there
are other forces which are always
perpendicular to the direction of
motion the kinetic and gravitational
potential energies may change but
the total mechanical energy remains
constant.
Pendulum
Highest
D
Point
A Highest
Point
B
Equilibrium Level
C
14.) At what point(s) (A,B,C,D) does the kinetic energy have its maximum value?
C
15.) At what point(s) (A,B,C,D) does the gravitational potential energy have its
maximum value?
A&D
16.At what point(s) (A,B,C,D) does the pendulum have both kinetic and potential
energy?
B
h. m
0
0
0
0
0
0
v, m/s
5
10.1
16.2
21.3
23.6
25.8
KE, J
63
255
656
1134
1392
1664
Horizontal Force
mass = 5 kg
GPE, J E, J
0
63
255
0
656
0
1134
0
1392
0
1664
0
F = 20 N
E, J
d, m F x d, J
0
***
***
9.7
192
194
29.7
594
593
53.7
1074
1071
1326
66.3
1329
1602
80.1
1601
%
***
1.0%
0.2%
0.3%
-0.2%
0.1%
Questions:
17.) Based on your table, as the object moves its kinetic energy: Increases
18.) Based on your table, as the object moves its gravitational potential energy:
Remains Constant
19.) Based on your table, as the object moves its total mechanical energy:
Increases
20.) As the object moves the applied force was in the SAME Same
OPPOSITE
direction as the motion.
2
kg

m
21.) The units of Force X Displacement are:
2
s
22.) The units of Force X Displacement are the same as the units of mechanical
energy: True
23. As the object moves the change in its total mechanical energy, E,
approximately equals the product of the applied force and the displacement.
True
Conclusion
When a net external force acts on
an object in the same direction as
its motion the total mechanical
energy increases.
The change in the object’s total
mechanical energy equals the
product of the net force and the
object’s displacement.
h, m
0
0
0
0
0
0
v, m/s
33
28.7
23.6
13.6
9.3
1.3
Frictional Force
mass = 3kg
KE, J GPE, J E, J
0
1634
1634
0
1236
1236
0
835
835
0
277
277
0
130
130
0
3
3
f = -20N
 J d, m F x d, J
***
***
0
-396
-398
19.8
-798
-799
39.9
-1357
67.8 -1356
-1504
75.2 -1504
-1631
81.6 -1632
%
***
-0.8%
-0.2%
-0.1%
0.0%
0.1%
Questions:
17.) Based on your table, as the object moves its kinetic energy: Decreases
18.) Based on your table, as the object moves its gravitational potential energy:
Remains Constant
19.) Based on your table, as the object moves its total mechanical energy:
Decreases
20.) As the object moves the frictional force was in the SAME
OPPOSITE
Opposite
direction as the motion.
23. As the object moves the change in its total mechanical energy, E,
approximately equals the product of the applied force and the displacement.
True
Conclusion
When a frictional force acts on an
object in the opposite direction as
its motion the total mechanical
energy decreases.
The change in the object’s total
mechanical energy equals the
product of the frictional force and
the object’s displacement.
Define :
F  d  work
Work is done when a force acts
Units :
kg  m
 Joule, J
2
s
on an object AND the object moves
parallel to that force. Work is a scalar!
If F & d are in the same direction
the work is positive and the work
is done ON the object.
If F & d are in opposite directions
the work is negative and
the work is done BY the object.
2
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.”
The Work-Energy Theorem can be
derived from Newton’s Second Law.
F  ma
multiply both sides by "d"
F  d = ma  d
Using : v 2f = vi2 + 2ad
a d =
2
vf
2
 vi
2
Substituting gives :
v 2  v 2 
F  d = m f 2 i 


Simplyfing gives :
2
1 mv2
Fd = 1
mv

f
i
2
2
 W  KE
Another Type of Energy
Define:
Gravitational Potential Energy is something an object
has because of its position.
Gravitational Potential Energy, GPE  m g h
GPE is a Scalar!
The operational definition of GPE
requires the choice of a " reference level"
Units :
from which " h" can be measured.
The choice is arbitrary but is usually kg  m 2
= Joule, J
2
chosen at the object' s initial position.
s
Let depends
the floor be
GPE
on the
where
reference
the objectlevel
is.
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