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Transcript
Elementary Qualifier Examination October 9, 2006 NAME CODE: [ ] Instructions: (a) Do any ten (10) of the twelve (12) problems on the following pages. (b) Indicate on this page (below right) which 10 problems you wish to have graded. (c) If you need more space for any given problem, write on the back of that problem’s page. (d) Mark your name code on all pages. (e) Be sure to show your work and explain what you are doing. (f) A table of integrals is available from the proctor. Possibly useful information: Planck constant ħ = h/(2) = 6.583 10-16 eV·sec = 1.05457 10-34 J·sec 1 eV = 1.602189 10-19 J Gauss’ Law: E dA q / Ampere’s Law: 0 Biot-Savart Law: dB 0 Ids rˆ 4 r 2 Speed of light, c = 3.00 10 8 m/sec Permeability, 0 = 410-7 Tm/A q of electron e = 1.6010-19 C me = 9.109 10-31 kg = 0.511 MeV/c2 m = 1.883 10-28 kg = 105.6 MeV/c2 m0 = 2.407 10-28 kg = 135.0 MeV/c2 Atomic weight N = 14.00674 u 1u = 1.660 10-27 kg = 931.5 MeV/c2 E di / dt Harmonic Oscillator k Inductance L - T 2 Relativistic kinematics E = g moc2 E2 = p2c2 + mo2c4 1 g v2 1- 2 c B ds 0 I Check the boxes below for the 10 problems you want graded Problem Number 1 2 3 4 5 6 7 8 9 10 11 12 Total Score m1 T1 m2 T2 m3 Problem 1 T3 Name code As shown above, three connected blocks are pulled to the right on a horizontal frictionless table by a force of magnitude T3 = 65.0 N. If m1 = 12.0 kg, m2 = 24 kg, and m3 = 31 kg, calculate: a. the magnitude of the system’s acceleration b. the tension, T1 c. the tension, T2 A hollow sphere of mass, M and outer radius a, and with an inner concentric cavity of radius b is shown in cross section below right. Problem 2 Name code a. Sketch a curve of the magnitude of the gravitational force F from the sphere on a mass m located a distance r from the center of the sphere, as a function of r over the range 0 r . Show any formulas used in determining the curve you draw. Consider r = 0, b, a, and in particular. M r m b a F b a r continued Problem 2 continued Name code b. Sketch the corresponding curve for the potential energy U(r) Show any formulas used in determining the curve you draw. Consider r = 0, b, a, and in particular. U(r) b a r k k m Problem 3 Name code Two springs are joined and connected to a block of mass m = 0.245 kg (see the figure above) that is set oscillating over a frictionless floor. The springs each have spring constant k =6430 N/m. a. Find an expression giving the total force acting on the mass when the two springs are elongated to give a total elongation x = x1 + x2. From this derive the effective spring constant of this system of two springs. b. What is the frequency of the oscillations? Problem4 1.00 mole of an ideal monatomic gas is taken through the cycle shown in the graph at right. Assume that p = 2p0, V = 2V0, p0 = 1.01 105 Pa, V0 = 0.0225 m3. b Pressure Calculate: (a) the work done during the cycle, and (b) the energy added as heat during the stroke abc. Name code c V0,p0 a Volume V,p d A B + + + + + + + + + + + + L Problem 5 a P Name code In the figure above a very thin nonconducting rod of length L carries a uniform linear positive charge density. The total charge of the rod is q. The left end of the rod is at position A and its right end at position B. Take the potential to be zero at infinity, and consider a point P on the line containing the rod, as indicated in the Figure. The distance between P and B is a. a. Which direction has the electric field at point P? Carefully explain your answer. b. Calculate the magnitude E of the electric field at point P in terms of L, q, and a. c. Calculate the electric potential VP at point P in terms of L, q, and a. continued A Problem 5 continued B + + + + + + + + + + + + L a P Name code For parts d. and e. assume the rod is L = 30 cm long and that point P is at a distance a = 30 cm from the rod. (You should not have used these values in parts a., b., and c.) d. Imagine a point charge with negative charge -q (i.e., equal in magnitude to the total charge of the rod). Where on the rod would we have to position this point charge to make the potential at point P zero? Is this position closer to A, closer to B, or exactly midway between A and B? Explain. e. Where would the point charge have to be positioned to make the electric field at point P zero? Is this position closer to A, closer to B, or exactly midway between A and B? Explain. In a simple model of the hydrogen atom (H), the electron orbits around the stationary proton in a circular orbit of radius a0 0.53 Å Problem 6 Name code a. Assuming the only force acting on the electron is the electrostatic force due to the stationary proton, show that the orbital period of the electron (the time it needs to revolve around the proton once) equals 1.5 10-16 s. b. Based on part a., calculate the time-averaged current i in the electron’s circular orbit. c. Calculate the magnitude B of the magnetic field at the location of the proton that is produced by this time-averaged current of the electron. Note: if you were unable to find the answer for part b., you may assume in this part c. that the current is i 3.0 mA (which is not the real answer for part b.). Problem 7 R = 47 W Name code E = 48 V + L= 37 mH S Circuit with DC battery, resistor and inductor. S is a switch. A solenoid with a self-inductance of L = 37 mH and a resistor with resistance R = 47 W are connected in series to a 48-volt DC battery as shown in the figure above. Initially, there is no current in the circuit. The switch S is closed at time t = 0. a. Calculate the current in the circuit immediately after the switch was closed. Explain. b. Calculate the current a very long time after the switch was closed. Explain. c. Calculate the amount of energy in the magnetic field of the solenoid a very long time after the switch was closed. d. Was there a time at which the potentials across the inductor and the resistor were equal? If yes, when was that? If no, explain. e. At what time was the power delivered from the battery to the solenoid greatest? What was this peak power? Problem 7 continued Name code plastic insulation (A) Problem 8 copper sheath dielectric Name code copper core (B) copper sheath copper core (A) Structure of a coaxial cable. (B) Coaxial cable as a cylindrical capacitor ra rb L dielectric The coaxial cable shown in Fig. (A) consists of a copper core (round wire), a dielectric insulator, and a cylindrical outer conductor (“copper sheath”), all coaxial with each other. For protection, the cable also has a plastic sheath. Figure (B) shows a piece of this cable of length L and with the plastic sheath removed. Consider this as a cylindrical capacitor: imagine there is a linear charge density + on the core and a linear charge density of - on the sheath, as indicated. These charges set up the radial field indicated by arrows in the gap between the two conductors. The core has a diameter of 0.812 mm. The dielectric has an outer diameter of 3.70 mm, and consists of cellular polyethylene (dielectric constant 2.25). Calculate the capacitance of this cable, in pF/ft (picofarad per foot). Note: 1 foot = 30.5 cm. Problem 8 continued Name code Problem 9 Two converging lenses, separated by 26 centimeters, bring to focus the image of an object set a distance of 24 cm before the 1st one as shown. The first lens has a focal length of 8-cm, the second lens 15-cm. Name code 26cm f1 f1 f2 f2 ___ The image produced by the first lens is a. real. b. virtual. ___ The image produced by the first lens is a. upright. b. inverted. ___ The final image seen through both lenses is a.upright & real. b.upright & virtual. c.inverted & real. d. inverted & virtual. What is the ratio (magnification factor) of the final image size to the object’s actual size? Show all work. A negative pion decays, at rest, to a muon and uncharged neutrino: - - Problem 10 Name code The muon and neutrino move relativistically. Assume the neutrino is massless. a. Find the momentum (in units of MeV/c) of the emitted muon. b. Find the speed of the emitted muon. c. Find the energy of the neutrino (in units of MeV). d. A pion is a spin-0 particle (it has no intrinsic angular momentum, but a muon is a fermion (spin-½ particle). What must the spin of the neutrino be? Below are energy shell levels of nucleons including the spin-orbit splitting of energy levels. Nucleons (neutrons and protons) are spin-½ fermions, so fill these energy levels just like electrons fill atomic orbitals. In exactly the same way, the letters s, p, d, f label each orbital’s angular momentum ( = 0, 1, 2, 3). Problem 11 Name code Fill in the boxes below, giving in the 2nd column the total angular momentum j for each level (see the “1s” and “1p”examples at bottom of the column). In the 3 rd column give the number of the degenerate states that fill each shell. 2p 1f 2p 2p 1f 1f 2s 1d Large energy gaps (see the grayed, dashed lines) show particularly stable configurations for nuclei when a nucleon total exactly matches one of several “magic numbers.” 2s 1d 1d This chart shows the 1st four magic numbers for nucleons are: 1p 1p1/2 1p3/2 1s 1s1/2 principal quantum level coupled with spin 2 Degeneracy (number of nucleons that fill each energy level) a. Express the rotational kinetic energy of a system classically in terms of its angular momentum L and its moment of inertia I. Problem 12 Name code b. Give the rotational energy EJ of a system quantum mechanically in terms of its moment of inertia I and total angular momentum quantum number J. From spectroscopic observations, the energy difference between the ground state and the second excited rotational state of the N2 molecule is known to be 14.9 10-4 eV. c. Deduce the moment of inertia of the molecule. d. Deduce the center- to-center distance between the N atoms.