Potential Energy

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

W J M Rankine coined the term potential energy 150 years ago.

gravitational potential energy

ΔU = mgΔh

• acceleration due to gravity is nearly constant
• height change is small compared to the separation between centers

the more general form will be dealt with later

U = −	Gm1m2
	r2

work-energy theorem, two possibilities

• conservative forces
  • work done is independent of path
  • W = ∮ F·dr = 0
  • can be associated with a potential energy function
• nonconservative forces
  • work done depends on path
  • W = ∮ F·dr > 0
  • cannot be associated with a potential energy function

force and potential energy

one-dimensional				three-dimensional							compact notation
F(r) = −	dU			F(r) = −	∂U	ˆi −	∂U	ˆj −	∂U	ˆk	F(r) = − ∇U
	dr				∂x		∂y		∂z

constant total energy, horizontal line above curve, kinetic energy is difference between line and curve

bounded and unbounded states, binding energy

stability of equilibrium

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

Summary

• Potential energy …
  • is energy associated with position in a field (a force that exists at many locations)
  • comes in four fundamental types, one for each of the fundamental forces, and several subtypes
    • gravitational
    • electromagnetic
      • electrostatic
      • chemical
      • elastic
    • strong
      • primary: between quarks, within nucleons (and other hadrons)
      • residual: between nucleons, within the nucleus
    • weak
• Gravitational potential energy
  • can be computed through one of two formulas

    the simplified formula	the more general formula
    ΔU = mgΔh	U = −  Gm1m2 r2
    assumes that	is dealt with in a later section of this book
    acceleration due to gravity is nearly constant height change is small compared to the separation between centers

• Forces and Potential Energy
  • nonconservative forces
    • work done depends on path
    • W = ∮ F·dr > 0
    • cannot be associated with a potential energy function
  • conservative forces
    • work done is independent of path
    • W = ∮ F·dr = 0
    • can be associated with a potential energy function
• Forces and Potential Energy
  • work is the force-displacement integral
    • this is the work energy theorem
  • a conservative force is the gradient of potential energy

  one-dimensional				three-dimensional							compact notation
  F(r) = −	dU			F(r) = −	∂U	ˆi −	∂U	ˆj −	∂U		F(r) = − ∇U
  	dr				∂x		∂y		∂z

• Potential energy curves (or surfaces, or their higher order equivalents) are useful problem solving tools
  • 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

Problems

practice

1. Write something.
   • Answer it.
2. Write something else.
   • Answer it.
3. Calculate the gravitational potential energy released by the collapse of the World Trade Center in New York City on 11 September 2001. Each 110 story tower had a mass of about 550,000,000 kg and a height of 415 m (not including the broadcast tower). Compare this to the energy released on 8 March 1993 when a truck carrying a fertilizer bomb exploded in the underground parking garage of this same complex. Assume an explosive yield equivalent to a half ton of TNT. (One ton of TNT has 4.184 × 10⁹ J of chemical potential energy.)

   Solution …

   Since the mass of the towers is distributed throughout their volume and is not concentrated at a point, this problem is best solved using calculus. Assume a uniform linear mass density and integrate the potential energy formula over the height of the towers. Not surprisingly, the results show that the center of mass of a tower lies at its geometric center, halfway up.

   h =	415 m
   m =	2(550,000,000 kg) = 1.1 × 109 kg
   	h			h
   Ug =	⌠ ⌡	ρgy dy =	mg	⌠ ⌡	y dy =	mg		h2	=	mgh
   			h			h		2		2
   	0			0
   Ug =	(1.1. × 109 kg)(9.8 m/s2)(415 m)
   	2
   Ug =	2.2 × 1012 J		1 ton TNT			= 530 ton TNT
   			4.184 × 109 J

   The gravitational energy released when the towers collapsed was thus about a thousand times greater than the chemical energy released when the truck bomb exploded.

4. Write something completely different.
   • Answer it.

numerical

1. Determine the gravitational potential energy of a guillotine if the mouton (the weighted blade) has a mass of 40 kg and is raised 2.5 m above the neck of the condemned.

algebraic

1. It's possible, but quite difficult, to balance an egg on it's end. It's also possible, but even more difficult, to balance an egg on an egg. Say you had the talent and patience to balance a dozen eggs of mass m and height h, one on top of the other. What potential energy would this column of eggs have?

calculus

1. The Great Pyramid of Cheops was built on the Ghiza Plateau just outside of what is now Cairo, Egypt some time between 2589 and 2566 BCE. It serves as the final resting place of the Pharaoh Cheops (also known as Khufu). As its name suggests, the Great Pyramid is the largest structure of its kind in the world. Determine the gravitational potential energy of the Great Pyramid with respect to its base given the following dimensions:
   • height: 136 m
   • base: square, 230 m on a side
   • top: square, 10 m on a side
   • number of blocks: 2.3 million
   • mass per block: 2.5 tons (average)
2. Fun Functions!

   a.		U(r) =  a  −  b r2 r		r > 0		b > a > 0		a combination of gravitational and centrifugal forces, used for satellite orbits
   b.		U(r) = a  ⎛ ⎝ r  −  b ⎞ ⎠ b r		r > 0		a, b > 0		partly inverse and partly linear, not used for anything, just for practice
   c.		U(r) = 4a  ⎡ ⎣ ⎛ ⎝ b ⎞12 ⎠  −  ⎛ ⎝ b ⎞6 ⎠ ⎤ ⎦ r r		r > 0		a, b > 0		the lennard-jones potential, a simplified model of interatomic forces
   d.		U(r) =  1  − 2(2r − 2)2 2r		r > 0				partly inverse and partly quadratic, not used for anything, just for practice
   e.		U(r) =  1  +  20 r (r − 3)2 + 2		r > 0				something like a dissociation reaction
   f.		U(x) = x3 − 3x2 − 6x + 8		− ∞ < x < + ∞				a cubic function, not used for anything, just for practice
   g.		U(x) =  4  −  6 x2 + 1 (4x)2 + 1		− ∞ < x < + ∞				attractive for short distances, repulsive for long distances
   h.		U(x) = −  1  −  1  −  1 (x + 2)2 (x + 0)2 (x − 2)2		− ∞ < x < + ∞				three atoms in a row

Resources

• general
  • The Mechanical Universe and Beyond (video on demand, login required)
    • Potential Energy, Potential energy provides a powerful model for understanding why the world has worked the same way since the beginning of time.

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