Chapter 8C Chapter 8C -- Conservation of EnergyConservation of Energy

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 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 this Objectives: After completing this module, you should be able to:module, you should be able to:

•• Define and give examples of Define and give examples of conservative conservative andand nonconservativenonconservative 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 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 = Work = mghmgh

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

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

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

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

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

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

U = 1960 JU = 1960 J

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

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

Work done by Work done by lifting forcelifting force F F

provides positiveprovides positive potential energypotential energy, , mghmgh, , for earthfor earth-- body system.body system.

Only Only external external forces can add or remove energyforces can add or remove energy.

mgh

F

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

mgh

FWeight is conservative.Weight is conservative.Work done by earth Work done by earth on the way up is on the way up is negative, negative, -- mghmgh

Work on return is Work on return is positive,positive, ++mghmgh

Net Work = - mgh + mgh = 0Net 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, -- ½½kxkx22..

On release, the spring doesOn release, the spring does positive work,positive work, + + ½½kxkx22 Fx

m

Fx

m

Net work = 0 (conservative)Net work = 0 (conservative)

Independence of PathIndependence of Path

Work done by conservative forces is independent of the path.

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

NonconservativeNonconservative ForcesForcesWork done by nonconservative forces cannot be restored. Energy is lost and cannot be regained. It is path-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 pathIt is path--dependent!dependent!

Friction forcesFriction forces are are nonconservativenonconservative forces.forces.

B

Af f

m

A B

Work of Conservative Forces Work of Conservative Forces is Independent of Path:is Independent of Path:

A

B

C

For gravitational force:For gravitational 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 greater for the longer path (BCD).

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

Stored Potential EnergyStored Potential EnergyWork 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 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 = 0

At y: Uo = mgy; Ko = ½mv2

At y=0: Uo = 0; Ko = ½mvf 2

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

Constant Total Energy Constant Total Energy for a Falling Bodyfor a Falling Body

vf

vy

K = 0h

0

TOP: E = U + K = TOP: E = U + K = mghmgh

AtAt any y: E = any y: E = mghmgh + + ½½mvmv22

mghmgh == mgymgy + + ½½mvmv2 2 = = ½½mvmvff 22

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

U = 0

Bottom: E = Bottom: E = ½½mvmv22

(Neglecting Air Friction)

Example 1:Example 1: A A 22--kgkg ball is released from ball is released from a height of a height of 20 m20 m. What is its velocity . What is its velocity when its height has decreased to when its height has decreased to 5 m5 m??

vv5m5m

v = 0v = 020m20m

00

mghmgh = = mgymgy + + ½½mvmv2 2

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

vv22 = 2g(h = 2g(h -- y) = 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 mTotal Etop = Total E at 5 m

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

Assume zero friction:Assume zero friction:At top: At top: U + K = U + K = mghmgh + 0+ 0

Bottom: Bottom: U + K = 0 + U + K = 0 + ½½mvmv22

Total energy is conservedTotal energy is conserved

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

mghmgh = = ½½mvmv22

v = 80 ft/sv = 80 ft/s

v = 2ghv = 2gh

Conservation of EnergyConservation of Energy in Absence of Friction Forcesin Absence of Friction Forces

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

mghmgh oo½½kxkx oo

22

½½mvmv oo 22

==mghmgh ff½½kxkx ff 22

½½mvmv ff 22

Height?Height?

Spring?Spring?

Velocity?Velocity?

Height?Height?

Spring?Spring?

Velocity?Velocity?

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

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

hh oo = 35 m; = 35 m; vv ff = 30 m/s= 30 m/s22

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

mghmgh oo½½kxkx oo

22

½½mvmv oo 22

Height?Height?

Spring?Spring?

Velocity?Velocity?

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

hh oo = 35 m; = 35 m; vv ff = 30 m/s= 30 m/s22

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

mghmgh ff½½kxkx ff 22

½½mvmv ff 22

Height?Height?

Spring?Spring?

Velocity?Velocity?

No (0 m)

No

Yes (vf )

Next choose END point at bottom of falls:

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

hh oo = 35 m; = 35 m; vv ff = 30 m/s= 30 m/s22

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

Total energy at top = Total energy at bottomTotal energy at top = Total energy at bottom

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 velocity A bicycle with initial velocity 10 10 m/sm/s coasts to a net height of coasts to a net height of 4 m4 m. What is . What is the velocity at the top, neglecting friction?the velocity at the top, neglecting friction?

4 m

vf = ?

vo = 10 m/s

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

EE toptop = = mghmgh + + ½½mvmv22

EE BotBot = 0 + = 0 + ½½mvmv oo 22

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 incline will the 2will the 2--kg block move after release? kg block move after release? The spring constant is 2000 N/m and it The spring constant is 2000 N/m and it is compressed by 8 cm.is compressed by 8 cm.

sshh

3030oo

BeginBeginEndEnd

mghmgh oo½½kxkx oo

22

½½mvmv oo 22

==mghmgh ff½½kxkx ff 22

½½mvmv ff 22

½½kxkx oo 22 = = mghmgh ffConservation of Energy:Conservation of Energy:

2 20

2

(2000 N/m)(0.08m)2 2(2 kg)(9.8 m/s )kxhmg

h = = 0.327 mh = = 0.327 m

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

sshh

3030oo

BeginBeginEndEndContinued:Continued:

hh = 0.327 m = 32.7 cm= 0.327 m = 32.7 cm

sinsin 3030oo ==hh

ss

ss = == =hh

sin sin 3030oo

32.7 cm32.7 cm

Sin 30Sin 30oo s = 65.3 cms = 65.3 cm

Energy Conservation and Energy Conservation and NonconservativeNonconservative Forces.Forces.

Work against friction forces must be accounted for. Energy is still conserved, but not reversible.

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

f

Conservation of Mechanical EnergyConservation of Mechanical Energy

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

Problem Solving StrategiesProblem Solving Strategies1. Read the problem; draw and label a sketch.1. Read the problem; draw and label a sketch.

2. Determine the reference points for 2. Determine the reference points for gravigravi-- tationaltational and/or spring potential energies.and/or spring potential energies.

3. Select a beginning point and an ending 3. Select a beginning point and an ending point and ask three questions at each point:point and ask three questions 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 (Continued)Problem Solving (Continued)

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

mghmgh oo½½kxkx oo

22

½½mvmv oo 22

==mghmgh ff½½kxkx ff 22

½½mvmv ff 22+

Work Work against against friction: friction:

ff kk xx

5. Remember to use the absolute (+) value 5. Remember to use the absolute (+) value of the work of friction. (Loss of energy)of the work of friction. (Loss of energy)

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

2gL 2gL -- 4gr = v4gr = v cc 22

r = L r = L -- dd

r = 20 m r = 20 m -- 12 m = 8 m12 m = 8 m

BL vc

rd

U = 0

A

vv cc 22 = 2(9.8 m/s= 2(9.8 m/s22)[20 m )[20 m -- (2)(8 m)](2)(8 m)]

vv cc 2 2 =2gL =2gL -- 4gr = 2g(L 4gr = 2g(L -- 2r)2r)

vv cc = = 2(9.8 m/s2(9.8 m/s22)(4 m) )(4 m) vc = 8.85 m/svv cc = 8.85 m/s= 8.85 m/s

Example 7Example 7: : A A 22--kgkg mass mass mm located located 10 m10 m above above the ground compresses a spring the ground compresses a spring 6 cm6 cm. The . The spring constant is spring constant is 40,000 N/m40,000 N/m and and kk = 0.4= 0.4. . What is the speed when it reaches the bottom?What is the speed when it reaches the bottom?

hh

2 kg2 kg

ss

3030oo mgmg

ff nnmg Sin 30mg Sin 30oo

mg Cos 30mg Cos 30oo

3030oo

BeginBegin

EndEnd

Conservation:Conservation: mghmgh + + ½½kxkx22 = = ½½mvmv22 + + ff kk xx

(Work)(Work) ff = (= (kk nn) x = ) x =

((mgmg Cos 30Cos 30oo)) xxContinued . . .Continued . . .

Example (Cont.)Example (Cont.): : A A 22--kgkg mass mass mm located located 10 m10 m above the ground compresses a spring above the ground compresses a 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 speed when it reaches the bottom?What is the speed when it reaches the bottom?

hh

2 kg2 kg

xx

3030oo

10 m10 m

ff kk xx = =

((mgmg Cos 30Cos 30oo)) xx

mghmgh + + ½½kxkx22 = = ½½mvmv22 + + ff kk xx

ff kk xx = (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

½½kxkx22 = = ½½(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 22--kgkg mass mass mm located located 10 m10 m above the ground compresses a spring above the ground compresses a 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 speed when it reaches the bottom?What is the speed when it reaches the bottom?

h

2 kg

x

30o

10 m

mghmgh + + ½½kxkx22 = = ½½mvmv22 + + ff kk xx

ff kk xx = 136 J= 136 J

mghmgh = 196 J= 196 J ½½kxkx22 = 72.0 J= 72.0 J

½½mvmv22 = = mghmgh + + ½½kxkx22 -- ff kk xx

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

Gravitational Potential EnergyGravitational Potential Energy

Spring Potential EnergySpring Potential Energy

Work Against FrictionWork Against Friction Work = fxWork = fx

Kinetic EnergyKinetic Energy 212K mv

Summary:Summary: Conservation of EnergyConservation of Energy

The basic rule for conservation of energy:

mghmgh oo½½kxkx oo

22

½½mvmv oo 22

==mghmgh ff½½kxkx ff 22

½½mvmv ff 22+

Work Work against against friction: friction:

ff kk xx

Remember to use the absolute (+) value of the work of friction. (Loss of energy)

CONCLUSION: Chapter 8CCONCLUSION: Chapter 8C Conservation of EnergyConservation of Energy