Physics 1020 Lecture 22 Chapter 8. Potential Energy and Conservative Forces 2 lectures in total

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Physics 1020 Lecture 22Chapter 8. Potential
Energy and Conservative Forces(2 lectures in
total)

• Conservative and Nonconservative Forces

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Kinetic Energy

• Energy associated with the motion of an object
• Work can be converted into kinetic energy

Example If someone would drive a car into a
tree, the kinetic energy of the car can do work
on the tree it can knock it over
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Potential Energy

• An object can store energy as the result of its
  position.
• Potential energy is associated with the position
  of the object within some system
• Gravitational potential energy
• h - change in position, m mass of the object

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Work done by gravity...
W NET W1 W2 . . . Wn
m
mg
?y

• Depends only on ?y,
• not on path taken!

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Work Done by Gravity on a Closed Path
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Work Done by Friction on a Closed Path
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Conservative and Nonconservative Forces

• Conservative - work depends only upon the initial
  and final positions of the object
• A conservative force is independent of the path
  taken and does zero total work on any closed path
• Can have a potential energy function associated
  with it
• Work and energy associated with the force can be
  recovered
• Nonconservative - work depends on the path taken
  by the object
• The forces are generally dissipative and work
  done against it cannot easily be recovered

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Conservative and Nonconservative Forces

• Conservative
• Gravity
• Spring force
• Nonconservative Forces
• Friction
• Tension
• Forces exerted by a motor (or muscles)

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Conservation of Energy

• The central feature of the energy approach is the
  notion that energy is conserved.
• This means that energy cannot be created or
  destroyed
• If the total amount of energy in a system
  changes, it can only be due to the fact that
  energy has crossed the boundary of the system by
  some method of energy transfer

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The Isolated System

• Conservation of Mechanical Energy
• Consider the work done by the book (system) as it
  falls from some height to a lower height
• Also the Work-Kinetic Energy Theorem gives
• So
• And, since the only thing moving is the book we
  can say

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Conservation of Mechanical Energy

• But we have seen this mgy construction before
• So,
• Expanding the ?s gives
• Rearranging yields

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Conservation of Mechanical Energy

• In general, we define the sum of the kinetic and
  potential energies of a system as the total
  mechanical energy of the system. The equation
• is, therefore, a statement of conservation of
  mechanical energy for an isolated system (that
  is, one for which there are no energy transfers
  across the boundary).

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Up and down the track
PE
PE
Kinetic Energy
If friction is not too big the ball will get up
to the same height on the right side.
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Conservation of Mechanical Energy - Example

• A ball of mass m is dropped from rest at a height
  h above the ground as shown. Ignore air
  resistance. (a) Determine the speed of the ball
  when it is at a height y above the ground.

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Conservation of Mechanical Energy - Example

• The ball and Earth do not experience any forces
  from the environment because we ignore air
  resistance.
• The ball-earth system is isolated ? use
  conservation of mechanical energy.
• At outset, system has PE but no KE.
• As the ball falls, the total ME remains constant.
  The PE of the system decreases and the KE of the
  system increases.

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Conservation of Mechanical Energy - Example

• At height h
• At height y
• ME is conserved so

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Conservation of Mechanical Energy - Example

• (b) Determine the speed of the ball at y if it is
  given an initial speed vi at the initial altitude
  h.
• Initial KE is not zero.
• We have then

Look familiar?
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Elastic Potential Energy

• The restoring force of a spring on an object
  attached to it is also conservative. (Recall
  Hookes Law Fs -kx).
• Recall that the work done by the spring force is
• Ws depends only on the initial (xi) and final
  (xf) positions and is 0 for a closed path (xixf)
• Force is conservative can associate a
  potential energy with the spring force analogous
  to that for the gravitational force.

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Elastic Potential Energy

• The elastic potential energy associated with the
  spring force is
• The elastic potential energy can be thought of as
  the energy stored in the deformed spring.
• The stored potential energy can be converted into
  kinetic energy.

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Elastic Potential Energy

• The elastic potential energy stored in a spring
  is zero whenever the spring is not deformed (Us
  0 when x 0).
• The energy is stored in the spring only when the
  spring is stretched or compressed.
• The elastic potential energy is a maximum when
  the spring has reached its maximum extension or
  compression.
• The elastic potential energy Us is always
  positive.
• x2 will always be positive

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Conservation of Energy and Nonconservative Forces
Consider a sliding block coming to rest because
of friction. Initially the system has KE but
afterward nothing is moving so the KE 0. The
friction force transforms mechanical energy into
internal energy (temperature of block and surface
are slightly warmer than before).

• Wtot Wc Wnc
• Wtot ?K
• Wc - ?U
• Wtot - ?U Wnc ?K i. e. Wnc ?K
  ?U
• Wnc ?E
• When nonconservative forces are present, the
  total mechanical energy of the system is not
  constant

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Conservation of Energy and Nonconservative Forces
The total energy (KE, PE, Eint) of an isolated
system is conserved, regardless of whether the
forces acting within the system are conservative
or nonconservative.

• DU is the change in all forms of potential energy
• Note that DK may represent more than one term if
  two or more parts of the system are moving
• In the absence of friction, this equation becomes
  the same as Conservation of Mechanical Energy
  (i.e., K U constant)

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Problem-Solving Strategy

• Determine if any nonconservative forces are
  involved.
• Remember that if friction or air resistance is
  present, mechanical energy is not conserved, but
  the total energy of an isolated system is
  conserved.
• Identify the configuration for zero potential
  energy
• Include both gravitational and elastic potential
  energies
• For each object that changes elevation, select a
  reference position that will define the zero
  configuration of gravitational PE for the system.
• For a spring, the zero configuration for elastic
  PE is when the spring is neither compressed nor
  extended from its equilibrium position.
• If more than one conservative force is acting
  within the system, write an expression for the
  potential energy associated with each force.

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Problem-Solving Strategy, cont.

• If a nonconservative force (for example, friction
  or air resistance) is present, the mechanical
  energy of the system is not conserved.
• First write expressions for the total initial and
  total final mechanical energies. The difference
  between the total final ME and the total initial
  ME the energy transformed to or from internal
  energy by the nonconservative forces.

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Problem-Solving Strategy, cont

• If the mechanical energy of the system is
  conserved, write the total energy as
• Ei Ki Ui for the initial configuration
• Ef Kf Uf for the final configuration
• Since mechanical energy is conserved, Ei Ef and
  you can solve for the unknown quantity

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Nonconservative Forces - Example 1

• The system is the child and the earth. The child
  is modeled as a point particle. The normal force
  does no work on the system because it is
  always perpendicular to the childs motion. With
  no friction present, there is no energy converted
  to internal energy so we can use the isolated
  system model for which

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Nonconservative Forces, Example 1

• Choose as the reference height for potential
  energy the bottom of the slide so that yi h,
  and yf 0.
• The same result as in free fall!

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Nonconservative Forces, Example 1

• B) Now if a friction force is present on the
  (say) 20 kg child and the child arrives at the
  bottom with a speed of 3.00 m/s. How much does
  the mechanical energy of the system decrease
  because of this force?
• Now we must define the system as the child, the
  earth and the slide. The mechanical energy is
  now NOT conserved!

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Nonconservative Forces, Example 1

• What saves us is that we know what the speed is
  at the bottom of the slide

• -ve means there is a reduction in mechanical
  energy
• The internal energy has increased by 302 J

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Nonconservative Forces - Example 2

• A block of mass 0.800 kg is given an initial
  velocity of vA1.20 m/s to the right and collides
  with a spring of force constant k 50.0 N/m.
• A) If the surface is frictionless, calculate the
  maximum compression of the spring.

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Nonconservative Forces - Example 2

• Our system is the block and the spring. There is
  no transfer of energy across the system boundary
  so we can use an isolated system model
• The block has kinetic energy when it is moving.
• The spring has potential energy when compressed.

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Nonconservative Forces - Example 2
All energy kinetic (due to moving block)
Mixture of kinetic and potential (due to moving
block and partially compressed spring)
All energy potential (due to compressed spring)
All energy kinetic again (due to moving block)
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Nonconservative Forces - Example 2

• Conservation of mechanical energy gives us
• And solving for xmax

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Nonconservative Forces - Example 2

• B) If friction is present (µk 0.500) and if the
  speed of the block just before hitting the spring
  is 1.20 m/s, what is the maximum compression of
  the spring?
• We define the system as the block, the spring and
  the surface. Mechanical energy is NOT conserved
  because of the presence of friction

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Nonconservative Forces - Example 2

• Friction is
• And the work that friction does is

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Nonconservative Forces - Example 2

• At this point we dont know xmax but we just
  carry on
• The change in mechanical energy is

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Nonconservative Forces - Example 2

• Substitute in the numbers
• Rearranging
• Find xmax -0.249 m or 0.0924 m

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Potential Energy Curves and Equipotentials
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A Contour Map