PHYSICS 113: Contemporary Physics Final Exam Solution Key (2016)

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1 PHYSICS 113: Contemporary Physics Final Exam Solution Key (2016) 1. [25 points] (5 points each) Short Answers (a) The central reaction that governs the weak nuclear reactions of the sun reduces to: 4 p 4 2 He + X where 4 2 He is a helium nucleus and X represents 1 or more particles. Using conservation of charge, lepton number, and baryon number, identify the missing particle(s). Baryon number is already conserved, but charge isn t. We need two extra positive charges on the right. The culprits are: X = 2e + + 2ν e where the neutrinos are required to counterbalance (b) A particular system has an energy diagram as given below. At points A,B, and C, please give the direction of the force on a particle. If the point is an equilibrium, please say whether it is stable or unstable. A: No force. Stable equilibrium B: Force to the left C: No force. Unstable equilibrium (c) The dwarf planet, Eris, has a semi-major axis of about 64 AU (64 times the distance from the earth to the sun.) If the earth has an orbital speed of 30 km/s, what is the orbital speed of Eris? 1

2 Remember: v 1 R Eris s orbital velocity is 1/8 that of earth s and thus: v c = 3.75 km/s (d) A 5kg ball moves at 9m/s and elastically collides head-on with a 10kg ball initially at rest. What are the velocities of the two balls after the collision? Noting: m 1 m 2 = 1 m 1 + m 2 3 ; 2m 1 = 2 m 1 + m 2 3 v 1f = 3 m/s v 2f = 6 m/s (e) A bicycle wheel has a radius of 0.2m and a mass of 1.2kg. A cyclist is riding at 5 m/s. What is the total kinetic energy of each wheel? First, note the moment of inertia: and the rotational velocity: I = MR 2 = kg m 2 ω = v R = 25 s 1 K rot = 1 2 Iω2 = 15 J but there s al a translational energy: the total kinetic energy is: K t = 1 2 mv2 = 15 J K = 30 J 2. [20 points] A 1.8 kg bob hangs from a 0.4 m massless rod. At me early time, a small 0.2 kg blob of putty is shot horizontally toward the bob at 8m/s. The blob strikes the bob and the two stick to one another, as shown. 2

3 (a) How fast does the combined mass move to the right after the two strike one another? You may want to think about what is conserved in this collision. First, note that the speed is non-relativistic. The initial momentum is: p i = mv i = 0.2 kg8 m/s = 1.6 kg m/s. In the final collision, the total mass is M = 2 kg. Thus, the final speed is: v f = p i M = 1.6 kg m/s 2 kg = 0.8 m/s (b) How much kinetic energy is lost to heat in the inelastic collision? You may want to compute the kinetic energy of the blob beforehand and the bob+blob afterwards. The initial kinetic energy is: K i = 1 2 mv2 i = 1 2 (0.2 kg)(8 m/s)2 = 6.4 J and the final kinetic energy is: K f = 1 2 Mv2 f = 1 2 (2 kg)(0.8 m/s)2 =.64 J Q = K f K i = 5.76 J Though I m not terribly concerned about the sign of your lution. 3

4 (c) What vertical height does the pendulum bob+blob on the upswing? The final kinetic energy is equal to the final potential energy on the upswing. So: 1 2 Mv2 f = Mgy y = v2 f (0.8 m/s)2 = 2g 2 10 m/s 2 = m (d) E.C. (2 points) What angle does it reach on the upswing? The vertical height (as we derived in class) can be expressed as: y = L(1 cos θ) cos θ = 0.92 θ rad (e) The bob+blob then oscillates back and forth. What is the angular frequency of oscillation? The angular frequency of oscillations is: ω = g 10m/s l = 2 0.4m = 5 s 1 4

5 3. [20 points] A 10kg crate is released from rests on an inclined plane, with an inclination to the horizontal, θ = 30, as shown. There is a coefficient of static friction between the crate and plane of 0.2. (a) Draw a free-body diagram for the crate. Please note that I have al included the induced force in this diagram which won t show up until part d. (b) What is the Normal force on the crate? This diagram will be useful for the rest of the lution: 5

6 As drawn, the vertical forces are: F N mg cos θ F N = 10 kg10 m/s 2 cos 30 = 86.6N (c) What is the net force down the plane? (Ignore the F on the diagram.) The forces down the plane are: F d = mg sin θ F N µ s = 50 NN 86.6 NN 0.2 = 32.7 N (d) In order to keep the crate from sliding, you apply a horizontal force, F. What is the minimum force you need to exert to keep the crate from sliding? Express your answer algebraically in terms of µ s, m, g, θ. Hint: This is a little trickier than it might appear, since the force will enter in multiple ways. If you re pressed for time, you may want to skip and return to this. Including the induced force, the net downward force is: F d = mg sin θ µ(f sin θ + mg cos θ) F cos θ = 0 or F (cos θ + µ sin θ) = mg sin θ mgµ cos θ sin θ µ cos θ F = mg cos θ + µ sin θ 4. [20 points] Consider a spaceship moving at 0.6c to the right, and 100 nls in length. It has a mass of 10 5 kg. 6

7 (a) What is the total energy in the the ship? First, we need to compute the γ factor: γ = 1 1 v2 /c = 1 = Al: E rest = mc 2 = 10 5 kg m 2 /s 2 = kg E = mc 2 γ = J (b) At me time, t = 0 (event A ) a light-beam is shot from the back of the ship to the front. It then bounces off the front (event B ) and comes to the back of the ship. Draw a spacetime diagram for those 3 events as seen within the ship. Both sets of diagrams, but only the left one is for this part (and only events A& B are for the next part). 7

8 (c) Draw a spacetime diagram for events A and B only as seen outside the ship. Be sure to label your axes. According to the folks outside the ship, how fast does the light-beam appear to travel toward the front of the ship? See the diagrams above. In the unprimed frame: x = 100 nls but in the primed frame: t = 100 ns and In the same frame: x = γ x + vγ t = 200 nls t = γ t + v γ x = 200 ns c2 u = x t = 200 nls 200 ns = c You could have al noted that the constant speed of light is one of Einstein s postulates and gotten full credit. (d) The time interval as seen between A and C is essentially the tick of a clock. How long does that tick take to meone outside of the ship? It s dilated by a factor of γ (see the diagram above). So instead of 200ns, it takes t = 250 ns Remember, x AC = 0. 8

9 5. [15 points] A particular quantum system has 4 possible energy states: 1) 0.4eV, 2) 1.2eV, 3) 2.0eV, 4) 2.8eV. (a) Please list the energy of all possible emitted photons from this system. The energies are degenerate since the difference between adjacent levels are all 0.8eV. Thus, the possible transitions are: 0.8 ev, 1.6 ev, 2.4 ev (b) For the n = 4 to n = 1 transition, what is the wavelength of the emitted photon? What waveband (e.g. IR, Vis, UV) does this correspond to? If it s visible, what s the approximate color? The wavelength is simply: λ = hc E = ( J s)( m/s) 2.4 ev = 517 nm This is toward the green end of the visible spectrum. ev J (c) Photons have a wavelength even though they are particles. Give at least 1 example of wavelike behavior that photons exhibit. I am looking for various interference phenomenon. In class, we focused on the double slit experiment and the many fringes that can be seen because the photons interference with one another. I will accept other correct phenomena as well. (d) E.C. (2 points) Based on the energy distribution, what type of quantum system might this be? As it follows the pattern: E n = hω(n + 1/2) (where hω = 0.8 ev), it s an oscillator. That means it might be a diatomic molecule. 9