Elementary Qualifier Examination - October 10, 2005

1. Elementary Qualifier Examination October 10, 2005 NAME CODE: [ ] • Instructions: • Do any ten (10) of the twelve (12) problems on the following pages. • Indicate on this page (below right) which 10 problems you wish to have graded. • If you need more space for any given problem, write on the back of that problem’s page. • Mark your name code on all pages. • Be sure to show your work and explain what you are doing. • A table of integrals is available from the proctor. Possibly useful information: Planck constant ħ = h/(2) = 6.583  10-16 eV·sec Stefan-Boltzmann constant  = 5.670  10-8 J·K-4m-2s-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/sec2 at the earth’s surface me = 9.109  10-31 kg = 0.511 MeV/c2 m = 1.883  10-28 kg = 105.6 MeV/c2 m0 = 2.407  10-28 kg = 135.0 MeV/c2 mK0 = 8.872  10-28 kg = 497.7 MeV/c2 Atomic weight N = 14.00674 u 1u = 1.660 10-27 kg = 931.5 MeV/c2 Inductance Harmonic Oscillator Relativistic kinematics E = g moc2 E2 = p2c2+mo2c4 hc = 1240 eV·nm Check the boxes below for the 10 problems you want graded Problem Number Score 1 2 3 4 5 6 7 8 9 10 11 12 Total

2. 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?

3. point mass roof left wall 5.5 m high right wall 3.5 m high ground house 6.5 m wide Problem 2 • A point mass with massm 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. • 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? Name code

4. Problem 2 continued Name code

5. Problem 3 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 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. 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? Name code

6. Problem 3 continued Name code

7. Problem4 • 38.6 grams of N2 (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. • How much work is produced during the expansion? • How much heat is produced during the expansion? Name code

8. I Problem 5 Name code b a P The wire semicircles shown above have radii a andb. 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.

9. 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 iin opposite directions. • Use Ampere’s law to find the magnetic field at any point in the volume between • the conductors. • 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). • 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. • Show that the inductance of a length l of the cable is . • Use • to calculate the energy stored in the magnetic field for a length l of the cable. dr b r a

10. Problem 6 continued Name code

11. Problem 7 Name code In the circuit at left, E = 10 V, R1 = 5.0 W, R2 = 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: • the current i1 through R1 • the current i2 through R2 • the current i • the potential difference across R2 • the potential difference across L • the rate of change di2 /dt • Calculate the same quantities, but at a very, very long time after the switch was closed: • the current i1 through R1 • the current i2 through R2 • the current i • the potential difference across R2 • the potential difference across L • the rate of change di2 /dt

12. Problem 7 continued Name code

13. Problem 8 E(t) =Em sin(d t) standardwall outlet 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?

14. Problem 8 continued Name code

15. 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. Multiple choice: For the first two questions select the best answer from the options that follow. Problem 9 Name code 1 2 • 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 • nair < n1 < n2. • nair < n2 < n1. • n1 < nair < n2. • nair < n1 = n2. • 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 n1 and the lens n2, what is the thinnest • possible the coating should be? Air nair MgF2n1 Glass n2 3 4

16. Problem 10 A neutral kaon decays, at rest, to a pair of neutral pions: a. Find the momentum (in units of MeV/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? Name code

17. Problem 11 Experimentally the equilibrium spacing between nuclei in the HClmolecule is found to be 1.27 Å (for the Cl35 isotope) and the force constant approximating its molecular bond is measured to be 513 N/m. a. What is its ground state vibrational energy (in eV) ? 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 (mR IR visible UV) does this lie? Name code

18. Problem 12 • The figure below right illustrates Compton scattering of a photon • off an electron. Using the labeled quantities in the figure • write an expression for the momentum of the scattered electron • (in terms of the photon momenta). • b. stating conservation of energy for this system of photon and electron. • Starting from a. above, find an expressions for pe2 (simplifying so that vector • quantities are expressed only with their amplitudes, i.e. . Name code me  p1  p2 continued

19. Problem 12 cont’d Name code d. 0.30 MeV X-rays are Compton scattered in the backward direction. What is the kinetic energy of the recoil electrons?