Potential Energy

Summary

• Potential energy is…
  • the energy associated with the arrangement of objects
  • the energy due to position of a quantity in a field
• Potential energy comes in four fundamental types, one for each of the fundamental forces, and several subtypes
  • Gravitational potential energy is…
    • the energy associated with the arrangement of masses
    • the energy of a mass in a gravitational field
  • Electromagnetic potential energy is…
    • the energy associated with the arrangement of charges
    • the energy of a charge in electric and magnetic fields
    • the origin of several subtypes of energy.
      • Chemical potential energy is associated with the arrangement of atoms in a molecule.
      • Elastic potential energy is associated with the deformation of elastic materials.
  • Strong nuclear potential energy is…
    • the energy associated with the arrangement of protons and neutrons is a nucleus
    • the energy associated with the arrangement of quarks in a meson or hadron
    • the energy of a color charge in a strong nuclear field
  • Weak nuclear potential energy is…
    • the energy associated with beta decay reactions
    • the energy associated with flavor change processes in fundamental particles
• Gravitational potential energy can be computed through one of two equations
  • The simplified equation…

    ∆U_g = mg∆h

    assumes that…
    • acceleration due to gravity is nearly constant
    • height change is small compared to the separation between centers
  • The more general equation is dealt with in a later section of this book

    Ug = −	Gm1m2
    	r

• Work and forces
  • For conservative forces, the work done…
    • is path independent (does not depend on the path taken)
    • can be associated with a potential energy function

    W = ∮F ⋅ dr = 0

  • For nonconservative forces, the work done…
    • is path dependent (depends on the path taken)
    • cannot be associated with a potential energy function

    W = ∮F ⋅ dr > 0

• Potential energy and forces
  • A conservative force is the gradient of a potential energy function for every location in space.
    • one dimensional equation

      F(r) = −	dU
      	dr

    • three dimensional equation, expanded notation

      F(r) = −	∂U	î −	∂U	ĵ −	∂U	k̂
      	∂x		∂y		∂z

    • three dimensional equation, compact notation

      F(r) = − ∇U

• Potential energy curves (or surfaces, or their higher order equivalents) are useful problem solving tools; for example…
  • the motion of a particle in a field
    • constant total energy, horizontal line above curve
    • kinetic energy is difference between line and curve
    • bound and unbound states, binding energy
  • stability of equilibrium
    
    

    stable equilibrium	unstable equilibrium	neutral equilibrium
    [diagram]	[diagram]	[diagram]
    local maximum	local minimum	constant potential energy