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Chapter 8Conservation of Energy

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Introduction

• POTENTIAL ENERGY• CONSERVATION OF ENERGY

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8-1 Conservative and Nonconservative Forces

• Forces can be categorized into two types: Conservative Nonconservative

• A Force is conservative if:

The work done by the force on an object moving from one point to another depends only on the initial and final positions and is independent of a particular path taken.

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WG = - mgh

x

y

Example 7-2

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Gravitational ForceGravitational force is a conservative force.WG = -mgh

Example 7-2

therefore

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Gravitational Force is Conservative

• Since (y2 – y1) is the vertical height h, the work done depends only on the vertical height and does not depend on the particular path taken.

• This gives an alternative definition of a conservative force:

A force is conservative if:

The net work done by the force on an object moving around any closed path is zero.

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Assume Conservative Force

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Nonconservative Forces

• Friction is a nonconservative force.• Work done moving a create across floor is equal to

the product of Ffr and the total distance traveled

• Friction force always operates in a direction opposite to motion.

• Therefore work done depends on path length.• Work done is negative. Why?• Thus in round trip, total work by friction is never

zero—always negative.

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Nonconservative Forces

The work done by a nonconservative force is not recoverable; it is lost forever.

• In the real world, ALL forces, except, apparently, forces at the atomic level, are nonconservative forces.

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Example Conservation and Nonconservative Forces

Conservative Forces

• Gravitation

• Elastic

• Electric

Nonconservative Forces*

• Friction

• Air resistance

• Tension in a cord

• Push or pull by a person

* Also known as dissipative forces forces

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8-2 Potential Energy

• Kinetic energy depends on velocity.• Potential energy is energy associated with the

position or configuration of objects.• Various types of potential energy can be defined.• Potential energy (U) only exists for conservative

forces. *

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Wext = mgh = U

WG = - mgh = -U

cos = 1

cos = -1

y

v = constant

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Wext = mgh = U

WG = - mgh = -U

cos = 1

cos = -1

y

v = constant

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Potential Energy For Any Conservative Force

The change in potential energy associated with a particular conservative force Fconservative is defined as the negative of the work done by that force.

U - Wconservative

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Example 8-1

Potential energy changes for a roller coaster.

A 1000-kg roller coaster car moves from point A to point B and then to point C. (a) What is the gravitational potential energy at B and C relative to A? That is, take y = 0 at point A. (b)What is the change in potential when it goes from B to C? (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point C.

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0

y

x

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0

y

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Elastic Potential Energy

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Potential Energy Summary

Gravitational Potential Energy: U = mgy

Elastic Potential Energy: U = ½ kx2

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Potential Energy Summarized

1. Potential energy is always associated with a conservative force, and the difference in potential energy between two points is defined as the negative of the work done by that force.

2. The choice of where U = 0 is arbitrary.

3. Since a force is always exerted by one body on another body, potential energy is not something a body “has” by itself, but rather is associated with the interaction of two or more bodies.

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Work Done by a Conservative Force

U = - Wconservative force

U = - Wgravitational force

U = - Welastic force (eg. spring)

VERY IMPORTANT!

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Example 8-2

Determine F from U.

Suppose U(x) = -ax/(b2 + x2), where a and b are constants. What is F as a function of x?

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8-3 Mechanical Energy and Its Conservation

• Consider a conservative system (i.e., only conservative forces do work.

• Energy is transferred from K to U and back.• Work-energy principle (Section 7-4):

Wnet K

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Total Mechanical Energy

• The total mechanical energy of the system is defined as the sum of the kinetic and potential energy at any particular moment.

E = K + U

K2 + U2 = K1 + U1

or E2 = E1 = constant.

• This holds for conservative forces only.

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Conservation of Mechanical Energy

• K + U remains constant in a closed conservative system.

• This is the principle of conservation of mechanical energy.

If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved.

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8-4 Problem Solving Using Conservation of Mechanical Energy

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Example 8-3

Falling rock.

If the original height of the stone in the figure is y1 = h = 3.0 m, calculate the stone’s speed when it has fallen to 1.0 m above the ground.

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Very Important Concept!

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Gravitational Potential Energy—Generic Situation

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Example 8-4Roller-coaster speed using energy

conservation.

Assuming the height of the hill in the figure is 40 m, and the roller coaster car starts from rest at the top, calculate (a) the speed of the roller coaster car at the bottom of the hill, and (b) at what height it will have half this speed. Take y = 0 (and U = 0) at the bottom of the hill.

Very important problem.

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y

x

y1 = 40 m

y2 = 0 m

U2 = mgy2 = 0

0

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y

x

y1 = 40 m

0

y2 = 30 m

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Conceptual Example 8-5

Speeds on two water slides.

Two water slides at a pool are shaped differently but start at the same height h. Two riders, Paul and Kathleen, start from rest at the same time on different slides. (a) Which rider is traveling faster at the bottom? (b) Which rider makes it to the bottom first? Ignore friction.

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vK = vP

vK > vP

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vK = vP

vK > vP

Kathleen reaches the bottom first because she is sliding at a greater speed for a longer time.

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Example 8-6

Pole vault.

Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter’s center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself.

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Example 8-7

Toy dart gun.

A dart of mass 0.100 kg is pressed against the spring of a toy dart gun as shown. The spring (with spring constant k = 250 N/m) is compressed 6.0 cm and released. If the dart detaches from the spring when the latter reaches its normal length (x = 0), what speed does the dart acquire?

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Spring is fully compressed, storing potential energy

Spring is completely uncompressed; all its potential energy has been changed into kinetic energy.

Dart has not yet started to move, so it has no kinetic energy.

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Example 8-8

Two kinds of potential energy.

A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring, which compresses (see figure) an amount Y = 15.0 cm. Determine the spring constant for the spring. Assume the spring has negligible mass. Measure all distances from the point where the ball first touches the uncompressed spring (y = 0 at this point).

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y

x0

Because ball has acquired K in falling, it will compress the spring an amount Y.

v3 = 0 m/s

v1 = 0 m/s

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Example 8-9

Bungee jump.

Dave jumps off a bridge with a bungee cord (a heavy stretchable cord) tied around his ankle. He falls 15 m before the bungee cord begins to stretch. Dave’s mass is 75 kg and we assume the cord obeys Hooke’s law, F= - kx, with k = 50 N/m. If we neglect air resistance, estimate how far below the bridge Dave will fall before coming to a stop. Ignore the mass of the cord (not realistic, however)

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All potential energy

Potential energy is transformed into both kinetic energy and elastic potential energy.

y

x

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Example 8-10A swinging pendulum.

The simple pendulum, shown in the figure, consists of a small bob of mass m suspended by a massless cord of length l. The bob is released (without a push) at t = 0. (a) Describe the motion of the bob in terms of kinetic energy and potential energy. (b) Determine the speed of the bob as a function of position as it swings back and forth. (c) Determine the speed of the bob at the lowest point of the swing. (d) Find the tension in the cord, FT. Ignore friction and air resistance.

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U = maximum, K = 0

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Summary

E = K + U *• E = total mechanical energy of the system at

a given time.

• Energy at any part in the process equals the energy at any other part of the process.

* True if no nonconservative forces are present.

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8-5 The Law of Conservation of Energy

• Take into account nonconservative forces such as friction.

• When non conservative forces are present in a system, K + U decreases.

• These forces are called dissipative forces.• Friction causes heat.• Heat, or thermal energy is now recognized as a form

of energy.

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Law of Conservation of Energy

K + U + [(energy due to NC forces)] = 0

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

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8-6 Energy Conservation with Dissipative Forces

Conservative Forces

• Gravitation

• Elastic

• Electric

Dissipative Forces*

• Friction

• Air resistance

• Tension in a cord

• Push or pull by a person

• Etc.

* Also known as nonconservative forces

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Work-Energy Principle with Nonconservative Forces

We write the total (net) work Wnet as a sum of the work done by conservative forces WC and nonconservative forces WNC.

Wnet = WC + WNC

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More on Wnet

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Problem SolvingConservation of Energy

1. Draw picture.2. Identify system: the objects and the forces acting.3. Identify all forces that do work.4. Ask yourself what quantity you are looking for, and

decide what the initial (point 1) and final (point 2) locations are.

5. If the body under investigation changes height during the problem, then choose a y = 0 level for gravitational potential energy. The lowest point is often the best.

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Problem SolvingConservation of Energy

6. If springs are involved, choose the unstretched spring position to be x (or y) = 0.

7. In no friction or other nonconservative (dissipative) forces act then apply conservation of mechanical energy: K2 + U2 = K1 + U1.

8. Solve for the unknown quantity.9. If friction or other nonconservative forces are

present and significant, then use: K2 + U2 = K1 + U1 + WNC.To be sure which side of the equation to put WNC or what sign to give it, use your common sense: is the total energy E increased or decreased in the process.

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Example 8-11

Friction on a roller coaster.

The roller coaster, which starts at a height y1 = 40 m, is found to reach a vertical height of only 25 m on the second hill before coming to a stop. It traveled a total distance of 400 m. Estimate the average friction force (assumed constant) on the car, whose mass is 1000 kg.

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0

y

x

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Example 8-12

Friction with a spring.

A block of mass m sliding along a rough horizontal surface is traveling at speed v0 when it strikes a massless spring head-on and compresses the spring a maximum distance X. If the spring has a stiffness constant k, determine the coefficient of kinetic friction between the block and surface.

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8-7 Gravitational Potential Energy and Escape Velocity

• For objects near the Earth’s surface, gravitational potential energy can be dealt with using F= mg.

• Far from the Earth’s surface, we must consider that the gravitational force exerted by the Earth on a particle of mass m decreases inversely as the square of the distance r from the Earth’s surface.

F = - G r mME

r2

[r > rE]

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m

Unit vector r

Gravitational force is attractive.

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Example 8-13

Package Dropped from a High Speed Rocket.

A box of empty film canisters is dumped from a rocket traveling outward from Earth at a speed of 1800 m/s when 1600 km above the Earth’s surface. The package eventually falls to the Earth. Estimate the speed just before impact. Ignore air resistance.

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Example 8-14

Escaping the Earth or the Moon.

(a) Compare the escape velocities of a rocket from the Earth and From the Moon. (b) Compare the energies required to launch the rockets. For the Moon, MM = 7.35 x 1022 kg and rM = 1.74 x 106 m, and for the Earth, ME = 5.97 x 1024 kg and rE = 6.38 x 106 m.

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Chapter 8-8 Power

• Power is defined as the rate at which work is done.• The average power, P, when an amount of work W

done in a time t is

P =

• The instantaneous power, P, is

P =

Wt

dWdt

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Power and Energy

• Whenever work is done, energy is transformed or transferred from one body to another:

P =

The units of power are J/s = the watt (W)• 1 W = 1 J/s• 1 horsepower = 746 watts.

dEdt

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Example 8-15

Stair-Climbing Power.

A 70-kg jogger runs up a long flight of stairs in 4.0 s. The vertical height of the stairs is 4.5 m. (a) Estimate the jogger’s power output in watts and horsepower. (b) How much energy did this require?

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Example 8-16

Power Needs of a Car.

Calculate the power required of a 1400-kg car under the following circumstances. (a) The car climbs a 10o hill at a steady 80 km/h; and (b) the car accelerates along a level road from 90 to 110 km/h in 6.0 s to pass another car. Assume the retarding force on the car is FR = 700 N throughout. The retarding force is friction.

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Efficiency

• Overall efficiency e, is defined as the ratio of the useful power output of the engine, Pout, to the power input, Pin:

e = Pout

Pin

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Homework Problem 5

In starting an exercise, a 1.70-m tall person lifts a 2.20-kg book off the ground so it is 2.40 m above the ground. What is the potential energy of the book relative to (a) the ground. And (b) the top of the person’s head? (c) How is the work done by the person related to the answers in parts (a) and (b)?

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Homework Problem 8

Air resistance can be represented by a force proportional to the velocity v of an object: F = - kv. Is this force conservative? Explain.

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Homework Problem 16

A roller coaster is pulled up to point A where it is released from rest. Assuming no friction, calculate the speed at points B, C, D.

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0

y

x

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Homework Problem 28

A 145-g baseball is dropped from a tree 12.0 m above the ground. (a) With what speed would it hit the ground if air resistance could be ignored? (b) If it actually hits the ground with a speed of 8.00 m/s what is the average force of air resistance exerted on it?

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Homework Problem 37

For a satellite of mass mS in a circular orbit of radius rS determine (a) its kinetic energy K, (b) its potential energy U (U = 0 at infinity), and (c) the ratio K/U.

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Homework Problem 39

Determine the escape velocity from the Sun for an object (a) at the Sun’s surface (r = 7.0 x 105 km, M = 2.0 x 1030 km), and (b) at the average distance of the Earth (1.50 x 108 km). Compare to the speed of the Earth in its orbit.

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Homework Problem 59

A driver notices that her 1000-kg car slows down from 90 km/h to 70 km/h in about 6.0 s on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 80 km/hr?

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*8-9 Potential Energy Diagrams; Stable and Unstable Equilibrium

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Unstable equilibrium

Stable equilibrium

Neutral equilibrium