Chapter 8. Potential Energy and Energy Conservation

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Chapter 8. Potential Energy and Energy
Conservation

• 8.1. What is Physics?      
• 8.2. Work and Potential Energy      
• 8.3. Path Independence of Conservative
  Forces      
• 8.4. Determining Potential Energy Values     
•  8.5. Conservation of Mechanical Energy      
• 8.6. Reading a Potential Energy Curve     
•  8.7. Work Done on a System by an External
  Force      
• 8.8. Conservation of Energy

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Introduction

• In Chapter 7 we introduced the concepts of work
  and kinetic energy. We then derived a net
  work-kinetic energy theorem to describe what
  happens to the kinetic energy of a single rigid
  object when work is done on it. In this chapter
  we will consider a systems composed of several
  objects that interact with one another.

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What is Physics? 

• (1) The system consists of an Earthbarbell
  system that has its arrangement changed when a
  weight lifter (outside of the system) pulls the
  barbell and the Earth apart by pulling up on the
  barbell with his arms and pushing down on the
  Earth with his feet

(2) The system consists of two crates and a
floor. This system is rearranged by a person
(again, outside the system) who pushes the crates
apart by pushing on one crate with her back and
the other with her feet
There is an obvious difference between these two
situations. The work the weight lifter did has
been stored in the new configuration of the
Earth-barbell system, and the work done by the
woman separating the crates seem to be lost
rather than stored away.
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• How do we determine whether the work done by a
  particular type of force is stored or used
  up.?

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The Path Independence Test for a Gravitational
Force

• The net work done on the skier as she travels
  down the ramp is given by

It does not depend on the shape of the ramp but
only on the vertical component of the
gravitational force and the initial and final
positions of her center of mass.
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Path Dependence of Work Done by a Friction Force

• The work done by friction along that path 1?2 is
  given by

• The work done by the friction force along path
  1?4?3?2 is given by

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Conservative Forces and Path Independence

• conservative forces are the forces that do path
  independent work
• Non-conservative forces are the forces that do
  path dependent work

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The work done by a conservative force along any
closed path is zero.
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• Test of a System's Ability to Store Work Done
  by Internal Forces the work done by a
  conservative internal force can be stored in the
  system as potential energy, and the work done by
  a non-conservative internal force will be used
  up

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EXAMPLE 1 Cheese on a Track

• Figure a shows a 2.0 kg block of slippery
  cheese that slides along a frictionless track
  from point 1 to point 2. The cheese travels
  through a total distance of 2.0 m along the
  track, and a net vertical distance of 0.80 m. How
  much work is done on the cheese by the
  gravitational force during the slide?

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Determining Potential Energy Values

• Consider a particle-like object that is part
  of a system in which a conservative force acts.
  When that force does work W on the object,

the change in the potential energy associated
with the system is the negative of the work done
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Gravitational Potential Energy
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GRAVITATIONAL POTENTIAL ENERGY

• The gravitational potential energy U is the
  energy that an object of mass m has by virtue of
  its position relative to the surface of the
  earth. That position is measured by the height h
  of the object relative to an arbitrary zero
  level
• SI Unit of Gravitational Potential Energy joule
  (J)

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Elastic Potential Energy
                                                                                                                                                       

or

• we choose the reference configuration to be
  when the spring is at its relaxed length and the
  block is at .

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Sample Problem 2

• A 2.0 kg sloth hangs 5.0 m above the ground
  (Fig. 8-6).
• a) What is the gravitational potential energy U
  of the slothEarth system if we take the
  reference point y0 to be (1) at the ground, (2)
  at a balcony floor that is 3.0 m above the
  ground, (3) at the limb, and (4) 1.0 m above the
  limb? Take the gravitational potential energy to
  be zero at y0.
• (b) The sloth drops to the ground. For each
  choice of reference point, what is the change
  in the potential energy of the slothEarth system
  due to the fall?

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What is mechanical energy of a system?

• The mechanical energy is the sum of kinetic
  energy and potential energies

For example,
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Conservation of Mechanical Energy

• In a system where (1) no work is done on it by
  external forces and (2) only conservative
  internal forces act on the system elements, then
  the internal forces in the system can cause
  energy to be transferred between kinetic energy
  and potential energy, but their sum, the
  mechanical energy Emec of the system, cannot
  change.

An isolated system is a system that there is no
net work is done on the system by external forces.
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Example 3
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Check Your Understanding 

• Some of the following situations are consistent
  with the principle of conservation of mechanical
  energy, and some are not. Which ones are
  consistent with the principle?
• (a) An object moves uphill with an increasing
  speed.
• (b) An object moves uphill with a decreasing
  speed.
• (c) An object moves uphill with a constant speed.
• (d) An object moves downhill with an increasing
  speed.
• (e) An object moves downhill with a decreasing
  speed.
• (f) An object moves downhill with a constant
  speed.

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Example 4  A Daredevil Motorcyclist

• A motorcyclist is trying to leap across the
  canyon shown in Figure by driving horizontally
  off the cliff at a speed of 38.0 m/s. Ignoring
  air resistance, find the speed with which the
  cycle strikes the ground on the other side.

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EXAMPLE 5 Bungee Jumper

• A 61.0 kg bungee-cord jumper is on a bridge
  45.0 m above a river. The elastic bungee cord has
  a relaxed length of L  25.0 m. Assume that the
  cord obeys Hooke's law, with a spring constant of
  160 N/m. If the jumper stops before reaching the
  water, what is the height h of her feet above the
  water at her lowest point?

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EXAMPLE 6

• In Fig., a 2.0 kg package of tamales slides along
  a floor with speed v14.0 m/s. It then runs into
  and compresses a spring, until the package
  momentarily stops. Its path to the initially
  relaxed spring is frictionless, but as it
  compresses the spring, a kinetic frictional force
  from the floor, of magnitude 15 N, acts on it.
  The spring constant is 10 000 N/m. By what
  distance d is the spring compressed when the
  package stops?

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Net Work on a system
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Internal Work on a single rigid object
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Internal Work on a system
Since Newton's Third Law tells us that
the internal work is given by the integral of y
the internal work on a system is not zero in
general
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Work-Energy Theorem
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Example 7  Fireworks

• A 0.20-kg rocket in a fireworks display is
  launched from rest and follows an erratic flight
  path to reach the point P, as Figure shows. Point
  P is 29 m above the starting point. In the
  process, 425 J of work is done on the rocket by
  the nonconservative force generated by the
  burning propellant. Ignoring air resistance and
  the mass lost due to the burning propellant, find
  the speed vf of the rocket at the point P.

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Reading a Potential Energy Curve
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Finding the Force Analytically
Solving for F(x) and passing to the differential
limit yield
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Reading a Potential Energy Curve

• Turning Points a place where K0 (because UE
  ) and the particle changes direction.
• Neutral equilibrium the place where the particle
  has no kinetic energy and no force acts on it,
  and so it must be stationary.
• unstable equilibrium a point at which . If the
  particle is located exactly there, the force on
  it is also zero, and the particle remains
  stationary. However, if it is displaced even
  slightly in either direction, a nonzero force
  pushes it farther in the same direction, and the
  particle continues to move
• stable equilibrium a point where a particle
  cannot move left or right on its own because to
  do so would require a negative kinetic energy

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Sample Problem

• A 2.00 kg particle moves along an x axis in
  one-dimensional motion while a conservative force
  along that axis acts on it. The potential energy
  U(x) associated with the force is plotted in Fig.
  8-10a. That is, if the particle were placed at
  any position between x0 and x7m , it would
  have the plotted value of U. At x6.5m , the
  particle has velocity v0(-4.0m/s)i . (a)
  determine the particles speed at x14.5m. (b)
  Where is the particles turning point located?
  (c) Evaluate the force acting on the particle
  when it is in the region 1.9mltxlt4.0m.

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General Energy Conservation
For a isolated system where Wext is zero, it
energy is conserved.

• THE PRINCIPLE OF CONSERVATION OF ENERGY Energy
  can neither be created nor destroyed, but can
  only be converted from one form to another.

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Example

•  In Fig. 8-58, a block slides along a path
  that is without friction until the block reaches
  the section of length L0.75m, which begins at
  height h2.0m on a ramp of angle ?30o . In
  that section, the coefficient of kinetic friction
  is 0.40. The block passes through point A with a
  speed of 8.0 m/s. If the block can reach point B
  (where the friction ends), what is its speed
  there, and if it cannot, what is its greatest
  height above A?