Theory of Energy

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Contents

Work

  • Energy is the capacity for doing work.
  • Work is the energy transferred as a force moves an object
  • britannica.com/science/work-physics
    • “Work is the measure of energy transfer that occurs when an object is moved over a distance by an external force. If the force is constant, work may be computed by multiplying the length of the path by the magnitude of the force.”
    • “Work done on a body is equal to the increase in the energy of the body, for work transfers energy to the body.}
    • “Work and energy are expressed in the same units, for example, joules and foot-pounds.”
  • Formal Definition
    • Work by a force on an object over a distance is the force exerted times the distance covered.
    • For a constant force in a straight line
      • W = F x D
    • For a variable force in a straight line from a to b
  • Dropping a bowling ball
    • If a sixteen pound bowling ball is dropped five feet, Earth’s gravity does work on the ball equalling:
      • 16 pounds of force x 5 feet travelled = 80 foot-pounds of work.
  • The cue ball, with 10.6 joules of kinetic energy, hits a stationary eight ball:
    • Before impact:
      • Cue ball has kinetic energy of 10.6 joules
      • Eight ball has zero.
    • At Impact: 
      • 10.6 joules of work is performed, transferring 10.6 joules of energy from the cue ball to the eight ball
    • After impact: 
      • The cue ball has zero kinetic energy
      • Eight ball has kinetic energy of 10.6 joules
  • The formula W = F x D is used to derive formulas for different forms of energy
    • To derive the formula for kinetic energy, for example, you calculate the amount of work required to move a mass m from rest to v meters per second.
    • That amount of work is ½ mv2 joules.

Forms of Energy

Forms
  • Total energy
    • The total energy of a system is the sum of the forms of energy in the system, each calculated by its formula.
  • Forms of energy include
    • Chemical energy
    • Elastic potential energy
    • Electrical energy
    • Gravitational potential energy
    • Kinetic energy
    • Mass energy
    • Nuclear energy
    • Radiant energy
    • Thermal energy
Formulas

Two Basic Forms

  • Kinetic Energy, the energy of motion
    • The kinetic energy of an object moving at a given speed is the work required to accelerate the object from rest to that speed.
    • KE = ½ mv2 is derived from W = FD and F = MA
  • Potential Energy, the energy of location (in a force field)
    • The potential energy of an object at a given location is the work required to move the object from a reference point to that location.
      • For example, the formula for gravitational potential energy, 
      • is derived from W = FD and Newton’s Law of Universal Gravitation.
    • Force and potential energy are intimately connected:
      • Force = change of potential energy divided by distance
    • For example, electric forces among atoms bend water molecules
    • In the diagram the forces push downhill, from higher potential energy to lower.

Pendulum

  • The total energy of a pendulum system at a time = the kinetic energy of the bob at the time + the gravitational potential energy of the system at the time.
  • The kinetic energy of the bob, calculated by ½ mv2, changes from zero to the total energy as the bob goes from its high point to its low point
    • m = mass of bob
    • v = velocity of bob
  • The gravitational potential energy, calculated by gmh, changes from the total energy to zero as the bob goes from its high point to its low point 
    • g = acceleration due to gravity
    • m = mass of bob
    • h = height of bob

Work and Heat
Forms of Energy Transfer
Not Forms of Energy

  • Work is a form of energy transfer, not a form of energy
    • Work is the energy transferred resulting from a force moving an object
  • Heat is a form of energy transfer, not a form of energy
    • Heat is the energy transferred resulting from a difference in temperature
  • The total energy of a system does not include work and heat.
  • Water Bucket Analogy
    • A first bucket has five quarts of water and a second two.  You pour one quart from the first bucket into the second. The total amount of water, seven quarts, remains the same but it’s now divided differently: 4 + 3 vs 5 + 2. 
    • The seven quarts does not include the one quart transferred.

Law of Conservation of Energy
Emmy Noether’s Proof

  • In 1915 Emmy Noether (EM-mee ΝEW-tar) proved Noether’s Theorem, which says:
    • If the laws of physics remain valid through transformation X, then quantity Y is conserved, where
      • X = translations in time and Y = energy
      • X = translations in space and Y = linear momentum 
      • X = rotations in space and Y = angular momentum
      • X = gauge transformations and Y = electric charge
  • Thus, Noether proved that the Law of the Conservation of Energy follows from the fact that the laws of physics remain valid through time.

Conservation of Energy in Classical and Quantum Mechanics

  • Proof of Conservation of Energy in Classical Mechanics
    • E = ½ mv2  + V(x)
      • The total energy of a system = kinetic energy + potential energy
    • [several steps of reasoning]
    • Therefore, dE/dt = 0
      • The rate of change of the total energy of the system = 0
  • Proof of Conservation of Energy in Quantum Mechanics
    • For any operator Q, d/dt ⟨Q⟩ = -i/ℏ ⟨[Q,H]⟩, where H represents the total energy of a system
      • The time rate of change of the expectation of an operator Q equals -i/ℏ times the expectation of the commutator of Q with H
    • [H,H] = 0
      • Every operator commutes with itself
    • Therefore,  d/dt ⟨H⟩ = -i/ℏ ⟨[H,H]⟩ = 0
      • The rate of change of the expectation of the Hamiltonian = 0