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Conservation of Energy Lecturer: Professor Stephen T. Thornton

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Conservation of EnergyLecturer: Professor Stephen T. Thornton

A) Mike produced

more power

B) Joe produced more

power

C) both produced the

same amount of

power

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

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

Because power = work / time, we see that Mike produced 0.5 W and Joe 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.

Last Time

Conservative 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

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.

Then the potential energy of the spring is:

Springs

The work required to

compress a spring is

The potential energy

of a spring is

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.

A)half the height

B) the same height

C) 2 times the height

D) twice the height

E) four times the height

A truck, initially at rest, rolls down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many 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

A truck, initially at rest, rolls down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be?

Use energy conservation:

- initial energy: Ei = PEg = mgH
- final energy: Ef = KE= mv2
Conservation of Energy:

Ei = mgH= Ef = mv2

therefore: gH = v2

So if v doubles, H quadruples!

A)half as much

B) the same amount

C) 2 times as much

D) twice as much

E) four times as much

x

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

x

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?

Use energy conservation:

initial energy: Ei = KE= mv2

final energy: Ef = PEs = kx2

Conservation of Energy:

Ei = mv2= Ef = kx2

therefore: mv2 = kx2

So if v doubles, x doubles!

A)4 m/s

B) 5 m/s

C) 6 m/s

D) 7 m/s

E) 25 m/s

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

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?

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

DPE = DKE = m(4)2

When starting from 3 m/s, the

final KE is:

KEf= KEi + DKE

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

= m(25)

= m(5)2

Speed is not the same as kinetic energy

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 Kof 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.

D) same speed

for all balls

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?

C

B

A

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?

D) same speed

for all balls

C

B

A

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

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

Law of Conservation of Energy

We discussed Conservation of Mechanical Energy last time.

Nonconservative, or dissipative, forces associated with:

Friction

Heat

Electrical energy

Chemical energy

and more

do not conserve mechanical energy. However, when these forces are taken into account, the total energy is still conserved:

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.

Height

New total energy

Height

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:

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.

Power

Power 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

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:

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!

A)Paul

B) Kathleen

C) both the same

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

A)Paul

B) Kathleen

C) both the same

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

Conservation of Energy:

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

A)Paul

B) Kathleen

C) both the same

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

A)Paul

B) Kathleen

C) both the same

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

Even though they both have the same final velocity, Kathleen is at a lower height than Paul for most of her ride. Thus, she always has a larger 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 2

B) only 3

C) 1, 2, and 3

D) only 1 and 3

E) only 2 and 3

1) 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 2

B) only 3

C) 1, 2, and 3

D) only 1 and 3

E) only 2 and 3

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

The gravitational PE depends upon the reference level, but the differenceDPE does not! The work done by gravity must be the same in the two solutions, so DPE and DKE should be the same.

Follow-up: Does anything change physically by the choice of y = 0?

A)hair dryer

B) microwave oven

C) both contribute equally

D) depends upon what you cook in the oven

E) depends upon how long each one is on

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

600 W

1500 W

A)hair dryer

B) microwave oven

C) both contribute equally

D) depends upon what youcook in the oven

E) depends upon how longeach one is on

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

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

600 W

1500 W

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

How does the work required to stretch a spring 2 cm compare with the work required to stretch it 1 cm?

The elastic potential energy is kx2. So in the second case, the elastic PE is four times greater than in the first case. Thus, the work required to stretch the spring is also four times greater.