Conservation of Energy Lecturer: Professor Stephen T. Thornton

Conservation of Energy

Lecturer: Professor Stephen T. Thornton

Mike performed 5 J of work in 10 secs. Joe did 3 J of work in 5 secs. Who produced the greater power?

A) Mike produced more power

B) Joe produced more power

C) both produced the same amount of power

Reading Quiz

Because power = work / time, we see that Mike Mike produced 0.5 Wproduced 0.5 W and Joe produced 0.6 WJoe produced 0.6 W of power. Thus, even though Mike did more work, he required twice the time to do the work, and therefore his power output was lower.

Reading QuizMike performed 5 J of work in 10 secs. Joe did 3 J of work in 5 secs. Who produced the greater power?

A) Mike produced more power

B) Joe produced more power

C) both produced the same amount of power

Last TimeConservative and nonconservative forces

Gravitational potential energy

Other kinds of potential energy

Conservation of mechanical energy

Today

Conservation of Energy

Escape velocity

Power

Potential energy diagrams

Copyright © 2009 Pearson Education, Inc.

Potential Energy

A spring has potential energy, called elastic potential energy, when it is compressed. The force exerted by the spring when compressed or stretched is

where k is called the spring constant, and needs to be measured for each spring.

SF kx=-

Copyright © 2009 Pearson Education, Inc.

Then the potential energy of the spring is:

22

S

1 0

2

( ) (0)

1( )

2

1( )

2

x

U U x U

F d kx dx kx

U x kx

D = -

=- × =- - =

=

ò ò

Springs

The work required to compress a spring is

The potential energyof a spring is

21

2W kx

21

2U kx

21

2W kx

Mass on Spring. When a mass m sits at rest on a spring, the spring is compressed by a distance d from its undeformed length. Suppose instead that the mass is released from rest when it barely touches the undeformed spring. Find the distance D that the spring is compressed before it is able to stop the mass.

Conceptual QuizConceptual QuizA truck, initially at rest, rolls A truck, initially at rest, rolls

down a frictionless hill and down a frictionless hill and

attains a speed of attains a speed of 20 m/s20 m/s at the at the

bottom. To achieve a speed of bottom. To achieve a speed of

40 m/s40 m/s at the bottom, how many at the bottom, how many

times higher must the hill be?times higher must the hill be?

A) half the height

B) the same height

C) 2 times the height

D) twice the height

E) four times the height

Conceptual QuizConceptual QuizA truck, initially at rest, rolls A truck, initially at rest, rolls

down a frictionless hill and down a frictionless hill and

attains a speed of attains a speed of 20 m/s20 m/s at the at the

bottom. To achieve a speed of bottom. To achieve a speed of

40 m/s40 m/s at the bottom, how many at the bottom, how many

times higher must the hill be?times higher must the hill be?

A) half the height

B) the same height

C) 2 times the height

D) twice the height

E) four times the height

Use energy conservation:

initial energy: EEii = PE = PEgg = mgH = mgH

final energy: EEff = KE = KE = = mv mv22

Conservation of Energy:Conservation of Energy:

EEii = mgH = mgH = E= Eff = mv = mv22

therefore: gHgH = = vv22

So if So if vv doubles, doubles, HH quadruples! quadruples!

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x

Conceptual QuizConceptual QuizA box sliding on a frictionless flat surface runs into a fixed spring, which compresses a distance x to stop the box. If the initial speed of the box were doubled, how much would the spring compress in this case?

A) half as much

B) the same amount

C) 2 times as much

D) twice as much

E) four times as much

x

Conceptual QuizConceptual Quiz

Use energy conservation:

initial energy: EEii = KE = KE = = mvmv22

final energy: EEff = PE = PEs s = = kxkx22

Conservation of Energy:Conservation of Energy:

EEii = = mvmv22 = E= Eff = kx = kx22

therefore: mvmv22 = kx = kx22

So if So if vv doubles, doubles, xx doubles! doubles!

A box sliding on a frictionless flat surface runs into a fixed spring, which compresses a distance x to stop the box. If the initial speed of the box were doubled, how much would the spring compress in this case?

A) half as much

B) the same amount

C) 2 times as much

D) twice as much

E) four times as much

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Conceptual QuizConceptual QuizA cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If

the cart were given an initial push, so its

initial speed at the top of the hill was 3 m/s,

what would be its speed at the bottom?

A) 4 m/s

B) 5 m/s

C) 6 m/s

D) 7 m/s

E) 25 m/s


When starting from rest, thecart’s PE is changed into KE:

PE = KEKE = m(4) m(4)22

A cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If

the cart were given an initial push, so its

initial speed at the top of the hill was 3 m/s,

what would be its speed at the bottom?

A) 4 m/s

B) 5 m/s

C) 6 m/s

D) 7 m/s

E) 25 m/s

When starting from 3 m/s, thefinal KE is:

KEKEff = KEi + KEKE

= m(3)2 + m(4) m(4)22

= m(25) m(25)

= m(5) m(5)22Speed is not the same as kinetic energy

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Conceptual Quiz:Two unequal masses are hung from a string that pass over an ideal pulley. What is true about the gravitational potential energy U and the kinetic energy K of the system after the masses are released from rest?

A) U > 0 and K < 0. B) U > 0 and K > 0. C) U > 0 and K = 0.

D) U = 0 and K = 0. E) U < 0 and K > 0.

Answer: E

Initially the system is at rest. Let the potential energy be zero at this point. Therefore the total mechanical energy is zero. If the system starts moving, then K > 0. Since E = 0, then U < 0.

Conceptual Quiz

Three balls of equal mass start from rest

and roll down different ramps. All ramps

have the same height. Which ball has the

greater speed at the bottom of its ramp?

A

D) same speed

for all balls

B C

All of the balls have the same initial same initial gravitational PEgravitational PE, since they are all at the same heightsame height (PE = mgh). Thus, when they get to the bottom, they all have the same final same final KEKE, and hence the same speedsame speed (KE = 1/2 mv2).

Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp?

A

D) same speed

for all ballsB C

Follow-up:Follow-up: Which ball takes longer to get down the ramp? Which ball takes longer to get down the ramp?

Law of Conservation of Energy

Nonconservative, or dissipative, forces associated with: Friction Heat Electrical energy Chemical energy and moredo not conserve mechanical energy. However, when these forces are taken into account, the total energy is still conserved:

We discussed Conservation of Mechanical Energy last time. 0K U

[change in all other forms of energy] = 0K UD +D +

Law of Conservation of Energy

The law of conservation of energy is one of the most important principles in physics.

The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one object to another, but the total amount remains constant.

Ball rolling on a frictionless track

Gravitational potential energy vs position for the previous track. See also kinetic and total energy.

Height

Gravitational potential energy vs position for the previous track. See also kinetic and total energy.

Height

New total energy

A Mass on a Spring

U

KE

21

2U kx

Potential Energy Diagrams; Stable and Unstable Equilibrium

This is a potential energy diagram for a particle moving under the influence of a conservative force. Its behavior will be determined by its total energy.

With energy E1, the object oscillates between x3 and x2, called turning points. An object with energy E2 has four turning points; an object with energy E0 is in stable equilibrium. An object at x4 is in unstable equilibrium.

Bath County, Virginia, pumped storage facility electrical power plant.

Day – water flows down from upper reservoir producing electricity.

Night – use power from other (nuclear) plants to pump water back up.

Gravitational Potential Energy

Far from the surface of the Earth, the force of gravity is not constant:

The work done on an object moving in the Earth’s gravitational field is given by:

Gravitational Potential Energy

Solving the integral gives:

Because the value of the integral depends only on the end points, the gravitational force is conservative and we can define gravitational potential energy:

( ) EGmMU r

r=-

2 1

E EGmM GmMW

r r= -

Gravitational Potential Energy and Escape Velocity

If an object’s initial kinetic energy is equal to the negative of the potential energy at the Earth’s surface, its total energy will be zero. The velocity at which this is true is called the escape velocity; for Earth:

Think about this. E = 0 at Earth’s surface; E = 0 at . At , U = 0 and K = 0.

r :r 0v

2 2At Earth's surface:

1

2E

escE

Eesc

E

GmMmv

GMv

rr

0 ; / EEKU Gm rE K U M

11.2 km/sescv =

PowerPower measures how fast work is done.

Average power = P = W/t

Instantaneous power

Power is so important that it also has its own unit. SI unit: watt

1 watt = 1 W = 1 J/s = 1 joule/sec

1 horsepower = 1 hp = 746 watt

dWP

dt

Power is also needed for acceleration and for moving against the force of friction.

The power can be written in terms of the net force and the velocity:

dW dP F F v

dt dt= = × = ×

Lance Armstrong was tested and could ride up the mountains in France during the Tour de France generating about 500 watts of power for 20 minutes. A typical college student could only do this for 30 s. (Lance has a large heart and low levels of lactic acid.)

Lance exerts 500 W x 1200 s = 600,000 J = W

Climbing: mgh = (70 kg)(9.8 m/s2 )h = W energy; h = 875 m = 2900 ft.

This is why he won the Tour de France seven consecutive years!

Conceptual Quiz

A) PaulB) KathleenC) both the same

Paul and Kathleen start from Paul and Kathleen start from rest at the same time on rest at the same time on frictionless water slides with frictionless water slides with different shapes. At the different shapes. At the bottom, whose velocity is bottom, whose velocity is greater?greater?

Conceptual Quiz


Paul and Kathleen start from Paul and Kathleen start from rest at the same time on rest at the same time on frictionless water slides with frictionless water slides with different shapes. At the different shapes. At the bottom, whose velocity is bottom, whose velocity is greater?greater?

Conservation of Energy:

Because they both start from the same heightsame height, they have the same same velocityvelocity at the bottom.

Conceptual QuizPaul and Kathleen start from Paul and Kathleen start from rest at the same time on rest at the same time on frictionless water slides with frictionless water slides with different shapes. Who different shapes. Who makes it to the bottom first?makes it to the bottom first?


Conceptual QuizPaul and Kathleen start from Paul and Kathleen start from rest at the same time on rest at the same time on frictionless water slides with frictionless water slides with different shapes. Who makes different shapes. Who makes it to the bottom first?it to the bottom first?

Even though they both have the same final velocity, Kathleen is at a lower height Kathleen is at a lower height than Paul for most of her than Paul for most of her rideride. Thus, she always has a larger velocitylarger velocity during her ride and therefore arrives earlier!


Space Shuttle. Early test flights for the space shuttle used a “glider” (mass of 980 kg including pilot). After a horizontal launch at 480 km/h at a height of 3500 m, the glider eventually landed at a speed of 210 km/h.

(a) What would its landing speed have been in the absence of air resistance?

(b) What was the average force of air resistance exerted on it if it came in at a constant glide angle of 12° to the Earth’s surface?

Ski Lift Power. A ski area claims that its lifts can move 47,000 people per hour. If the average lift carries people about 200 m (vertically) higher, estimate the maximum total power needed.

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

A) only 2B) only 3C) 1, 2, and 3D) only 1 and 3E) only 2 and 3

Conceptual Quiz

1) skier’s PE 2) skier’s change in PE 3) skier’s final KE1) skier’s PE 2) skier’s change in PE 3) skier’s final KE

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

A) only 2B) only 3C) 1, 2, and 3D) only 1 and 3E) only 2 and 3

The gravitational PE depends upon the reference gravitational PE depends upon the reference levellevel, but the differencedifference PE does notPE does not! The work done by gravity must be the same in the two solutions, so PE and PE and KE should be the sameKE should be the same.

Conceptual Quiz

1) skier’s PE 2) skier’s change in PE 3) skier’s final KE1) skier’s PE 2) skier’s change in PE 3) skier’s final KE

Follow-up:Follow-up: Does anything change Does anything change physicallyphysically by the choice of by the choice of yy = 0? = 0?

Conceptual QuizWhich contributes more to the cost of your electric bill each month, a 1500-Watt hair dryer or a 600-Watt microwave oven?

A) hair dryerB) microwave ovenC) both contribute equallyD) depends upon what you

cook in the ovenE) depends upon how long

each one is on

1500 W1500 W

600 W600 W

We already saw that what you actually pay for is energyenergy. To find the energy consumption of an appliance, you must know more than just the power rating—you have to know how long it you have to know how long it was runningwas running.

Conceptual QuizWhich contributes more to the cost of your electric bill each month, a 1500-Watt hair dryer or a 600-Watt microwave oven?

1500 W1500 W

600 W600 W

A) hair dryerB) microwave ovenC) both contribute equallyD) depends upon what you

cook in the ovenE) depends upon how long

each one is on

How does the work required to

stretch a spring 2 cm compare

with the work required to

stretch it 1 cm?

A) same amount of work

B) twice the work

C) four times the work

D) eight times the work


How does the work required to

stretch a spring 2 cm compare

with the work required to

stretch it 1 cm?

A) same amount of work

B) twice the work

C) four times the work

D) eight times the work

The elastic potential energy is kxkx22. So in the second case,

the elastic PE is four times greaterelastic PE is four times greater than in the first case. Thus,

the work required to stretch the spring is also four times work required to stretch the spring is also four times

greatergreater.


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Conservation of Energy Lecturer: Professor Stephen T. Thornton