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Elementary Qualifier Examination October 10, 2005 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 e. V·sec hc = 1240 e. V·nm Stefan-Boltzmann constant = 5. 670 10 -8 J·K-4 m-2 s-1 Speed of light, c = 3. 00 10 8 m/sec Permeability, 0 = 4 10 -7 T m/A Gas constant R = 8. 3144 J /(mol K) ag = 9. 8 m/sec 2 at the earth’s surface me = 9. 109 10 -31 kg = 0. 511 Me. V/c 2 m = 1. 883 10 -28 kg = 105. 6 Me. V/c 2 m 0 = 2. 407 10 -28 kg = 135. 0 Me. V/c 2 m. K 0 = 8. 872 10 -28 kg = 497. 7 Me. V/c 2 Atomic weight N = 14. 00674 u 1 u = 1. 660 10 -27 kg = 931. 5 Me. V/c 2 Inductance Harmonic Oscillator Relativistic kinematics E = g mo c 2 E 2 = p 2 c 2+mo 2 c 4 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
Problem 1 Name code A bucket half-filled with water (total mass 10 kg) is swung smoothly in a circular path of radius 0. 8 meters at a uniform rate of 8 revolutions per 10 seconds. a. Calculate and compare the amount of force the person holds the bucket with just to keep it moving through the very top and bottom positions along this circular path. b. What minimum uniform rotational speed is necessary to keep the water from falling from the bucket? With what force must the person hold the bucket then at the top and bottom positions?
A point mass with mass m is initially held at rest on the roof of a house at the position shown (see figure below). When released it slides down the roof and then falls landing on the ground to the right of the house. Neglect any friction. a. b. c. d. Problem 2 Name code What is the speed of the point mass when it leaves the roof? What is its speed when it hits the ground? How much time does the point mass spend on the roof, until it falls off the roof? How far to the right of the right wall will it hit the ground? point mass roof left wall 5. 5 m high right wall 3. 5 m high ground house 6. 5 m wide
Problem 2 continued Name code
Problem Imagine a straight tunnel from the North to the South Pole, passing right through the center of the Earth (assumed to be a perfect ball of radius R, mass M and uniform mass Name code density, ). From the South Pole you drop a brick of mass m down the tunnel, releasing it from rest at time t=0. Ignore frictional forces. 3 a. Find the magnitude of the Earth’s gravitational acceleration in the tunnel as a function of distance from the center of the Earth. Consider both distances less than and greater than the radius of the earth. b. Find where the gravitational acceleration is largest and calculate its value there. c. Referring to the form of your result from part a. , describe the resulting motion of the brick. d. How much time after you dropped it will the brick return to where you dropped it?
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Problem 4 38. 6 grams of N 2 (a real gas that behaves like an ideal gas) is allowed to expand reversibly, and isothermally. The temperature during expansion is 425 K and the volume changes from 14 to 27 liters. a. How much work is produced during the expansion? b. How much heat is produced during the expansion? Name code
I Problem 5 Name code b P a The wire semicircles shown above have radii a and b. Calculate the net magnetic field (give both the magnitude and direction) that the current in the wires produces at point P. Assume the current is positive in the direction indicated.
Problem 6 b Name code a A small solid conductor with radius a is centered along the axis of a thin-walled hollow tube with inner radius b ( b > a ). The inner and outer conductors carry equal currents i in opposite directions. a. Use Ampere’s law to find the magnetic field at any point in the volume between the conductors. b. Write the expression for the flux d B through a narrow strip of length l parallel to the axis, of width dr, at a distance r from the axis of the cable and lying in a plane containing the axis (see illustration at right). dr b r c. Integrate your expression from part b over the volume between the two conductors to find the total flux produced by a current i in the central conductor. d. Show that the inductance of a length l of the cable is . e. Use to calculate the energy stored in the magnetic field for a length l of the cable. a
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Problem 7 Name code In the circuit at left, E = 10 V, R 1 = 5. 0 W, R 2 = 10 W, and L = 5. 0 H. Initially, there are no currents. The switch S is closed at time t = 0. Calculate the values of the following quantities immediately after the switch was closed: a. the current i 1 through R 1 b. the current i 2 through R 2 c. the current i d. the potential difference across R 2 e. the potential difference across L f. the rate of change di 2 / dt Calculate the same quantities, but at a very, very long time after the switch was closed: g. the current i 1 through R 1 h. the current i 2 through R 2 i. the current i j. the potential difference across R 2 k. the potential difference across L l. the rate of change di 2 / dt
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E(t) = Em sin( d t) standard wall outlet Problem 8 Name code R A simple light dimmer. The filament in the light bulb has resistance R. The inductance L of the inductor can be adjusted, as is indicated by the arrow through the symbol. The system is powered from a standard wall outlet. L (adjustable) A simple way to dim a light bulb is to connect an adjustable inductor in series to it as shown in the figure. A plug connects this circuit to a standard wall outlet (fd = 60 Hz, Em, RMS = 120 V, Em = 170 V). The light bulb is rated at 100 watt. a. Calculate the resistance of the light bulb. b. Draw a phasor diagram including the following four phasors: I, the total current through the prongs of the plug; Em, the emf at the wall outlet; VL, the voltage across the inductor; and VR, the voltage across the light bulb’s resistance. Carefully label each phasor. c. Find the impedance Z of the LR circuit. d. To make the light bulb burn at maximum power (100 W), should the inductance L be made very large or very small? Why? e. What inductance should the inductor be given to make the light bulb burn at 30 W?
Problem 8 continued Name code
Binoculars and microscopes are frequently made with coated optics. A thin layer of transparent material is added to the lens surface as shown below right. Problem 9 Name code Multiple choice: For the first two questions select the best answer from the options that follow. 1 The desired condition is for a. constructive interference between light rays 1 and 2. b. the coating be more transparent than the lens. c. destructive interference between rays 3 and 4. d. the speed of light in the coating to be less than in the lens. e. destructive interference between rays 1 and 2. A necessary condition is for a. nair < n 1 < n 2. b. nair < n 2 < n 1. c. n 1 < nair < n 2. d. nair < n 1 = n 2. Air nair Mg. F 2 n 1 Glass n 2 Show your work in calculating the following: If denotes the central value of the wavelengths of incident light (in air), and the coating has a refractive index of n 1 and the lens n 2, what is the thinnest possible the coating should be? 3 4 2
A neutral kaon decays, at rest, to a pair of neutral pions: Problem 10 Name code a. Find the momentum (in units of Me. V/c) of each pion. b. Find the speed of each pion. c. The mean lifetime of a pion at rest is 8. 4 10 -17 seconds. What mean path length would the pions coming from this kaon decay be expected to travel in the lab frame?
Experimentally the equilibrium spacing between nuclei in the HCl molecule is found to be 1. 27 Å (for the Cl 35 isotope) and the force constant approximating its molecular Name code bond is measured to be 513 N/m. Problem 11 a. What is its ground state vibrational energy (in e. V) ? b. What is the separation of its vibrational energy levels ? c. Transitions between vibrational levels result in the emission of what frequency of light? Approximately where in the electromagnetic spectrum ( R IR visible UV) does this lie?
The figure below right illustrates Compton scattering of a photon off an electron. Using the labeled quantities in the figure Problem 12 Name code a. write an expression for the momentum of the scattered electron (in terms of the photon momenta). p 1 b. stating conservation of energy for this system of photon and electron. me p 2 c. Starting from a. above, find an expressions for pe 2 (simplifying so that vector quantities are expressed only with their amplitudes, i. e. . continued
Problem 12 cont’d Name code d. 0. 30 Me. V X-rays are Compton scattered in the backward direction. What is the kinetic energy of the recoil electrons?