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1 Physics 123, Section 2 (majors/minors), Fall Semester 2014 Instructor: Mark Transtrum Final Exam Dec MC EC Total Name points Class ID # This is the final exam and consists of 20 multiple choice problem and 8 free response problems, each with multiple parts. You may use your textbook, previous homework assignments, or any of the course materials on the class website to help you complete this test. You may NOT use any other internet resources or get help from other members of the class or TAs. If you need clarification about something, contact Dr. Transtrum directly. When you are finished, sign the statement below and submit your solutions to the usual homework boxes before the building closes on December 17. GOOD LUCK! I affirm that the solutions on the following pages of this exam are my own work and that I did not use any unauthorized resources while taking this exam. Signature: CID:

2 Multiple-choice questions 1. Boron trifluoride is a flat, triangular molecule. The fluorine atoms are at the corners of the equilateral triangle and the boron atom is at the center of the triangle. Assume that the bonds are very stiff (like in a typical diatomic gas). At room temperature, how many degrees of freedom will this gas have (per molecule)? (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, (f) 6, (g) 7, (h) 8, (i) 9, (j) 0 2. If the temperature of a gas is doubled, the average speed of the molecules increases by a factor of (a) 2 (b) 2 (c) 3/2 (d) 2/3 (e) 3 (f) 4 3. An engine has an efficiency of e. When the same cycle is run in reverse as a refrigerator, the COP is (a) e (b) 1/e (c) 1/e 1 (d) 1/e + 1 (e) 1 1/e 4. An ideal gas is allowed to undergo a free expansion. If the initial volume if V 1 and its final volume is V 2, the change in entropy is (a) nr ln(v 2 /V 1 ) (b) nrt ln(v 2 /V 1 ) (c) nk ln(v 2 /V 1 ) (d) 0 (e) nrv 2 /V 1 5. As you speed down I-15 at 90 mph, a police car pursues you at 110 mph. You observe a frequency from the siren of 500 Hz. What is the apparent wavelength of the sound waves from the police siren as observed by you, assuming you could somehow observe the wavelength? (1 mph = ft/s and the speed of sound in air is 767 mph) (a) 2.25 (b) 2.32 ft (c) 2.18 ft (d) 1.99 ft (e) 2.71 ft 6. The sound of a trumpet playing a note is qualitatively different than the sound of a flute playing the same note. Why is that? (a) The two notes have different amplitudes. (b) The two notes have different durations. (c) The two notes have different fundamental frequencies. (d) The two notes have different phases. (e) The two notes have different strengths of harmonics. 7. To correct near sightedness, you need to choose a lens that will create a image at of an object at. (a) real;, the near point; the far ponit (b) real; the far point; infinity (c) real; infinity;, the far point (d) real; infinity; 25 cm (e) virtual; the near point; 25 cm (f) virtual; the far point; infinity (g) virtual; infinity; the far point (h) virtual; infinity; 25 cm 8. Two waves are described by y 1 = 3 sin(4x 700t)

3 y 2 = 3 sin(4x 700t 2). What is the amplitude of the superposition of these two waves? (a) 8.0 (b) 4.3 (c) 6.0 (d) 3.2 (e) Which of the following best describes the image formed by a diverging lens when the object is at a distance of twice the focal length of the lens? (Assume a positive object distance.) (a) real, inverted, and enlarged (b) real, inverted, and reduced (c) real, upright, and enlarged (d) real, upright, and reduced (e) virtual, inverted, and enlarged (f) virtual, inverted, and reduced (g) virtual, upright, and enlarged (h) virtual, upright, and reduced 10. Imagine an empty cavity inside a thick plate of glass. Viewed from outside the glass (in air), the cavity will appear: (a) deeper in the glass than it really is (b) at the same depth as it actually is (c) closer to the surface (air/glass interface) than it really is (d) it will not appear any differently because the cavity is empty (n = 1 inside the cavity) 11. Upon moving a mirror in an Michelson interferometer a distance x, you see a shift of m fringes. What is the wavelenth of the light? (A fringe is defined as from dark to dark or from bright to bright) (a) x/2m (b) x/m (c) 2x/m (d) m/x (e) m/2x 12. Two flat mirrors are at a 60 degree angle to one another. An light ray is incident on the first mirror such that it is parallel to the second mirror. What is the angle of reflection from the second surface (measured with respect to the normal of the second surface)? (a) 30 (b) 60 (c) 45 (d) 53 (e) At what angle is the sun above the horizon if its light is found to be completely polarized when it is reflected from the surface of a still lake (n = 1.33)? (a) θ < 10, (b) 10 < θ 20, (c) 20 < θ 30, (d) 30 < θ 40, (e) 40 < θ 50, (f) 50 < θ 60, (g) 60 < θ 70, (h) 70 < θ 80, (i) 80 < θ While driving your car at 0.5c (relative to the road) you see another car going in the opposite direction at 0.3c (relative to the road). In your reference frame, how fast does the second car appear to be going? (a) 0.7c (b) 0.8c (c) 0.2c (d) 0.24c (e) None of the above 15. An astronaut traveling with a speed 0.9 c holds a meterstick in his hand. If he measures its length, he will obtain a value of (a) 1 m (b) 2.3 m (c) 0.19 m (d) 0.43 m (e) 0.81 m 16. A pole vaulter runs with a pole 10 ft long at a speed of 10 m/s. How much shorter (in feet) does the pole appear to a spectator? (Hint: 1 x 1 x/2) (a) (b) (c) (d) (e) 10 17

4 17. The half-life of a muon is 2.2µ s. How fast must it be moving as a fraction of the c if its half-life doubles? (a) 0.87 (b) 0.75 (c) 0.97 (d) 0.72 (e) Assume a gram of a substance marketed as Pure Energy is annihilated by a gram of a second substance Anti-Pure Energy. For how many hours would the energy released power a city which uses 10 9 Watts of power? (a) 40 (b) 25 (c) 50 (d) 65 (e) 50, An electron has a kinetic energy that is twice its rest energy. What is its speed as a fraction of c? (a) 0.76 (b) 0.81 (c) 0.94 (d) 0.54 (e) A 30-year-old woman takes a trip on a rocket, leaving her 20-year-old brother behind. She travels at a speed of 0.8 c and is gone 20 years according to the younger brother. When she returns, how many years older/younger is she than her brother? (a) 2 years younger (b) 2 years older (c) 3 years older (d) 10 years older (e) 8 years older

5 Written questions 1. (12 points) Give short answers/expalanations to the following questions: (a) Why is there a limit to how efficient a heat engine can be? (Don t make a statement like Because of the 2nd law of thermodynamics. Explain what physically prevents you from constructing a more efficient engine.) (b) In the context of waves, what are beats? Give an example of beats from everyday life. (c) What is a quantum mechanical particle? Is it really a particle? Or is it a wave?

6 (d) In the context of special relativity, what is an event? What is a reference frame? Give an example of both. (e) What is the principle of Galilean relativity and how is it at odds with the postulates of special relativity? (f) In class we said that time dilation and length contraction are two sides of the same coin. What did we mean by this? Explain how this works with an example.

7 2. (10 points) A tank of cross-sectional area A is filled with water. A tightly fitting piston, with a total mass M rests on top of the water in the figure. A small hole of area a (a A) is opened at a depth of h below the water level of the tank. M h (a) What is the speed at which water comes out the hole? (b) How fast does the piston initially move downward? That is, find dh/dt.

8 (c) Extra Credit: If A = m 2, M = 10 kg, a = 1 cm 2. How long does it take the piston to reach the height of the hole if was initially 60 cm above? Assume the density of water is ρ = 1000 kg/m 3.

9 3. (13 points) A sample of a monatomic ideal gas goes through a 3-step cycle. Initially the gas has a volume of 0.2 m 3 and a pressure of 400 kpa (point A in a P-V diagram). In the first step, the gas expands isothermally to 0.8 m 3 (point B in a P-V diagram). Then, it is compressed isobarically to a point C in a P-V diagram. Finally it is further compressed adiabatically to the original volume and pressure. (a) Draw a picture of the P-V diagram, marking the points A, B, and C. Make sure the relative slopes of your processes are correct. (b) What is the difference of internal energies ( E int ) between points A and B? Between points B and C?

10 (c) What is the entropy change for a cycle? (d) How much work is done in this cycle? (e) If this cycle is used as a heat engine, what is its efficiency?

11 (8 points) Two snapshots of a one-dimensional wave traveling to the right are shown to the right. The solid line 4. occurred at t = 0 seconds. The dotted line occurred at t = 3 seconds. Between these snapshots, exactly one trough passed through the origin (a) Write down a complex function whose real part represents this wave. (b) Now, instead of thinking of the two figures as snapshots in time of a single spatial wave, imagine that they represent two oscillations in time (that is, the horizontal axis now represents time, not space). Using complex numbers, find a single periodic function in time whose real part represents the superposition of these two oscillations. (Your answer should be an expression of time only, not space.)

12 5. (7 points) Two speakers emitting 651 Hz are separated by a distance of 2.0 m. A student is positioned just in front of the first speaker. As she walks away from the first speaker along a path 90 from the line that joins the two speakers, she notices minima in the sound level. Use 343 m/s for the speed of sound. (a) How far is she from the speaker for the first minimum she encounters? (b) How many minima could she experience if she walks far away?

13 6. (7 points) Coherent monochromatic light with a wavelength λ passes through three parallel slits spaced evenly from each other with a distance d. What is the intensity of the resulting interference pattern at a height y on a screen a distance L from the slits?

14 7. (10 points) Consider two lenses with focal lengths f and f placed right next to each other so that the distance between them is zero. (a) This setup is equivalent to that of a single lens with an effective focal length f eff satisfying 1 p + 1 q = 1, f eff where p is the position of the initial object and q is the position of the final image. Find an expression for f eff in terms of f and f. (Hint: Consider the thin lens equation for each lens and use the fact that the distance between the lenses is zero implies that the image distance from the first lens is the negative object distance for the second lens.)

15 (b) Consider the specific case where f = 10 cm and f = 5 cm and an object is placed at p = 15 cm. Draw the ray diagrams for each lens. Graphically check that your final image appears where the formula you derived in part (a) predicted. (c) What happens if f = f?

16 8. 13 points A photon collides with an electron that is at rest. The photon imparts momentum to the electron, and a new photon recoils in the opposite direction (exactly backward relative to the incoming photon). Show that the momentum of the recoiling photon p photon is related to the momentum of the incident photon p photon by the equation 1 p photon 1 p photon = 2 m e c where m e is the mass of the electron. This equation is known as the Compton shift. (b) Sketch the wordlines for the electron and the photons in the reference frame in which the electron is initially at rest. Let the origin correspond to the collision event. What do the worldlines look like in the frame in which the electron is at rest after the collision?

17 9. (Extra Credit) The continuity equation from fluid dynamics A 1 v 1 = A 2 v 2 is an equation that is not invariant under Galilean relativity. One way to see this is by transforming into the frame in which v 1 0 and v 2 v 2 v 1 so the left hand side of the equation is zero but the right hand side is not. What is the special reference frame in which this equation is valid?