1
Energy Conservation
Energy Conservation
1. Conservative/NonconservativeForces���� Work along a path
(Path integral)���� Work around any closed path
(Path integral)
2. Potential Energy
Mechanical Energy Conservation
Energy Conservation
1. Gravitational Force (Conservative Force)���� Work along a path
(Path integral)
���� Work around any closed path(Path integral)
2. Gravitational Potential Energy Function
Part I
Mechanical Energy Conservation
2
Energy Conservation
Work Done bythe Gravitational Force (I )
Consider you hold a box (mass m) and br ing it up slowly.
Find the work done by the gravitational force (FG).
Assume FG = mg
l
Energy Conservation
Near the Ear th’s sur face
Work Done bythe Gravitational Force (I I )
y
(Path integral)
dl
3
Energy Conservation
Wg < 0 if y2 > y1
Wg > 0 if y2 < y1
The work done by the gravitational
force depends only on the initial andfinal positions.
Work Done bythe Gravitational Force (I I I )
Energy Conservation
Work Done bythe Gravitational Force (IV)
Wg(A����B����C����A)
= Wg(A����B) +Wg(B����C) +Wg(C����A)
= mg(y2 - y1) +0 +mg(y1 - y2)
= 0
dl
A
BC
4
Energy Conservation
Work Done bythe Gravitational Force (V)
Wg = 0 for a closed path
The gravitational force is a conservative force.
Energy Conservation
Near the Ear th’s sur face
Gravitational Potential Energyto estimate Work Energy
mgyy
where
yy
mgymgy
=
−=−=
)(
)()(
21
21
����
��������
W
dl
y
Gravitational Potential Energy at y = y1 relative to at y = y2
5
Energy Conservation
Example 1A
A 1000-kg roller -coaster car moves from
point A, to point B and then to point C.What is its gravitational potential energy
at B and C
relative topoint A?
Energy Conservation
Example 1B
A 1000-kg roller -coaster car moves from
point A, to point B and then to point C.Find the work done by the gravitational
force between
A and C.
Hint: Wg(A����B����C) = Wg(A����B) + Wg(B����C)
6
Energy Conservation
Work-Energy Theorem ����Conservation of
Mechanical Energy (K+U)Wconservative (A����C) = UA – UC
I f Wnet = Wconservative , then
Wnet (A����C) = KC – KA =
∴
Energy Conservation
Example 2
A
A roller coaster sliding without friction alonga circular vertical loop (radius R) is to remain
B
on the track at all times. Find
the minimum release height h.
C
7
Energy Conservation
Recap:Gravitational Force
���� Conservative Force�Potential Energy Function
�Use the P.E. function to estimate the work done by the conservative forces
�Mechanical energy conservation with Work-Energy theorem
Energy Conservation
1. Spr ing Force (Conservative Force)���� Work along a path
(Path integral)���� Work around any closed path
(Path integral)
2. Elastic Potential Energy Function
Part I I
Mechanical Energy Conservation
8
Energy Conservation
Spr ing Force (Hooke’s Law)
FS(x) = −−−− k x
FPFS
Natural Length x > 0
x < 0
Spr ing Force(Restor ing Force):The spring exerts its force in thedirection opposite the displacement.
x
Energy Conservation
Work Done by a Spr ing
W S = FS(x) dx = −−−− (1/2) k x2
FS(x) = −−−− k x
Natural Length
FPFS
9
Energy Conservation
k = 2.50 � 103 N/m
x1x2
Elastic Potential Energy: US(x)
x2 = 0WS = FS(x) d x
x1 = −−−−0.030 m
= (1/2) k x12 –(1/2) k x2
2
= US(x1) – US(x2)= ++++1.13 J
x
Energy Conservation
Work Done bythe Spr ing Force
The work done by the spr ing forcedepends only on the initial and final
positions! ! !
WS = 0 for a closed path
The spr ing force is a conservative force.
10
Energy Conservation
Spr ing Force (Hooke’s Law)
FS(x) = −−−− k x+ b x2
FPFS
Natural Length x > 0
x < 0
x
Spr ing Force:restor ing force term
plus extra term(s)
Energy Conservation
Work Done by a Spr ing?
W S = FS(x) dx = −−−− (1/2) k x2 + (1/3) b x3
FS(x) = −−−− k x+ bx2
Natural Length
FPFS
11
Energy Conservation
k = 2.50 � 103 N/m
x1x2
x2 = 0WS = FS(x) d x
x1 = −−−−0.030 m
= [ (1/2) k x12 –(1/3) b x1
3 ]−−−−[ (1/2) k x2
2 –(1/3) b x23 ]
= US(x1) – US(x2)= ++++1.17 J
Potential Energy: US(x)
b = 5.00 � 103 N/m2
Energy Conservation
Part I I I
1. How to define nonconservativeForces���� Work around any closed path
(Path integral)
2. Potential Energy + Thermal Energy
Mechanical Energy Conservation
12
Energy Conservation
Work Done by F f (I )����A block (mass m) slides on a circular hor izontal
track in a circle of radius R. I ts initial speed is v0, but after one revolution the speed has dropped because of fr iction Ff = µµµµk FN = µµµµk mg. The work done by fr iction force is:
)(
)()(
)(
)(
)(
mgl
lmg
lFW
R
Rl
l
Rl
lff
µµµµ
µµµµ
ππππ
ππππ
ππππ
2
0
2
0
2
0
2
1
2
1
d
d
−−−−====
−−−−====
••••====
����
����====
====
====
====
��
dlF f
Closed Path
Energy Conservation
The work done by the fr iction forcedepends on the path length.
The fr iction force:(a) is a non-conservative force;
(b) decreases mechanical energy of the system.
Wf = 0 (any closed path)
Work Done by F f (I I )
13
Energy Conservation
Work Done by F f (I I I )
=−=
•= �=
=
)(
d
0
)(
)0(
mgl L
Ll
l
2
1
µ
lFW ff
��(Path integral)
−−−− mg L
LB
LA
L depends on the path.
Path A
Path B
Energy Conservation
Glossary
1. K: Energy associated with the motion of an object.
2. U: Energy stored in a system of objects� Can either do work or be converted to K.
3. Q: Thermal Energy (Internal Energy)� The energy of atoms and molecules that make
up a body.
14
Energy Conservation
Using Diagrams (corrected)
h/2
�
U K
����
U = Ug = mg y U = Ug + Uel
Energy Conservation
Steps in Building a Solution
1. Draw F.B.D. for each body
2. CalculateWork for individual force:1. W done by each FC using a path integral or U
2. W done by each FNC using a path integral
3. Wnet = Kf – Ki
Ki + Ui + WNC = Kf + Uf
Ki + Ui = Kf + Uf
4. Solve for the quantity algebraically with symbols.
if no FNC
15
Energy Conservation
Work Done by Fg using Ug(h)
Wg = Ug(hi ) – Ug(hf )
where:
Ug(h) = m g h
h1
h2
h3 h4 = 0
(near the Earth’s sur face)
Energy Conservation
Example 3
(4) W-E Theorem to
find �2 (= 1.93 m/s).
motion
� 1= 0
� 2= ?
(1) F.B.D.(2) W by each force
(3) Wnet
d = 5 m
µµµµk= 0.100
16
Energy Conservation
Example 4
Energy Conservation
x
y
Uel(x)
(((( ))))
)()(
21
21
21
21
21
]))(())(
d
21
22
21
21
22
2
2
1
SS
2
1
2
1
2
1
xUxU
kxkx
kxkx
x
x
xdxkx
dlFdlF
lFW
elel
x
x
yy
l
lxx
l
l
−−−−====
��������
������������
����−−−−��������
������������
����====
��������
������������
���� −−−−−−−−��������
������������
���� −−−−====
��������
������������
���� −−−−====
−−−−====
++++====
••••====
����
����
����
[(
��
Work Done by FS
Work Done by FS using Uel
0! 0!
17
Energy Conservation
Example 4
d
�f = 0
µµµµk = ?
Known:kmgv0
vf = 0d
Energy Conservation
Example 5A: Using UG (r)……
rC
rB
s s
b
rA
�B
�A
�C
� �������
�B = m/s�C = m/s
Inputs:RE = 6380 kmME = 5.97�1024 kgTsat = 2.02�104 s�A = 8650 m/s
18
Energy Conservation
Example 5B
�C = ?
�A = ?How much energy mustthe satellite’s enginesprovide to moveits satellite (mass m = 300 kg) froma circular orbit ofradius rA = 8000 km about the Ear th toanother circular orbitof radius rC = 3 rA?
Hint: EC = EA + ∆∆∆∆Esatellite
Energy Conservation
Part IV
1. How can we find the potential energy (P.E.) function for a general form of conservative force?
2. How can we find the conservative force for a general form of P.E. function?
19
Energy Conservation
How to Find U
���� ••••−−−−≡≡≡≡ lFU��
d
Energy Conservation
Near the Ear th’s sur face
Gravitational Potential Energyto estimate Work Energy
mgyy
where
yy
mgymgy
=
−=−=
)(
)()(
21
21
�� ��
�� ���� ��W
dl
y
Gravitational Potential Energy at y = y1 relative to at y = y2
(((( ))))
mgy
dymg
jdyidxjmg
lFU gg
====
−−−−−−−−====
++++••••−−−−−−−−====
••••−−−−====
����
����
����
]ˆ)(ˆ)[(]ˆ)[(
d ��