conservation of energy lecturer: professor stephen t. thornton
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Conservation of Energy Lecturer: Professor Stephen T. Thornton. Reading Quiz. A) Mike produced more power B) Joe produced more power C) both produced the same amount of power. - PowerPoint PPT PresentationTRANSCRIPT
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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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
Conceptual QuizConceptual Quiz
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
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?
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?
A) PaulB) KathleenC) both the same
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!
A) PaulB) KathleenC) both the same
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
Conceptual QuizConceptual Quiz
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.
Conceptual QuizConceptual Quiz
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