Chapter 8 Conservation of Energy

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Physics is fun!
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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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Example 7-2
y
x
WG - mgh
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Gravitational Force
Example 7-2
Gravitational force is a conservative force.
therefore
WG -mgh
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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 zeroalways 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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y
Wext mgh U
cos q 1
v constant
WG - mgh -U
cos q -1
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y
Wext mgh U
cos q 1
v constant
WG - mgh -U
cos q -1
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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.
• DU - 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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y
x
0
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y
0
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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

• 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.
• The choice of where U 0 is arbitrary.
• 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

• DU - Wconservative force
• DU - Wgravitational force
• DU - 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 DK

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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
constantit 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
stones 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 EnergyGeneric Situation
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Example 8-4

• Roller-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
y1 40 m
y2 0 m
x
0
U2 mgy2 0
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y
y1 40 m
y2 30 m
x
0
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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 gt vP
vK vP
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Kathleen reaches the bottom first because she is
sliding at a greater speed for a longer time.
vK gt vP
vK vP
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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 vaulters
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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Dart has not yet started to move, so it has no
kinetic energy.
Spring is fully compressed, storing potential
energy
Spring is completely uncompressed all its
potential energy has been changed into 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
v1 0 m/s
x
0
v3 0 m/s
Because ball has acquired K in falling, it will
compress the spring an amount Y.
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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. Daves mass is 75
kg and we assume the cord obeys Hookes 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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y
Potential energy is transformed into both kinetic
energy and elastic potential energy.
All potential energy
x
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Example 8-10

• A 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 q 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
U 0, K maximum
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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

• DK DU D(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

• Draw picture.
• Identify system the objects and the forces
  acting.
• Identify all forces that do work.
• Ask yourself what quantity you are looking for,
  and decide what the initial (point 1) and final
  (point 2) locations are.
• 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

• If springs are involved, choose the unstretched
  spring position to be x (or y) 0.
• In no friction or other nonconservative
  (dissipative) forces act then apply conservation
  of mechanical energy K2 U2 K1 U1.
• Solve for the unknown quantity.
• 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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y
x
0
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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 Earths surface,
  gravitational potential energy can be dealt with
  using F mg.
• Far from the Earths 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 Earths
  surface.
• F - G r

mME
r gt rE
r2
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Gravitational force is attractive.
m
Unit vector r
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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 Earths 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 joggers 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 persons 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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y
x
0
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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 Suns 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
Neutral equilibrium
Stable equilibrium