Energy

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Energy
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• Analyzing the motion of an object can often get
  to be very complicated and tedious requiring
  detailed knowledge of the path, frictional
  forces, etc.
• There has to be an easier way
• It turns out that there is it is done by
  analyzing the objects energy.

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Energy

• The something that enables an object to do work
  is energy.
• Energy is measured in Joules (J).
• Forms of Energy
• Mechanical (kinetic and potential)
• Thermal (heat)
• Electromagnetic (light)
• Nuclear
• Chemical

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Mechanical Energy

• Mechanical energy is the form of energy due to
  the position or the movement of a mass.

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Kinetic Energy

• Kinetic energy is the energy of motion it is
  associated with the state of motion of an object.
• The faster an object is moving, the greater its
  kinetic energy an object at rest has zero
  kinetic energy.
• For an object of mass m, we will define kinetic
  energy as
• The SI unit of kinetic energy is the Joule (J).

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Kinetic Energy

• If we do positive work on an object by pushing on
  it with some force, we can increase the objects
  kinetic energy (and thereby increasing its
  speed).
• We can account for the change in kinetic energy
  by saying that the force transferred energy from
  you to the object.
• If we do negative work on an object by pushing on
  it with some force in the direction opposite to
  the direction of motion, we can decrease the
  objects kinetic energy (and decrease its
  speed).
• We can account for the change in kinetic energy
  by saying that the force transferred energy from
  the object to you.

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Kinetic Energy

• Whenever we have a transfer of energy via a
  force, we say that work is done on the object by
  the force.
• Work W is energy transferred to or from an object
  by means of a force acting on that object.
• Energy transferred to the object is positive
  work.
• Energy transferred from the object is negative
  work.
• Work is nothing more than transferred energy it
  therefore has the same units as energy and is
  also a scalar quantity.
• Note that nothing material is transferred.
• Think of it like the balance in two bank
  accounts when money is transferred the number
  for one account goes down by some amount and the
  number for the other account goes up by the same
  amount.

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Work-Energy Theorem

• Suppose we have an bead which is constrained to
  move only along the length of a frictionless
  wire.
• We then supply a constant force F on the bead at
  some angle ? to the wire.
• Because the force is constant, we know that the
  acceleration will also be constant.

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Work-Energy Theorem

• But because of the constraint, only the force in
  the x direction matters, thus Fx max where m
  is the beads mass.
• We can relate the beads velocity at some
  distance down the wire to the acceleration using

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Work-Energy Theorem

• Solving for ax, substituting into the Fx
  equation, multiplying both sides by d, and
  distributing the ½m throughout the equation

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Work-Energy Theorem

• But we can see that the right side of the
  equation is no more than the kinetic energy after
  the force has been applied minus the kinetic
  energy before the force was appliedand that
  by definition is the work done

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Work-Energy Theorem

• When calculating the work done on an object by a
  force during a displacement, use only the
  component of the force that is parallel to the
  objects displacement.
• where ? is the angle between the force F and the
  horizontal.
• The force component perpendicular to the
  displacement does no work.

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Work-Energy Theorem

• Work-Energy Theorem the net work done on an
  object is equal to the change in kinetic energy
  of the object.
• W Kf Ki Fd 0.5m(vf2 - vi2)
• A net force causes an object to change its
  kinetic energy because a net force causes an
  object to accelerate, and acceleration means a
  change in velocity, and if velocity changes,
  kinetic energy changes.

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Gravitational Potential Energy

• Potential energy (U) an object may store energy
  because of its position. Energy that is stored
  is called potential energy because in the stored
  state it has the potential to do work.
• Work is required to lift objects against Earths
  gravity.
• Potential energy due to elevated positions is
  gravitational potential energy.
• The amount of gravitational potential energy
  possessed by an elevated object is equal to the
  work done against gravity in lifting it.
• Ug Fwh mgh

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Work and Potential Energy

• When we throw a tomato up in the air, negative
  work is being done on the tomato which causes it
  to slow down during its ascent.
• As a result, the kinetic energy of the tomato is
  reduced eventually to zero at the highest
  point.
• But where did that energy go???

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Work and Potential Energy

• Where it went was into an increase in the
  gravitational potential energy of the tomato.
• The reverse happens when the tomato begins to
  fall down.
• Now the positive work done by the gravitational
  force causes the gravitational potential energy
  to be reduced and the tomatos kinetic energy to
  increase.

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Work and Potential Energy

• From this we can see that for either the rise or
  fall of the tomato, the change ?U in the
  gravitational potential energy is the negative of
  the work done on the tomato by the gravitational
  force
• In equation form we get

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• Only changes in potential energy have meaning it
  is important that all heights be measured from
  the same origin.
• In many problems, the ground is chosen as the
  zero level for the determination of the height.
• As the ball falls from A to B, the potential
  energy at A is converted to kinetic energy at B.
  
• The amount of potential energy of the ball at
  point A will equal the amount of kinetic energy
  of the ball at point B.

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Elastic Potential Energy

• Stretching or compressing an elastic object
  requires energy and this energy is stored in the
  elastic object as elastic potential energy.
• The work required to stretch or compress a spring
  is dependent on the force constant k.
• The force constant will not change for a
  particular spring as long as the spring is not
  permanently distorted (which occurs when the
  elastic limit of the spring is exceeded).
• F kx

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Elastic Potential Energy

• The force required to stretch/compress an elastic
  object is not a constant force. The work needed
  varies with the amount of stretch/compression.
• W 0.5kx2
• The potential energy of an elastic object is
  equal to the work done on the elastic object.
• Ue 0.5kx2
• In actual practice, a small fraction of the work
  in stretching/compressing an elastic object is
  converted into heat energy in the spring.

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Work and Elastic Potential Energy

• If we give the block a shove to the right, the
  kinetic energy of the block is transferred into
  elastic potential energy as the spring
  compresses.
• The work done in compressing the spring is the
  negative of the change in the blocks kinetic
  energy.
• And of course the reverse happens when the spring
  stretches back out potential energy gets
  transformed back into kinetic energy.

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Conservation of Mechanical Energy

• Conservative forces all the work done is stored
  as energy and is available to do work later.
  Example gravitational forces, elastic forces.
• Nonconservative (dissipative) forces the force
  generally produces a form of energy that is not
  mechanical.
• Friction is a nonconservative (dissipative) force
  because it produces heat (thermal energy, not
  mechanical).
• The total amount of energy in any closed system
  remains constant.

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Conservation of Mechanical Energy

• The sum of the potential and kinetic energy of a
  system remains constant when no dissipative
  forces (like friction) act on the system.
• Law of Conservation of Energy energy cannot be
  created or destroyed it may be changed from one
  form to another or transferred from one object to
  another, but the total amount of energy never
  changes.

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Conservation of Mechanical Energy

• In a closed system in which gravitational
  potential energy and kinetic energy are involved,
  the potential energy at the highest point is
  equal to the kinetic energy at the lowest point.
• mgh 0.5mv2

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• Notice that the sum of the potential energy (PE)
  and kinetic energy (KE) at every point is 40000
  J. Energy is conserved.

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• The velocity at the lowest point can be
  determined by the height

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Conservation of Mechanical Energy

• The change in velocity due to a change in height
  can also be determined
• The height can be determined by the initial
  velocity (vf 0 m/s)

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Conservation of Mechanical Energy

• In a closed system in which elastic potential
  energy and kinetic energy are involved, the
  potential energy at the maximum distance of
  stretch/compression is equal to the kinetic
  energy at the equilibrium (rest) position.

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Conservation of Mechanical Energy

• Conservation of Energy Equation
• W done (by applied force) Ugravitational before
  Uelastic before K before Ugravitational
  after Uelastic after K after W done
  (usually by friction)
• Fappliedd m g hi 0.5 k xi2 0.5 m vi2
  m g hf 0.5 k xf2 0.5 m vf2 FFd

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• If there is no change in height m g hi and m
  g hf drop out of the equation.
• If there is no spring or elastic object 0.5
  k xi2 and 0.5 k xf2 drop out of the equation.
• If there is no change in velocity 0.5 m vi2
  and 0.5 m vf2 drop out of the equation.
• If there is no applied force (a push/pull that
  you supply) Fappliedd drops out of the
  equation.
• If there is no friction FFd drops out of the
  equation.

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Helpful Online Links

• Work Energy Theorem
• The Work-Energy Theorem
• Elastic Constant k
• Hookes Law Applet