1. Elementary Qualifier Examination January 13, 2003 NAME CODE: [ ] Instructions: (a)Do any ten (10) of the twelve (12) problems of 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: g = 9.8 m/sec 2 Planck constant h = 6.626  10 -34 J·sec = 4.136  10 -15 eV·sec Speed of light, c = 3.00  10 8 m/sec Permeability,  0 = 4  10  7 T  m/A Relativistic kinematics E =  m o c 2 E 2 = p 2 c 2 +m o 2 c 4 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. Name code Problem 1 Starting from rest, a small block of mass m slides down frictionlessly from the top of a quarter circle path of radius R cut into the corner of a large block of mass M. Initially at rest as well, the large block moves without friction across the table that supports it. Find the velocity of each immediately after the small block leaves the quarter circle path. M R m

3. Name code Problem 2 An amusement park ride consists if a vertical cylinder of radius R that rotates about its vertical axis. Riders stand initially on a floor which drops away once the ride starts and leaves them suspended, pressed against the wall. Given the final rotational speed of the cylinder, , derive the minimum coefficient of static friction to ensure the riders’ safety. 

4. Name code Problem 3 A 300-kg block hangs at the end of a 100-kg 10-meter long horizontal beam. A supporting cable is attached to the beam a distance ℓ out from the wall and anchored to that wall 10 meters above the beam.. A. Draw the free body diagram of the long bar. B. Find the value for ℓ such that the force exerted by the beam and block on the wall has no vertical component. 300-kg ℓ

5. Name code Problem 4 The pressure versus volume graph for 1 mole of an ideal monatomic gas undergoing a cyclic thermodynamic process. A.When a gas undergoes a sufficiently rapid change, it approaches the ideal adiabatic process. In such a process (circle all that apply) a.  T = 0.b.  W = 0. c.  Q = 0.d.  U = 0 B. Show that for the Carnot cycle graphed above a T1T1 PaPa b d T2T2 VaVa V c VbVb P PbPb PdPd PcPc a  bIsothermal b  cAdiabatic c  dIsothermal d  aAdiabatic VdVd VcVc

6. Name code Problem 5 a.Consider the circuit shown in the figure. Calculate the equivalent resistance between points A and B. b.Find the current through the 7  resistor. c.If this circuit is on for 24 hours, find the electrical energy used in the 6  resistor. Give your answer in kwh. E F C D A B 6v + - 44 55 33 66 22 77

7. Name code Problem 6 b.If an electron is released from rest at point P in the figure above, calculate the kinetic energy it gains by the time it hits one of the plates. Express your answer in eV. c. Which plate does the electron hit? (circle one) AB a. Consider the uniform electric field between the parallel plates A and B as shown in the figure at left. Calculate the potential difference V A  V B. E=120V/m P A B 5 cm

8. Name code Problem 7 a.Calculate its charge to mass (q/m) ratio. What is the sign of this charge? b.A proton is moving through this region parallel to the direction of the magnetic field (directed out of the page) at a velocity v = 7.5  10 4 m/sec. Calculate the magnitude and direction of the force on the proton. An unknown particle undergoes circular motion in the presence of a 0.5  10 -4 T magnetic field directed outward as shown in the figure at right. 5 cm v = 5  10 3 m/sec B out = 0.5  10 -4 T

9. Name code Problem 8 A copper wire (negligible resistance) is bent into a circular shape of radius 0.5 m. A gap separates the ends B and D. The radial section BC, fixed in place, has a resistance of 3 . A copper bar AC sweeps clockwise at an angular speed of 15 rad/sec. The loop lies in a uniform magnetic field of 3.8  10 -3 T. The field is perpendicular to the loop and points into the page. Find the magnitude and direction of the current induced in the loop ABC. A B C D

10. Name code Problem 9 A muon, created by a collision between a primary cosmic ray and an atmospheric nuclei at an altitude of 40 km, travels straight down with a speed ~0.99c. Given the mean lifetime for a muon at rest is 2.2  sec, find the probability that it survives to reach the earth’s surface classically (i.e., without the effects of time dilation) relativistically, taking into account its time dilation.

11. Name code Problem 10 A.The wave function of a hydrogen atom depends on the 3 quantum numbers n, ℓ, and m. Each is associated with a boundary condition on a 3 dimensional geometric variable. Identify that variable for each: n is associated with the boundary condition on ___. ℓ is associated with the boundary condition on ___. m is associated with the boundary condition on___. B.The energy level of a free hydrogen atom depends solely on a single quantum number, .  is actually which of the above? _______ C. Each quantized energy level E  depends on that quantum number in proportion to (circle one)a.  d. 1/  2 b.  2 e. e  c. 1/  f. e  D.The 1 st excited state, n=2 has what possible values of ℓ? and for each ℓ, what possible values of m? E.For one of these states, Sketch the probability distribution P (r) for this wave function. Solve for the normalization constant, . Find the most probable radius for an electron in this state. x, y, z r, , 

12. Name code Problem 11 Questions A-F: An observer peers into a highly reflective silvered, spherical Christmas ornament of radius R as shown below. E marks the ornament’s center. B and G are points on the surface. D, and F are halfway between the center and surface. ___ A. The observer sees an ___ image of himself. a. inverted, enlarged, and real b. upright, enlarged, and real c. upright, enlarged, and virtual d. upright, reduced, and virtual e. upright, reduced, and real f. inverted, reduced, and virtual ___ B. The observer’s eye is approximately a distance R from the surface of the ornament. An image of his eye forms approximately at point a. A b. B c. C d. D e. E f. F C. Verify your answer by solving for the image’s location algebraically. D. Graphically confirm your answer by drawing a neat ray diagram in the figure above showing the formation of an image of any point of the face. ___ E. The image of distant objects behind him, appear somewhere between a. AB b. BC c. CD d. DE e. EF f. FG ___ F. Moving so close his nose touches the ornament, he sees the reflected image of his nose a. roughly life-size, just behind the surface of the ornament. b. reduced to a point at the center. c. reduced to a point halfway between the center and surface. d. magnified at a point beyond the center of the ornament. R

13. Name code Problem 12 According to Einstein’s interpretation of the photoelectric effect (Annalen der Physik, Vol. 17; 1905 ) the slope of the line fit above should be given precisely by what simple ratio of fundamental physical constants? What is the photoelectric threshold wavelength, th for the metal Millikan used in the experiment producing the plot above? th = What is the work function  for this metal? Find the maximum speed of photoelectrons liberated from this metal by visible light of wavelength 4000A. 30 40 50 60 70 80 90 100 110 120 43.9  10 13 Hz Stopping potential, volts Frequency (  10 13 Hz) 321 0321 0 Plot from: Millikan, Phys.Rev.7,362 (1916)

14. Name code Problem 11 Questions A-F: An observer peers into a highly reflective silvered, spherical Christmas ornament of radius R as shown below. E marks the ornament’s center. B and G are points on the surface. D, and F are halfway between the center and surface. ___ A. The observer sees an ___ image of himself. a. inverted, enlarged, and real b. upright, enlarged, and real c. upright, enlarged, and virtual d. upright, reduced, and virtual e. upright, reduced, and real f. inverted, reduced, and virtual ___ B. The observer’s eye is approximately a distance R from the surface of the ornament. An image of his eye forms approximately at point a. A b. B c. C d. D e. E f. F C. Verify your answer by solving for the image’s location algebraically. D. Graphically confirm your answer by drawing a neat ray diagram in the figure above showing the formation of an image of any point of the face. ___ E. The image of distant objects behind him, appear somewhere between a. AB b. BC c. CD d. DE e. EF f. FG ___ F. Moving so close his nose touches the ornament, he sees the reflected image of his nose a. roughly life-size, just behind the surface of the ornament. b. reduced to a point at the center. c. reduced to a point halfway between the center and surface. d. magnified at a point beyond the center of the ornament. R d c 1 R 1 i 2 R - = +