PHYSICS 109 FINAL EXAMINATION

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Oswald Gerard Dean
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1 PRINTED NAME: PHYSICS 109 FINAL EXAMINATION Problem January 24, :30 11:30 am Jadwin A09 Score 1 /20 2 /10 3 /20 4 /20 5 /20 6 /10 7 /20 Total /120 When you are told to begin, check that this examination booklet contains all the numbered pages from 2 through 13. The exam contains 7 problems with unequal point values as listed above. Do not panic or be discouraged if you cannot do every problem; there are both easy and hard parts in this exam. Keep moving and finish as much as you can! Read each problem carefully. You must show your work the grade you get depends on how well I can understand your solution even when you write down the correct answer. Please BOX your answers. DO ALL THE WORK YOU WANT GRADED IN THIS EXAMINATION BOOKLET! Rewrite and sign the Honor Pledge: I pledge my honor that I have not violated the Honor Code during this examination. Signature

2 MISCELLANEOUS CONSTANTS AND FORMULAE R = 8.3J/molK N A = mol 1 k = J/K 1atm = Pa ρ water = 1000kg/m 3 v sound = 340m/s g = 9.8m/s 2 G = N m 2 /kg 2 1 4πɛ 0 = Nm 2 /C 2 µ 0 4π = 10 7 N/A 2 c = m/s e = C m e = kg = 511keV/c 2 m p = kg = 938MeV/c 2 h = Js hc = 1240eV nm h m e c = hc m e c 2 = m = nm Giga = 10 9 Mega = 10 6 kilo = 10 3 centi = 10 2 milli = 10 3 micro = µ = 10 6 nano = 10 9 pico = 10 12

3 Physics 109 Final Exam 24-Jan-2002 Page 2 1. An ideal heat engine operates with four moles of a diatomic gas on the cycle shown in the diagram. At point A, the temperature is T A = 300K and the volume is V A = 0.04m 3. A B is a constant volume process in which the pressure is doubled. B C is an adiabatic process. C A is an isothermal process. (a) What is the pressure of the gas at A? (2 points) (b) What are the pressure and temperature of the gas at B? (2 points) (c) What is the temperature at C? (1 point) (d) What is the volume at C? (3 points)

4 Physics 109 Final Exam 24-Jan-2002 Page 3 Problem 1. Continued (e) How much work is done by the engine in each of the three processes, A B, B C, and C A? (5 points) (f) How much heat is taken in by the engine in each of the three processes, A B, B C, and C A? (5 points) (g) What is the efficiency of this engine? (2 points)

5 Physics 109 Final Exam 24-Jan-2002 Page 4 2. Water fills a reservoir, open to the atmosphere, to height H = 3 m above the centerline of a horizontal exit pipe at the bottom of the reservoir. The first section of the pipe has radius r 1 = 1cm, the unknown velocity there is v 1 and this section of the pipe has a manometer in which the water rises to an unknown height h above the centerline of the pipe. The second section of the pipe has radius r 2 = 0.5cm and the water has unknown velocity v 2 in this section of the pipe. This is also the speed with which the water leaves the pipe. Assume the flow is incompressible, frictionless, irrotational, and steady. Also, assume the reservoir is so large compared to the pipe, that the water level in the reservoir is constant, even though water is flowing out the exit pipe. (a) What is the speed, v 2, of the water leaving the exit pipe? (3 points) (b) What is the speed, v 1, in the first section of the pipe? (2 points) (c) What is the height, h, of the water in the manometer? (3 points) (d) What is the volume flow rate of the water flowing through the exit pipe? (2 points)

6 Physics 109 Final Exam 24-Jan-2002 Page 5 3. Two (musical?) instruments are made from pipes closed at one end and open at the other. Pipe A has a fundamental (lowest resonant frequency) f 1 = 150Hz. Pipe B is slightly out of tune and has a fundamental frequency f 1 = 155Hz. An observer is located D = 1000m away. (a) What is the length of pipe A? (4 points) (b) What are the second and third lowest resonant frequencies, f 2 and f 3 for pipe A? (4 points) (c) Pipe A begins to play. How many seconds later is the sound first heard by the observer located D = 1000 m away? (2 points)

7 Physics 109 Final Exam 24-Jan-2002 Page 6 Problem 3. Continued (d) If the observer at distance D = 1000m hears a sound level of 50dB, what sound level is heard by an observer who is 10 times farther away at d = 10, 000 m? Assume the pipe radiates sound uniformly in all directions. (3 points) (e) Now pipe B begins to play along with pipe A. Since the frequencies are slightly different, the observer hears beats which the observer thinks are annoying, so he wonders how to make the frequencies he hears the same to get rid of the beats. One way is to change the length of pipe B. Must pipe B be made longer or shorter to make it have the same frequency as pipe A? Why? (2 points) (f) Another way is to move pipe B towards or away from the observer at the correct speed. In what direction (left or right in the diagram) and with what speed must pipe B be moved in order to make the frequency heard by the observer be the same as that from pipe A? (5 points)

8 Physics 109 Final Exam 24-Jan-2002 Page 7 4. A small, low power microscope is made from two convex lenses, each having a focal length of 2.5 cm. The lenses are held in a tube such that their separation is 15.0 cm. To use, the object to be observed is placed to the left of the left lens (the objective lens) and the observer looks from the right of the right lens (the eyepiece). The microscope is placed so that the objective forms an image of the object at the focal point of the eyepiece. (a) How far to the left of the objective must the object be placed so that its image formed by the objective is located 12.5cm to the right of the objective (in other words, at the focal point of the eyepiece)? (4 points) (b) If the object is an evil bacterium disguised as an arrow, 1mm tall, how tall is its image formed by the objective? Is the image erect or inverted? Is the image real or virtual? (4 points)

9 Physics 109 Final Exam 24-Jan-2002 Page 8 Problem 4. Continued (c) The eyepiece forms an image of the image. Where is this second image located? (3 points) (d) You could look at the image formed by the objective directly, but you probably can t get your eye closer to the image than about 20cm. What then is the angular magnification of the eyepiece? (4 points) (e) What is the overall magnification of this microscope (assuming that without the microscope you could get your eye no closer than 20cm to the evil bacterium)? (2 points) (f) Suppose you wanted to increase the magnification of this microscope by changing the length of the tube which separates the two lenses (but you d keep the same two lenses). Would you make the tube longer or shorter? Why? (3 points)

10 Physics 109 Final Exam 24-Jan-2002 Page 9 5. A grating has 200 slits per millimeter (so the slit separation is d = 5000nm). The grating is illuminated by blue, λ = 400 nm, light at normal incidence. The diffraction pattern appears on a very wide screen at a distance L = 10m from the grating. (a) What is the largest order which can appear on the screen (let the screen be as wide as you need to intercept the largest order!) (5 points) (b) Suppose the slit width is a = 1250nm. Some orders will be missing. List all the orders that actually appear on the screen. The grating is still illuminated by the blue light of part (a). (5 points)

11 Physics 109 Final Exam 24-Jan-2002 Page 10 Problem 5. Continued (c) Suppose the incident light contains two different wavelengths, λ 1 = 400nm and λ 2 = nm. Also suppose that the incident beam illuminates 1 cm of the grating. Can the two wavelengths be resolved by this setup? If so, what is the minimum order (of those that actually appear) in which the two wavelengths are resolved? (5 points) (d) Suppose the entire setup is placed in water with an index of refraction n = The illuminating light still has a wavelength in air of λ air = 400nm. Does the number of orders on the screen increase, stay the same, or decrease? Why? (5 points)

12 Physics 109 Final Exam 24-Jan-2002 Page Positively charged kaons (K + ) at rest have a mean lifetime τ 0 = s before they decay, usually into a muon and a neutrino. Kaons can be produced by collisions generated at particle accelerators. Suppose a beam of kaons travelling at v = 0.99c is produced in an accelerator experiment. (a) What is the mean lifetime of the kaons in the beam as observed by an observer at rest? (4 points) (b) How far do the kaons travel (assuming they don t decay first) in the mean lifetime you calculated above as seen by an observer at rest. (2 points) (c) In the rest frame of the kaons, how far does the observer of parts (a) and (b) appear to travel in one rest frame lifetime? Is there an inconsistency with your answer to part (b)? Why or why not? (4 points)

13 Physics 109 Final Exam 24-Jan-2002 Page As a model for an inner electron in a heavy element like lead or uranium, we can take an electron to be confined in a very small three-dimensional, cubical box, of side L. That is, the potential energy is 0 inside the box and infinite outside the box. The energy states in the box then depend on 3 quantum numbers n 1, n 2, and n 3. (a) Obtain an expression (don t plug in numbers) for the energy of the electron as a function of the quantum numbers, L, and whatever other physical constants you may need. Assume the electron is non-relativistic. (5 points) (b) What are the quantum numbers of the ground state and the first excited state? Note that there may be several sets of quantum numbers that correspond to the first excited state (that is, the state may be degenerate). Be sure to list all the sets of quantum numbers that apply! (4 points)

14 Physics 109 Final Exam 24-Jan-2002 Page 13 Problem 7. Continued (c) The electron is in the first excited state and makes a transition to the ground state by emitting a photon. If the energy of the photon is 100keV = 100,000eV, what is the size, L, of the box? (Here, a number, with units, is desired.) (3 points) (d) What is the wavelength of the 100keV photon? (3 points) (e) The photon scatters straight backwards from a free electron at rest. What is the energy of the scattered photon? (5 points)

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