Name _________________
Final Exam
December 12, 2014
This test consists of five parts. Please note that in parts II through V, you can skip one question
of those offered.
Part I: Multiple Choice (mixed new and review questions) [50 points]
For each question, choose the best answer (2 points each)
1. In what way is gravity different from other forces, like electromagnetism, and this clue
helped Einstein come up with his general relativity?
A) Electromagnetism is much stronger, in general, than gravity
B) Electromagnetism has vector fields like the electric and magnetic fields, unlike gravity
C) Electromagnetism binds atoms together; gravity does not
D) Gravity falls off with the square of distance, electromagnetism has a different power
E) All objects accelerate exactly the same way in a gravitational field, unlike from other
forces
2. What is the strongest piece of evidence demonstrating the presence of dark matter in spiral
galaxies?
A) The obstructed view of large portions of the galaxy, the light blocked by the dark matter
B) Measuring the rotational velocities of gas using the 21 cm line of hydrogen
C) Measuring the orbital periods of stars around the center of the galaxy
D) The focusing of light by MACHO’s in the galactic halo
E) The clumping of stars caused by concentrations of dark matter
3. What is the significance of the event horizon, or Schwarzschild radius, for a black hole?
A) It is the point of no return; beyond this, nothing can leave the black hole
B) It is the last radius at which an object can orbit in a circle around the black hole
C) It is the point at which tidal forces would rip any physical object apart
D) It is the point at which gravitational forces become infinite
E) It is the point at which spacetime around a black hole begins to be curved
4. Under what circumstance can you tell that event A occurred before event B, as agreed by all
observers?
A) If the time of A is before the time of B
B) If the time of A is before the time of B, and the events are timelike separated
C) If the time of A is before the time of B, and the events are spacelike separated
D) If the time of A is before the time of B, and they occur at the different locations
E) It is impossible to ever determine which of two events occurs first
5. If an object of mass m is accelerated to near the speed of light, what is the upper limit of its
total energy?
A) 12 mc 2
B) mc 2
C) 32 mc 2
D) 2mc 2
E) 
6. The largest amount of our galaxy’s mass is in the
A) Disk
B) Bulge
C) Halo
D) Nucleus
E) Fast food restaurants
7. In one half-life, fifty percent of the original atoms are gone. What percentage of the original
atoms are gone after two-half lives?
A) 0%
B) 25%
C) 50%
D) 75%
E) 100%
8. What assumption did Planck have to make to avoid the ultraviolet catastrophe?
A) The number of polarizations of light at high energy had to be modified
B) An upper limit had to be put on the frequencies of light that could occur
C) The energy in each mode had to come in multiples of a constant times the frequency
D) The energy in each mode had to come in multiples of a constant times the wavelength
E) Light had to be in circular orbits around atoms
9. The stress-energy tensor measures the presence of matter and other sources of gravity.
According to general relativity, what is the stress energy tensor proportional to?
A) The metric, or local distance formula
B) The geodesic, or formula for the path of a particle
C) The Einstein tensor, a measure of space-time curvature
D) The local velocity of light
E) The torsion field, or measure of the twisting of space-time
10. What property of the hydrogen atom made it possible to factor the wave function into an
angular part and a radial part, reducing a 3D problem to a 1D problem?
A) The potential is spherically symmetric
B) The force is an inverse-square law
C) The electron mass is much less than the nuclear mass
D) The nucleus is much smaller than the size of the atom
E) The atom is net neutral, so the nuclear charge cancels the electron’s charge
11. An  particle contains ___ protons and ___ neutrons
A) 0, 0
B) 0, 2
C) 2, 0
D) 2, 2
E) 2, 4
12. Why is it X-rays that produce diffraction patterns when they scatter off of crystals, rather
than some other wavelength?
A) X-rays have a wavelength comparable to the interatomic spacing in crystals
B) Only X-rays have enough energy to knock an inner electron out of the crystal
C) Only X-rays can penetrate deep enough in the crystal to scatter from it
D) Only X-rays feel the effects of the atoms, other wavelengths are immune
E) Only X-rays carry enough momentum to bounce off the atomic layers
13. What is the superiority of the formula c 2 d 2  c 2  dt    dx    dy    dz  used in general
2
2
2
2
relativity compared to the usual formula, c 2  2  c 2  t    x    y    z 
2
A)
B)
C)
D)
E)
2
2
2
It contains curvature, allowing one to move to general relativity
It allows one to calculate proper time along curved paths, not just straight ones
It allows for both spacelike and timelike paths, not just one or the other
It makes it easier to quantize the theory
It has the Latin letter d, which is easier to write than the Greek letter 
14. Which of the following is a correct uncertainty principle?
A) xp  12 
B) xp  12  C) xt  12  D) xt  12 
E) pt  12 
15. How bright are the brightest active galaxies?
A) A bit brighter than our Sun, but not as bright as a hundred Suns
B) Brighter than a hundred Suns, but not as bright as a million Suns
C) Brighter than a million Suns, but much dimmer than our galaxy
D) About as bright as our entire galaxy
E) Considerably brighter than our entire galaxy
16. Which of the following is a surprising but true fact that comes from special relativity?
A) One can determine which of two observers is actually moving by measuring the relative
rate of their clocks
B) One can determine which of two observers is actually moving by measuring the relative
length of their measuring sticks
C) There are no objects, even fundamental particles, that travel at the speed of light
D) There is no such thing as a truly rigid object
E) Matter and energy are the same thing
17. How do we hope to detect gravity waves in the near future, or have they been detected in the
past?
A) By carefully measuring oscillations in the distance of two arms of a large interferometer
B) By studying the rotation of sensitive gyroscopes in orbit around the Earth
C) By studying the change in the orbit of planets and/or satellites
D) By measuring the bending of starlight around the Sun
E) It is not conceivable that gravity waves will be directly detected in the near future
18. 12B contains twice as many neutrons as protons. The most likely decay for this type of
nucleus would be
B) +
C) 
D) 
E) electron capture
A) –
19. In one version of the barn-pole paradox, a runner with a pole the same length as a barn runs
through a barn, and the farmer then closes the doors both simultaneously, proving the pole is
shorter than the barn. How does this resolve or not resolve the paradox?
A) It resolves it by demonstrating that the runner is truly moving, not the barn
B) It fails to resolve it, because the farmer will fail to close both doors, since the pole is
actually longer
C) It fails to resolve it, because the farmer will see both doors close, but the runner will say
he didn’t close them
D) It fails to resolve it, because the runner will say that the doors were not closed at the same
time
E) It fails to resolve it, because the runner will claim the pole shrank briefly, but only during
the time he was inside the barn
20. Which of the following is close to the mass of a 120Sn atom (atomic number 50)?
A) 50 u
B) 70 u
C) 120 u
D) 190 u
E) None of the preceding
21. Which of the following is true when we do quantum mechanics in 3D?
A) There are separate wave functions for each of the three dimensions
B) The wave function is usually a sum of three wave functions, one for each dimension
C) The energies are usually the product of energies in each of the three dimensions
D) There wave function is never a function of angles, just the distance from the origin
E) The wave function becomes a function of x, y, and z, rather than just x.
22. Why did J.J.Thomson put electrons he liberated in a magnetic field?
A) The field kept the particles trapped long enough that he could study them
B) Magnetic fields allowed him to separate electrons from extraneous particles he wasn’t
interested in
C) Electrons were attracted to the magnet, allowing him to collect them
D) The curvature told him the mass to charge ratio of the electrons
E) He was trying to explain the Earth’s magnetic field, and thought studying electrons might
clarify this
23. At the right is a crude sketch of our galaxy. Where do we
live?
A
B CD E
24. Which types of galaxies typically contain gas, dust, and young stars?
A) Elliptical (only)
B) Spiral (only)
C) Barred spiral (only)
D) Elliptical and spiral, but not barred spiral
E) Spiral and barred spiral, but not elliptical
25. Classically, the minimum energy for a particle in a finite box is zero. When we include
quantum mechanics, the minimum energy is
A)
B)
C)
D)
E)
Negative (always)
Zero (always)
Positive (always)
Positive or negative, but not zero
Positive, negative, or zero
Part II: Short answer (review material) [20 points]
Choose two of the following three questions and give a short answer (1-3 sentences) (10
points each).
26. If I fire a laser at rest, the light moves at speed c. If I am running forwards at 0.5c, the laser
will fire at speed 1.5c, as viewed by an observer at rest. Explain what, if anything, is wrong
with this conclusion.
27. Explain what it means for a wave function to be normalized, giving the relevant equation.
Suppose we have a solution to Schrödinger’s equation that is not normalized, but it does have
a finite integral. Explain how to fix the problem.
28. When a wave with energy E comes across a step potential V0 > E, explain qualitatively what
happens. In particular, what is the probability that it is reflected, and is there any wave
function in the classically forbidden region.
Part III: Short answer (new material) [30 points]
Choose three of the following four questions and give a short answer (1-3 sentences)
(10 points each).
29. Explain qualitatively why the most stable nuclei have approximately equal numbers of
protons and neutrons. Now explain why in heavy nuclei, there is a preference for neutrons.
30. In – decay, explain at the particle level which particles are being destroyed and produced. It
is observed that in – decay, the light negatively charged particle that comes out has a range
of energy, rather than coming out as a single energy. How do you account for this range of
energy?
31. Explain, giving any relevant equations, why the evidence that certain active galactic nuclei
vary very quickly tells us that they are not large in size.
32. What is a geodesic? When do particles follow geodesics?
Part IV: Calculation (review material) [40 points]
Choose two of the following three questions and perform the indicated calculations (20
points each)
A


A

33. A standard playing card is about 8.9 cm  5.7 cm (3½ in  2¼ in) in size. A card thrower
attempts to throw the card in such a way that to a stationary observer, it appears square.
(a) How fast would the card have to be moving to look square?
(b) Which direction would it have to be thrown?
(c) According to an observer moving along with the card, what would
the dimensions of the card be?
(d) The card contains a timed explosive, designed to go off one
second after it is thrown (1.00 s in the card’s frame of reference). How long, according
to a stationary observer, after it is thrown does the card explode?
34. Sound waves, just like light waves, are quantized, and come in discrete amounts. It also
satisfies the frequency-wavelength relation as light if we replace c by the speed of sound,
typically c  343 m/s . A speaker is putting out 1.00 W = 1.00 J/s of sound power at middle
C, a frequency of 262 Hz.
(a) What is the wavelength for this sound?
(b) What is the energy and momentum for one phonon (sound quantum) of this sound, in J?
(c) How many phonons per second are coming out of this speaker? What is the total
momentum per second coming from this speaker?
35. An electron is in a hydrogen atom in the n = 3 state.
(a) What are the possible values of the angular momentum quantum number l? What are the
corresponding values of the angular momentum squared L2?
(b) The actual value of L2 is measured, and discovered to take on its maximum value. What
are the possible values of m? What are the corresponding values of the z-component of
angular momentum Lz?
(c) The atom then loses energy when the electron falls the smallest possible amount to a
different value of n. How much energy, in eV, will the resulting light have?
Part V: Calculation (new material): [60 points]
Choose three of the following four questions and perform the calculations (20 points each)
36. Plutonium is normally thought of as being manmade, but it is likely there is some natural
plutonium present even today. 244Pu is the most stable isotope, with a half-life of 8.1107 y.
Suppose that when the Earth was formed, we had a lump of 1.00 kg of pure 244Pu.
(a) What is the approximate weight, in u, of 244Pu? How many atoms would be in 1.00kg?
(b) What is the decay constant  for 244Pu in y-1?
(c) The Earth is now 4.54109 y old. How many atoms of 244Pu would be left now?
(d) How many radioactive decays per year would be coming from this lump today? How
many decays would have occurred from this lump since
mode Daughter Q (MeV)
you were born, if you are 21 years old?
e.c.
37. Photocopied with the equations on the next page is a portion

of Appendix A from the text. 236Np is an unstable nucleus

which decays first by electron capture, then -decay, then
-decay again, and then – decay, one after the other. Keep
–
in mind that for each step, the parent isotope is the daughter
from the previous step
(a) What are the isotopes after each of these steps?
(b) What is the Q-value for each of these steps?
(c) On the first step, why didn’t it do + decay instead?
38. A ring of atomic hydrogen gas is observed orbiting the
center of a distant galaxy. Normally, this gas produces a
spectral line with a wavelength of 21.10 cm, but in this case
it is observed that the spectral line is shifted to a wavelength
that depends on how far out the gas is from the center, as
plotted at right.
(a) Find the red shift for gas at a distance of 200 Tm from
the center, and repeat for gas at a distance of 800 Tm. For each of these locations, find
the velocity of the gas in km/s.
(b) Estimate the mass of the source of gravity for each of the distances in part (a) (1 Tm =
1012 m). You may leave your answer in kg.
(c) Based on your computations from part (b), is the source of gravity a single object at the
center, or is it a diffuse source of gravity?
39. A scientist is studying a black hole. She stays safely far away from it, while a probe travels
to a distance of R = 100.00 km. It then sends a radio signal with a frequency f0 = 137.0 MHz
up to the scientist, where she observes it at a frequency of f = 112.0 MHz.
(a) What is the mass of the black hole, in solar masses (MSun = 1.9891030 kg)?
(b) There is a timer on the probe so it is supposed to return after one hour. However, an hour
later, the probe is still near the black hole. What went wrong, and when should the
scientist expect the probe to return?
(c) The scientist now sends the probe to the Schwarzschild radius. What is this distance?
h  6.626 10
Constants:
34
Equations
J  s  4.136 1015 eV  s
u  931.494 MeV / c 2
  1.055 1034 J  s  6.582 1016 eV  s
u  1.661 1027 kg
G  6.673 1011 m3 / kg / s 2
2me c 2  1.022 MeV
M He  4.002602 u
N A  6.022 1023
Orbits:
 13.6 eV  Z 2
Hydrogen-like atoms energy: En 
n2
GM
v
r
Gravitational time dilation:
Red Shift:


  0 1 
  t 1
2GM 

c2r 
1/2
2GM
c2r
Schwarzschild radius: RS 
1/2
,
 2GM 
f  f 0 1  2 
cr 

Isotope Masses
2GM
c2