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Name _________________ Final Exam December 10, 2012 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. When we look at clusters of galaxies, we often see distorted images of galaxies far behind them. What is causing this distortion? A) The gravity from the cluster is actually bending the light, like a lens B) The gas within the cluster causes the light to refract C) The dust absorbs the light from the distant galaxy, then reemits it to produce a distorted image D) We are actually seeing a reflected version of our own galaxy E) Photon-photon scattering by the light from the intermediate cluster is causing the image to get distorted 2. Why do astronauts float around inside the space station? A) There is nothing holding the space station up, so they are falling together B) They are so far from the Earth the gravity is negligible C) The Earth’s gravity is screened out by the Earth’s atmosphere D) The rocket engines constantly hold it up E) The space station is rotated to create an artificial gravity that counters Earth’s gravity 3. The way we map out the spiral arms of our galaxy is A) Using the 21 cm line caused by electron spins flipping B) Using the light from hot hydrogen regions, where there is ionized hydrogen C) Mapping the infrared light produced by dust clouds D) Reflecting light off of planetary nebulas E) Taking external pictures using space probes that have left the galaxy 4. According to Newton, a planet orbiting a point mass source will move in an ellipse. According to Einstein, what changes about this prediction? A) Nothing B) It still goes around an ellipse, but the speed is different C) Only circular orbits are possible D) The ellipse precesses, so the direction of the ellipse is constantly changing E) The plane of the ellipse moves up and down, so the Sun does not remain in that plane 5. In what way is the absolute future different from the naïve future? A) The absolute future has time differences that exceed the uncertainty relation B) The absolute future is places that are at the same place, but at a different time C) The absolute future demands only that the absolute value of the time difference is positive D) All observers will agree on the absolute future, unlike the naïve future E) The absolute future is restricted to events that have lightlike separation 6. Probably the most important thing that determines the morphology (shape or classification) of a galaxy is A) Its total mass B) How strong the magnetic fields are C) Whether it is composed of light or heavy stars D) Whether it has a source of cool gas flowing in E) What its hydrogen/helium ratio is 7. According to quantum mechanics, what objects have both particle and wave properties? A) Photons, but not electrons B) Electrons, but not photons C) Photons and electrons, but not other elementary particles D) Elementary particles and atoms, but not larger objects E) Everything 8. What distinguishes a “good” coordinate transformation, one in which the laws of physics remain unchanged, from a “bad” coordinate transformation? A) Good ones leave the equation F ma unchanged B) Good ones are restricted to space translations and rotations around an axis C) Good ones leave the four-dimensional distance formula unchanged D) Good ones leave volumes unchanged E) Good ones do not mix time with space 9. Antimony (Sb) is element number 51 on the periodic table. How many neutrons does 121Sb have? A) 19 B) 51 C) 70 D) 121 E) 172 10. In the photoelectric effect, why does (typically) visible light not eject any photons from some metals, but ultraviolet light does? A) The light has to match the resonance frequency for extracting electrons B) Ultraviolet lights are much brighter than visible light C) The photons of ultraviolet light have enough energy to extract an electron; visible light photons do not D) Metals reflect visible light but are transparent to ultraviolet E) Ultraviolet light has a wavelength short enough it can slip between atoms and pull out the electrons 11. Which of the following might be a good summation of Einstein’s general theory of relativity? A) Things are even more out of whack than you think B) Matter tells space how to curve; space tells matter how to move C) Light tells objects how fast they can go; objects determine the speed of light D) Gravitational force is proportional to one over the distance squared; force equals mass times acceleration E) Planetary orbits precess in gravitational fields; time slows down near a black hole 12. The best evidence that galaxies contain large amounts of dark matter spread throughout galaxy haloes is A) We often cannot see much of the galaxy, since it is blocked by dark objects B) Gravitational lensing shows most of the mass is in the form of massive compact halo objects (MACHOs) C) The total mass as measured by gravity is much less than the total mass as measured by light D) There are too few stars to create all the light, so there must be a dark source of light E) The rotation curves measured by Doppler shift do not fall off with distance 13. How do the three types of radioactive decay rank, from most penetrating to least penetrating? A) ,, B) ,, C) ,, D) ,, E) ,, 14. The central engine that powers an active galaxy is almost certainly a very massive A) conventional star B) black hole C) neutron star D) supernova E) star cluster 15. The momentum operator in quantum mechanics is 2 d 2 A) B) x C) D) V x i x 2m dx 2 E) 2 d 2 V x 2m dx 2 16. What is the approximate ratio of the size of an atom divided by the size of a nucleus? A) 300,000 B) 3,000 C) 30 D) 0.03 E) 0.0003 17. When you excite the electrons in hydrogen, why do you only get very specific wavelengths back out? A) Hydrogen forms a lattice of atoms, and only those wavelengths that fit evenly between the atoms are possible B) The electrons can only take on discrete energy values, and when the electron moves from one level to another, it emits a definite amount of energy C) Hydrogen is very good at absorbing almost all wavelengths, so you only see those wavelengths that aren’t absorbed D) The interaction takes a very specific amount of time to take place, and the time it takes times the speed of light c yields the wavelength E) Actually, all wavelengths come out, but the diffraction gratings used to measure the wavelengths cause interference for most other wavelengths 18. Which of the following formulas is still true in special relativity? A) F ma B) W F d C) E 12 mv 2 D) p mv E) None of these 19. What fraction of a heavy nucleus (A ~ 200) might be protons? A) 10% B) 40% C) 50% D) 60% E) 90% 20. In general relativity, the gravitational source (cause of gravity) is A) Mass (only) B) Energy (only) C) Momentum (only) D) The stress-energy-momentum tensor E) Invisible elves that push everything around 21. Radio galaxies typically produce most of their radio power A) Right in the center, at the nucleus B) Spread throughout the disk of the galaxy C) Coming from one or two lobes that stick well out of the galaxy D) From a sphere that completely surrounds the visible galaxy; the halo E) Coming from a ring in the plane of the galaxy that is near the rim 22. The name of the galaxy we live in is A) Andromeda B) Coma C) Milky Way D) Fornax E) Whirlpool 23. If a wave function satisfies Schrodinger’s equation, but doesn’t satisfy the normalization condition, what do you do to fix it? A) Try changing the energy, and see if that can fix the problem B) Adjust the potential function V to make it work out C) Rescale the distance x to make it work out D) Multiply (or divide) by a constant to make the normalization work out E) Normalization is not ever necessary or useful, so ignore the problem 24. Which of the following can be found in significant quantities in spiral galaxies like ours? A) Cool gas (only) B) Dust (only) C) Young stars (only) D) Cool gas and dust, but not young stars E) Cool gas, dust, and young stars 25. Which of the following is the correct statement of the uncertainty principle? p 1 x 1 A) x 12 B) p 12 C) 2 D) E) xp 12 p x 2 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. Why is it that you would expect there to be a fourth component of momentum p in special relativity? What does the fourth component of the conservation of momentum rule turn out to be? 27. Explain in terms of the uncertainty principle why it is that confining a particle into a small space gives it energy. 28. Under what circumstances does it make sense to describe the wave function of a particle as the product of a radial and angular part, R r Y , ? 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 (in terms of the neutron/proton ratio) which sort of nuclei are likely to undergo - decay and which undergo + decay. 30. Explain qualitatively how it might be possible that different categories of active galaxies might actually be more similar than they appear. 31. What are the four different categories of galaxies? 32. What is a geodesic? Under what circumstances does an object follow geodesics, according to general relativity? Part IV: Calculation (review material) [40 points] Choose two of the following three questions and perform the indicated calculations (20 points each) 33. A Borg Collective Cube ship is in the shape of a cube 3040 m on a side. The ship is then accelerated to a speed such that at least one of its dimensions is now 2000 m, as viewed by a stationary observer.1 (a) What are the other dimensions of the ship? (b) How fast is it traveling? (c) How long would it take, in years, at this speed, to travel the 4.2 light years from -Centauri C to the Earth? (d) According to a clock on the Borg ship, how long would it take to go this distance? 34. A harmonic oscillator with angular frequency 1.20 1014 s 1 is in the state n = 11. (a) What is the energy of this electron? (b) If the electron shifts to the lowest possible energy state, what is its new energy? Will it emit or absorb a photon to make this transition? (c) What is the energy of the photon in part (b)? (d) What is the frequency and wavelength of the photon in part (c)? 35. A neutral hydrogen atom is not in the ground state, but rather it has principal quantum number n = 4. (a) What are the possible values of l? What would be the corresponding possible values of L2? (b) What are the possible values of m? What would be the corresponding possible values of Lz? (c) What are the possible values of s? What would be the corresponding possible values of S2? (d) What are the possible values of ms? What would be the corresponding possible values of Sz? 1 Assume that “warp drive” does not exist. In other words, assume relativity applies. Part V: Calculation (new material): [60 points] Choose three of the following four questions and perform the indicated calculations (20 points each) 36. The world produces about 400 g of tritium, 3H, per year, which has a half-life of 12.32 y. (a) Approximately how many atoms of tritium are there in 400 g? (b) What is the decay rate in s-1 for this isotope? A year is 3.156107 s. (c) How many decays per second are there when this much tritium is fresh? Each decay produces 18,590 eV of energy. What is the power, in W, of all this tritium? 1 eV = 1.60210-19 J. (d) How long do you have to wait until you only have one atom of tritium left? 37. Photocopied with the equation on the next page is a portion of Appendix A from the text. 7Be is an unstable atom which decays with a half-life of 53.3 days. (a) What would be the resulting isotope if this isotope underwent decay? What if it underwent electron capture? What if it underwent + decay? (b) What is the Q-value for each of these processes? (c) Cosmic rays from deep space, many light years away, sometimes contain 7Be+4 nuclei, which have no electrons. With such a short half-life, how is this possible? 38. A certain galaxy, viewed edge on, has the H- line measured, which normally has a wavelength of 656.3 nm . It is found to vary as a function of distance from the center, as sketched at right. (a) How fast is each side of the galaxy moving, compared to us, in km/s? In each case, is it towards us or away from us? (b) How fast is the galaxy moving towards or away from us, on average? (c) How fast is this galaxy rotating around its center, and which side is rotating towards us or away from us? 39. A black hole of mass M 20.0 M ( M 1.989 1030 kg ) has a clock lowered to a distance R from the black hole, and left there for one hour. It is then found that the clock has the incorrect time by four minutes. (a) What is the Schwarzschild radius for this star, in km? (b) Will the clock be four minutes fast or four minutes slow? (c) What is the radius R, in km? Equations h 6.626 1034 J s 4.136 1015 eV s u 931.494 MeV / c 2 1.055 1034 J s 6.582 1016 eV s Constants: G 6.673 1011 m3 / kg / s 2 N A 6.022 10 2me c 2 1.022 MeV 23 M He 4.002602 u 1 y 3.156 107 s Gravitational time dilation: u 1.661 1027 kg 2GM t 1 2 cr Red Shift: Harmonic Oscillator: E n 12 , n 0,1, Schwarzschild radius: RS 2GM c 2 Isotope Masses 2GM 0 1 2 cr 1/ 2