The Physics
Hypertextbook
Opus in profectus

# Potential Energy

## Problems

### 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. A region of space has the following two-dimensional potential energy function…

U(x, y) = x4 + y4 + 2x2y2 − 8x2 + 8y2 + 16

Find the…
1. points of stable and unstable equilibrium
2. range of bound states (not the right term for what I'm thinking of)

### numerical

1. Determine the gravitational potential energy added to a guillotine when the 40 kg mouton (the weighted blade) is raised 2.5 m above the neck of the condemned.
2. In about 5 billion years, the sun will no longer be able to fuse hydrogen into helium and release energy. The delicate balance between the outward push of thermal motion and the inward pull of gravity will be upset. The outermost layers will blow away forming a planetary nebula spanning several light years. The innermost layers will collapse in on themselves forming a white dwarf about as big as the Earth. As a first approximation, assume that about half the current mass of the sun becomes a planetary nebula and half becomes a white dwarf.
1. What is the acceleration due to gravity on the surface of the white dwarf that will form from the sun?
2. How much gravitational potential energy does a 7.5 g marshmallow have relative to the surface of the white dwarf if it is raised to the height of a typical table (85 cm)? State your answer in joules and tons of TNT.
3. Stars that are 1.5~3.0 times the mass of the sun end their lives as neutron stars. Pressures in these dead stars are so great that the normally fluffy electron clouds that surround atoms get squeezed down into the nucleus. In these tight quarters, electrons and protons interact via the weak nuclear force to form neutrons and neutrinos. The heavyweight neutrons stay behind to battle it out for the remaining elbow room, while the lightweight neutrinos escape at nearly the speed of light. The resulting star is essentially a giant nucleus made entirely of neutrons. Since most of the mass of an atom is located in its nucleus and the size of a nucleus is roughly 1/100,000 the volume of an atom, neutron stars are incredibly dense. A 1.5 solar mass neutron star is about 21 km in diameter, which is about the same as the length of Manhattan.
1. What is the acceleration due to gravity on the surface of the neutron star described above?
2. How much gravitational potential energy does a 7.5 g marshmallow have relative to the surface of this neutron star if it is raised to the height of a typical table (85 cm)? State your answer in joules and tons of TNT.

### 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 100 eggs of mass m and height h, one on top of the other. What potential energy would this column of eggs have? (Let the potential energy of the bottommost egg be zero.)

### 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! (Let abr > 0 and − ∞ < x < + ∞.)
1. satellite orbits (gravity and centrifugal force)
 U(r) = a − b r2 r
2. partly inverse, partly linear, just for practice
 U(r) = a ⎛⎝ r − b ⎞⎠ b r
3. interatomic forces (the Lennard-Jones potential)
 U(r) = 4a ⎡⎣ ⎛⎝ b ⎞12⎠ − ⎛⎝ b ⎞6⎠ ⎤⎦ r r
4. partly inverse, partly quadratic, just for practice
 U(r) = 1 − 2(2r − 2)2 2r
5. something like a dissociation reaction
 U(r) = 1 + 20 r (r − 3)2 + 2
6. a cubic function, just for practice
 U(x) = x3 − 3x2 − 6x + 8
7. short range attraction, long range repulsion
 U(x) = 4 − 6 x2 + 1 (4x)2 + 1
8. three wells in a row
 U(x) = − 1 − 1 − 1 (x + 2)2 (x + 0)2 (x − 2)2
9. interatomic forces (the Buckingham potential)
 U(r) = ae−br − c r6