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1 Energy Conservation Energy Conservation 1. Conservative/Nonconservative Forces 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

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Page 1: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 2: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 3: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 4: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 5: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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)

Page 6: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 7: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 8: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 9: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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.

Page 10: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force 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

Page 11: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 12: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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 )

Page 13: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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.

Page 14: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 15: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 16: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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!

Page 17: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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

Page 18: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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?

Page 19: Energy Conservation - Texas A&M Universitypeople.physics.tamu.edu/kamon/teaching/phys218/slide/lec08_1x2_031016.pdfEnergy Conservation Recap: Gravitational Force Conservative Force

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 ��