practice

  1. Write something.
  2. Write something else.
  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.)
  4. Write something completely different.

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:
  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