Chapter 8C - Conservation of EnergyChapter 8C - Conservation of Energy

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

A waterfall in Yellowstone Park provides an example of energy in nature. The potential energy of the water at the top is converted into kinetic energy at the bottom.

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• Define and give examples of Define and give examples of

conservative conservative andand nonconservative nonconservative forces.forces.

• Define and apply the concept of Define and apply the concept of conservation of mechanical energyconservation of mechanical energy for for conservative forces.conservative forces.

• Define and apply the concept of Define and apply the concept of conservation of mechanical energy conservation of mechanical energy accounting for accounting for friction lossesfriction losses..

Potential EnergyPotential EnergyPotential EnergyPotential Energy is the ability to do work by virtue of position or condition.

Potential EnergyPotential Energy is the ability to do work by virtue of position or condition.

Earth

mgh

mExample:Example: A mass held a distance h above the earth.

If released, the earth can do work on the

mass:Work = mghWork = mgh

Is this work + or - ?Is this work + or - ?Positive!

Gravitational Potential Gravitational Potential EnergyEnergyGravitational Potential Energy UGravitational Potential Energy U is is equal to the work that can be done equal to the work that can be done BYBY gravity due to height above a specified gravity due to height above a specified point.point.

Gravitational Potential Energy UGravitational Potential Energy U is is equal to the work that can be done equal to the work that can be done BYBY gravity due to height above a specified gravity due to height above a specified point.point.

U = mghU = mgh Gravitational P. E.Gravitational P. E.

Example:Example: What is the potential energy What is the potential energy when a 10 kg block is held 20 m above when a 10 kg block is held 20 m above the street?the street?

U = mgh = U = mgh = (10 kg)(9.8 m/s(10 kg)(9.8 m/s22)(20 )(20 m)m)

U = 1960 J

U = 1960 J

The Origin of Potential The Origin of Potential EnergyEnergyPotential energyPotential energy is a property of the is a property of the Earth-body system. Neither has Earth-body system. Neither has potential energy without the other.potential energy without the other.

Potential energyPotential energy is a property of the is a property of the Earth-body system. Neither has Earth-body system. Neither has potential energy without the other.potential energy without the other.

Work done by Work done by lifting forcelifting force F F

provides provides positivepositive potential potential

energyenergy, , mghmgh, , for earth-body for earth-body

system.system.Only Only external external forces can add or remove forces can add or remove energyenergy.

mgh

F

Conservative ForcesConservative ForcesAA conservative forceconservative force isis one that one that does zero work during a round does zero work during a round trip.trip.

AA conservative forceconservative force isis one that one that does zero work during a round does zero work during a round trip.trip.

mgh

FWeight is Weight is

conservative.conservative.Work done by Work done by earth on the way earth on the way up is negative, up is negative, - - mghmghWork on return is Work on return is positive,positive, +mgh+mgh

Net Work = - mgh + mgh = 0

Net Work = - mgh + mgh = 0

The Spring ForceThe Spring Force

The force exerted by aThe force exerted by a springspring is alsois also conservative.conservative.

When stretched, the spring When stretched, the spring does negative work, does negative work, - -

½kx½kx22..On release, the spring doesOn release, the spring does

positive work,positive work, + ½kx+ ½kx22 Fxm

Fx

m

Net work = 0 (conservative)

Net work = 0 (conservative)

Independence of PathIndependence of Path

Work done by Work done by conservative forcesconservative forces is independent of the path.is independent of the path.

Work done by Work done by conservative forcesconservative forces is independent of the path.is independent of the path.

A

C

B

C

A B

Force due to gravitymg

Work (A C) = Work (A B C) Why?Because only the vertical

component of the weight does work against gravity.

Nonconservative ForcesNonconservative ForcesWork done byWork done by nonconservativenonconservative forces forces cannot be restored. Energy is lost and cannot be restored. Energy is lost and cannot be regained.cannot be regained. It is path-It is path-dependent!dependent!

Work done byWork done by nonconservativenonconservative forces forces cannot be restored. Energy is lost and cannot be restored. Energy is lost and cannot be regained.cannot be regained. It is path-It is path-dependent!dependent!

Friction forcesFriction forces are nonconservative are nonconservative forces.forces.

B

Af f

m

A B

Work of Conservative Work of Conservative Forces is Independent of Forces is Independent of

Path:Path:

A

B

C

For gravitational For gravitational force:force:

(Work)(Work)ABAB= -(Work)= -(Work)BCABCA Zero net workZero net work

For friction force:For friction force:(Work)(Work)AB AB --

(Work)(Work)BCABCA

The work done against friction is The work done against friction is greater for the longer path (BCD).greater for the longer path (BCD).The work done against friction is The work done against friction is

greater for the longer path (BCD).greater for the longer path (BCD).

Stored Potential EnergyStored Potential Energy

Work done by a conservative force isWork done by a conservative force is storedstored inin the system as potential energy.the system as potential energy.

m

xox

F(x) = kx to compress

Displacement is x

212U Work kx 212U Work kx Potential energy

of compressed spring:

The potential energy is equal to the work done in compressing the spring:

The potential energy is equal to the work done in compressing the spring:

Conservation of Energy Conservation of Energy (Conservative forces)(Conservative forces)

In the absence of friction, the sum of the potential and kinetic energies is a constant, provided no energy is added to system.

In the absence of friction, the sum of the potential and kinetic energies is a constant, provided no energy is added to system.

vf

vy mg

v = 0h

0

At top: Uo = mgh; Ko = 0At y: Uo = mgy; Ko =

½mv2

At y=0: Uo = 0; Ko = ½mvf 2

E = U + K = ConstantE = U + K = Constant

Constant Total Constant Total Energy Energy

for a Falling Bodyfor a Falling Body

vf

v

y

K = 0h

0

TOP: E = U + K = mghTOP: E = U + K = mgh

AtAt any y: E = mgh + ½mvany y: E = mgh + ½mv22

mgh =mgh = mgy + ½mvmgy + ½mv2 2 = = ½mv½mvff

22

Total E is same at any Total E is same at any point.point.

U = 0

Bottom: E = ½mvBottom: E = ½mv22

(Neglecting Air Friction)

Example 1:Example 1: A A 2-kg2-kg ball is released ball is released from a height of from a height of 20 m20 m. What is its . What is its velocity when its height has velocity when its height has decreased to decreased to 5 m5 m??

vv5m5m

v = 0v = 020m20m

00

mgh = mgy + ½mvmgh = mgy + ½mv2 2

2gh = 2gy + v2gh = 2gy + v22

vv22 = 2g(h - y) = = 2g(h - y) = 2(9.8)(20 - 2(9.8)(20 - 5)5)

v = v = (2)(9.8)(15)(2)(9.8)(15) v = 17.1 m/sv = 17.1 m/s

Total Etop = Total E at 5 m

Total Etop = Total E at 5 m

Example 2:Example 2: A roller coaster boasts A roller coaster boasts a maximum height of a maximum height of 100 ft100 ft. What . What is the speed when it reaches its is the speed when it reaches its lowest point?lowest point?

Assume zero friction:Assume zero friction:

At top: At top: U + K = mgh + 0U + K = mgh + 0Bottom: Bottom: U + K = 0 + U + K = 0 +

½mv½mv22

Total energy is Total energy is conservedconserved

v = v = (2)(32 ft/s(2)(32 ft/s22)(100 ft))(100 ft)

mgh = mgh = ½mv½mv22

v = 80 ft/sv = 80 ft/s

v = 2ghv = 2gh

Conservation of EnergyConservation of Energyin Absence of Friction in Absence of Friction

ForcesForces

Begin: Begin: (U + K)(U + K)oo = End: = End: (U + (U + K)K)f

mghmghoo

½kx½kxoo22

½mv½mvoo22

==mghmghff

½kx½kxff22

½mv½mvff22

Height?Height?

Spring?Spring?

VelocityVelocity??

Height?Height?

Spring?Spring?

VelocityVelocity??

The total energy is constant for a The total energy is constant for a conservative system, such as with conservative system, such as with gravity or a spring.gravity or a spring.

The total energy is constant for a The total energy is constant for a conservative system, such as with conservative system, such as with gravity or a spring.gravity or a spring.

Example 3.Example 3. Water at the bottom of a Water at the bottom of a falls has a velocity of 30 m/s after falling falls has a velocity of 30 m/s after falling 35 ft. 35 ft.

hhoo = 35 m; v = 35 m; vff = 30 = 30 m/sm/s22

What is the water What is the water speed at the top of speed at the top of the falls?the falls?

mghmghoo

½kx½kxoo22

½mv½mvoo22

Height?Height?

Spring?Spring?

VelocityVelocity??

Yes (35 m)

No

Yes (vo)

First look at beginning point—top of falls. Assume y = 0 at bottom for

reference point.

Example 3 (Cont.)Example 3 (Cont.) Water at the bottom Water at the bottom of falls has a velocity of 30 m/s after of falls has a velocity of 30 m/s after falling 35 ft. falling 35 ft.

hhoo = 35 m; v = 35 m; vff = 30 = 30 m/sm/s22

What is the water What is the water speed at the top of speed at the top of the falls?the falls?

mghmghff

½kx½kxff22

½mv½mvff22

Height?Height?

Spring?Spring?

VelocityVelocity??

No (0 m)

No

Yes (vf)

Next choose END point at bottom of falls:

Example 3 (Cont.)Example 3 (Cont.) Water at the bottom Water at the bottom of falls has a velocity of 30 m/s after of falls has a velocity of 30 m/s after falling 35 ft. falling 35 ft.

hhoo = 35 m; v = 35 m; vff = 30 = 30 m/sm/s22What is the water What is the water speed at the top of the speed at the top of the

falls?falls?

Total energy at top = Total energy at Total energy at top = Total energy at bottombottom

2 2 2 20 2 (25.8 m/s) 2(9.8 m/s )(33.2 m)fv v gh

2 20 14.9 m /sv vo = 3.86 m/svo = 3.86 m/s

2 202 fgh v v 2 21 1

02 20 fmgh mv mv

Example 4.Example 4. A bicycle with initial A bicycle with initial velocity velocity 10 m/s10 m/s coasts to a net height coasts to a net height

of of 4 m4 m. What is the velocity at the . What is the velocity at the top, neglecting friction?top, neglecting friction?

4 m

vf = ?

vo = 10 m/s

E(Top) = E(Bottom)E(Top) = E(Bottom)

EEtoptop = mgh + ½mv = mgh + ½mv22

EEBotBot = 0 + ½mv = 0 + ½mvoo22

2 21 102 2fmv mgh mv 2 21 1

02 2fv v gh 2 2 2 2

0 2 (10 m/s) 2(9.8 m/s )(4 m)fv v gh

2 221.6 m /sfv vf = 4.65 m/svf = 4.65 m/s

Example 5:Example 5: How far up the 30How far up the 30oo--incline will the 2-kg block move incline will the 2-kg block move after release? The spring constant after release? The spring constant is 2000 N/m and it is compressed is 2000 N/m and it is compressed by 8 cm.by 8 cm.

sshh

3030oo

BegiBeginn

EndEndmghmghoo

½kx½kxoo22

½mv½mvoo22

==mghmghff

½kx½kxff22

½mv½mvff22

½kx½kxoo22 = mgh = mghff

Conservation of Conservation of Energy:Energy:

2 20

2

(2000 N/m)(0.08m)

2 2(2 kg)(9.8 m/s )

kxh

mg h = = 0.327

mh = = 0.327

m

Example (Cont.):Example (Cont.): How far up the How far up the 3030oo-incline will the 2-kg block -incline will the 2-kg block move after release? The spring move after release? The spring constant is 2000 N/m and it is constant is 2000 N/m and it is compressed by 8 cm.compressed by 8 cm.

sshh

3030oo

BegiBeginn

EndEndContinued:Continued:hh = 0.327 m = 32.7 = 0.327 m = 32.7

cmcm

sinsin 30 30oo = =hh

ss

ss = = = =hh

sin sin 3030oo

32.7 32.7 cmcm

Sin 30Sin 30oo

s = 65.3 cms = 65.3 cm

Energy Conservation and Energy Conservation and Nonconservative Forces.Nonconservative Forces.

Work against Work against frictionfriction forces must be forces must be accounted for. Energy accounted for. Energy is still conserved, but is still conserved, but notnot reversible. reversible.

Work against Work against frictionfriction forces must be forces must be accounted for. Energy accounted for. Energy is still conserved, but is still conserved, but notnot reversible. reversible.

f

Conservation of Mechanical Conservation of Mechanical EnergyEnergy

(U + K)o = (U + K)f + Losses (U + K)o = (U + K)f + Losses

Problem Solving Problem Solving StrategiesStrategies1. Read the problem; draw and label a 1. Read the problem; draw and label a

sketch.sketch.

2. Determine the reference points for 2. Determine the reference points for gravi- tational and/or spring gravi- tational and/or spring potential energies.potential energies.

3. Select a beginning point and an 3. Select a beginning point and an ending point and ask three questions ending point and ask three questions at each point:at each point:

a. Do I have height?a. Do I have height?

U = mghU = mgh

b. Do I have velocity?b. Do I have velocity? K = ½mv2K = ½mv2

c. Do I have a spring?c. Do I have a spring? U = ½kx2U = ½kx2

Problem Solving Problem Solving (Continued)(Continued)

4. Apply the rule for Conservation of 4. Apply the rule for Conservation of Energy.Energy.

mghmghoo

½kx½kxoo22

½mv½mvoo22

==mghmghff

½kx½kxff22

½mv½mvff22

+

Work Work against against

friction: friction:

ffkk x x5. Remember to use the absolute (+) 5. Remember to use the absolute (+)

value of the work of friction. (Loss of value of the work of friction. (Loss of energy)energy)

Example 6Example 6: : A mass A mass mm is connected to a is connected to a cord of length cord of length LL and held horizontally as and held horizontally as shown. What will be the velocity at point shown. What will be the velocity at point BB? (d = 12 m, L = 20 m)? (d = 12 m, L = 20 m)

BL vc

rd

1. Draw & label.2. Begin A and end

B.3. Reference U = 0.

U = 0(U + K)o =(U + K)f + loss0

mgL + 0 = mg(2r) + ½mvc2 (Multiply by 2,

simplify)

2gL - 4gr = vc2 Next find r from

figure.

A

Example (Cont.)Example (Cont.): : A mass A mass mm is connected is connected to a cord of length to a cord of length LL and held horizontally and held horizontally as shown. What will be the velocity at as shown. What will be the velocity at point point BB? (d = 12 m, L = 20 m)? (d = 12 m, L = 20 m)

2gL - 4gr = 2gL - 4gr = vvcc

22

r = L - dr = L - d

r = 20 m - 12 m = 8 r = 20 m - 12 m = 8 mm

BL vc

rd

U = 0

A

vvcc22 = 2(9.8 m/s = 2(9.8 m/s22)[20 m - (2)(8 )[20 m - (2)(8

m)]m)]

vvcc2 2 =2gL - 4gr = 2g(L - =2gL - 4gr = 2g(L -

2r)2r)

vvcc = = 2(9.8 m/s2(9.8 m/s22)(4 )(4 m) m)

vvcc = 8.85 = 8.85 m/sm/s

vvcc = 8.85 = 8.85 m/sm/s

Example 7Example 7: : A A 2-kg2-kg mass mass mm located located 10 m10 m above the ground compresses a spring above the ground compresses a spring 6 6 cmcm. The spring constant is . The spring constant is 40,000 N/m40,000 N/m and and kk = 0.4 = 0.4. What is the speed when it . What is the speed when it reaches the bottom?reaches the bottom?

hh

2 kg2 kg

ss

3030oo mgmg

ff nnmg Sin mg Sin

3030oomg Cos mg Cos

3030oo3030oo

BeginBegin

EndEnd

Conservation:Conservation: mgh + ½kxmgh + ½kx22 = ½mv = ½mv22 + + ffkkxx (Work)(Work)ff = ( = (kknn) x = ) x = ((mg Cos mg Cos

3030oo)) xx Continued . . .Continued . . .

Example (Cont.)Example (Cont.): : A A 2-kg2-kg mass mass mm located located 10 m10 m above the ground compresses a above the ground compresses a spring spring 6 cm6 cm. The spring constant is . The spring constant is 40,000 N/m40,000 N/m and and kk = 0.4 = 0.4. What is the . What is the speed when it reaches the bottom?speed when it reaches the bottom?

hh

2 2 kgkg

xx

3030oo

10 m10 m

ffkkx = x = ((mg Cos 30mg Cos 30oo)) xx

mgh + ½kxmgh + ½kx22 = ½mv = ½mv22 + + ffkkxx

ffkkxx = (0.4)(2 kg)(9.8 m/s2)(0.866)(20 m) = 136 J

x = = 20 m10 m

Sin 30o

mgh = (2 kg)(9.8 m/s2)(10 m) = 196 J

½kx½kx22 = ½= ½(40,000 N/m)(0.06 m)(40,000 N/m)(0.06 m)22 = = 72.0 J72.0 J

Example (Cont.)Example (Cont.):: A A 2-kg2-kg mass mass mm located located 10 m10 m above the ground compresses a above the ground compresses a spring spring 6 cm6 cm. The spring constant is . The spring constant is 40,000 N/m40,000 N/m and and kk = 0.4 = 0.4. What is the . What is the speed when it reaches the bottom?speed when it reaches the bottom?

h

2 kg

x

30o

10 m

mgh + ½kxmgh + ½kx22 = ½mv = ½mv22 + + ffkkxx

ffkkxx = 136 J= 136 J

mghmgh = 196 J= 196 J ½kx½kx22 = 72.0 = 72.0 JJ

½mv½mv22 = mgh + ½kx = mgh + ½kx22 - - ffkkxx

½½(2 kg) (2 kg) vv22 = = 196 J + 72 J - 136 J = 132 J 196 J + 72 J - 136 J = 132 J

v =11.4 m/sv =11.4 m/s

Summary: Summary: Energy Gains or Losses:Energy Gains or Losses:

U = mghU = mgh

212U kx 212U kx

Gravitational Potential Energy

Gravitational Potential Energy

Spring Potential EnergySpring Potential Energy

Work Against FrictionWork Against Friction Work = fxWork = fx

Kinetic EnergyKinetic Energy 212K mv 21

2K mv

Summary:Summary:Conservation of EnergyConservation of Energy

The basic rule for conservation of energy:

mghmghoo

½kx½kxoo22

½mv½mvoo22

==mghmghff

½kx½kxff22

½mv½mvff22

+

Work Work against against

friction: friction:

ffkk x xRemember to use the absolute (+) value of the work of friction. (Loss of energy)

CONCLUSION: Chapter 8CCONCLUSION: Chapter 8CConservation of EnergyConservation of Energy