Chapter 8 Potential Energy and Conservation of Energynsmn1.uh.edu/rbellwied/classes/spring2013/ch8-notes.pdf · 8-3 Conservation of Mechanical Energy ... Summary of Chapter 8 •

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Chapter 8

Potential Energy andConservation of Energy


Units of Chapter 8

• Conservative and NonconservativeForces

• Potential Energy and the Work Done byConservative Forces

• Conservation of Mechanical Energy

• Work Done by Nonconservative Forces

• Potential Energy Curves andEquipotentials


8-1 Conservative and NonconservativeForces

Conservative force: the work it does is stored inthe form of energy that can be released at a latertime

Example of a conservative force: gravity

Example of a nonconservative force: friction

Also: the work done by a conservative forcemoving an object around a closed path is zero;this is not true for a nonconservative force



Work done by gravity on a closed path is zero:



Work done by friction on a closed path is notzero:



The work done by a conservative force is zeroon any closed path:


8-2 The Work Done by ConservativeForces

If we pick up a ball and put it on the shelf, wehave done work on the ball. We can get thatenergy back if the ball falls back off the shelf; inthe meantime, we say the energy is stored aspotential energy.

(8-1)



Gravitational potential energy:


Example: What is the change in gravitational potentialenergy of the box if it is placed on the table? The table is1.0 m tall and the mass of the box is 1.0 kg.

( )( )( )( ) J 8.9m 0m 0.1m/s 8.9kg 0.1 2 +=!=

!="=" ifg yymgymgU

First: Choose the reference level at the floor. U = 0 here.


Example continued:

( )( )( ) ( )( ) J 8.9m 0.1 m0.0m/s 8.9kg 1 2 +=!!=

!="=" ifg yymgymgU

Now take the reference level (U = 0) to be on top of thetable so that yi = −1.0 m and yf = 0.0 m.

The results do not depend onthe location of U = 0.



Springs: (8-4)


8-3 Conservation of Mechanical EnergyDefinition of mechanical energy:

(8-6)

Using this definition and considering onlyconservative forces, we find:

Or equivalently:


8-3 Conservation of Mechanical EnergyEnergy conservation can make kinematicsproblems much easier to solve:


Example (text problem 6.28): A cart starts from position 4with v = 15.0 m/s to the left. Find the speed of the cart atpositions 1, 2, and 3. Ignore friction.

( ) m/s 5.202

2

1

2

1

34

2

43

2

33

2

44

3344

34

=!+=

+=+

+=+

=

yygvv

mvmgymvmgy

KUKU

EE


( ) m/s 0.182

2

1

2

1

24

2

42

2

22

2

44

2244

24

=!+=

+=+

+=+

=

yygvv

mvmgymvmgy

KUKU

EE

( ) m/s 8.242

2

1

2

1

14

2

41

2

11

2

44

1144

14

=!+=

+=+

+=+

=

yygvv

mvmgymvmgy

KUKU

EE

Or useE3=E2

Or useE3=E1

E2=E1

Example continued:


Example (text problem 6.84): A roller coaster car is about toroll down a track. Ignore friction and air resistance.

40 m

20 m

y=0

m = 988 kg

(a) At what speed does the car reach the top of the loop?

( ) m/s 8.192

2

10

2

=!=

+=+

+=+

=

fif

ffi

ffii

fi

yygv

mvmgymgy

KUKU

EE


(b) What is the force exerted on the car by the track at the topof the loop?

N w

y

x

N 109.24

2

2

2

!="=

=+

"="=""=#

mgr

vmN

r

vmwN

r

vmmawNF ry

Example continued:

FBD for the car:

Apply Newton’s Second Law:


(c) From what minimum height above the bottom of the trackcan the car be released so that it does not lose contact withthe track at the top of the loop?

Example continued:

2

min

2

10 mvmgymgy

KUKU

EE

fi

ffii

fi

+=+

+=+

=

Using conservation of mechanical energy:

g

vyy fi

2

2

min+=Solve for the starting height


Example continued:

v = vmin when N = 0. This means that

grv

r

vmmgw

r

vmwN

=

==

=+

min

2

min

2

What is vmin?

m 0.255.22

22

2

min ==+=+= rg

grr

g

vyy fi

The initial height must be


8-4 Work Done by Nonconservative Forces

In the presence of nonconservative forces, thetotal mechanical energy is not conserved:

Solving,

(8-9)


8-4 Work Done by Nonconservative Forces

In this example, thenonconservative forceis water resistance:


8-5 Potential Energy Curves andEquipotentials

The curve of a hill or a roller coaster is itselfessentially a plot of the gravitationalpotential energy:



The potential energy curve for a spring:



Contour maps are also a form of potentialenergy curve:


Summary of Chapter 8

• Conservative forces conserve mechanicalenergy

• Nonconservative forces convert mechanicalenergy into other forms

• Conservative force does zero work on anyclosed path

• Work done by a conservative force isindependent of path

• Conservative forces: gravity, spring


Summary of Chapter 8• Work done by nonconservative force on closedpath is not zero, and depends on the path

• Nonconservative forces: friction, airresistance, tension

• Energy in the form of potential energy can beconverted to kinetic or other forms

• Work done by a conservative force is thenegative of the change in the potential energy

• Gravity: U = mgy

• Spring: U = ½ kx2


Summary of Chapter 8

• Mechanical energy is the sum of the kinetic andpotential energies; it is conserved only insystems with purely conservative forces

• Nonconservative forces change a system’smechanical energy

• Work done by nonconservative forces equalschange in a system’s mechanical energy

• Potential energy curve: U vs. position

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Chapter 8 Potential Energy and Conservation of Energynsmn1.uh.edu/rbellwied/classes/spring2013/ch8-notes.pdf · 8-3 Conservation of Mechanical Energy ... Summary of Chapter 8 •