Elementary Qualifier Examination February 16 2004 NAME CODE

Elementary Qualifier Examination February 16, 2004 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 h = 6. 626 10 -34 J·sec = 4. 136 10 -15 e. V·sec 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) He = c. P/c. V = 5/3 Check the boxes below for the 10 problems you want graded mp = 1. 67 10 -27 kg 1 u = 1. 66 10 -27 kg d = dx dy dz = (r sin d )(r d ) dr = r 2 sin d d dr Relativistic kinematics E = g mo c 2 E 2 = p 2 c 2+mo 2 c 4 Problem Number 1 2 3 4 5 6 7 8 9 10 11 12 Total Score

Problem 1 Name code A stuntman is launched by a cannon at an angle of 45 o into the air. He must grab the catch bar suspended below a plane flying 30 m/sec, horizontally overhead at an altitude of 45 m. (you can assume g = 10 m/s 2). a. At what speed should the stuntman be launched? b. Where should the plane be at the moment that the stuntman is launched?

Problem 2 Name code vcar 30 o A ball can be shot straight up from a moving cart (perpendicular to its forward motion) with a speed vball. When placed on an inclined plane making an angle of 30 o with respect to the ground, the ball is launched as the cart rolls freely down the plane. a. If the cart reaches a speed vcar when the ball is launched, give the horizontal and vertical components of speed for the cart and ball at that moment. Horizontal speed Vertical acceleration speed acceleration Cart Ball b. Re-express all the components now in terms of the new coordinate system selected at right: y x parallel to ramp vx ax perpendicular to ramp vy ay Cart Ball Problem 2 continued

Problem 2 cont’d Name code c. Neglecting frictional forces, do you expect the ball to land (circle one) i. behind the rolling cart? d. Justify your answer. ii. in iii. in front of

Problem 3 Name code 1 kg Fpush 3 kg 2 kg Three blocks of 1 , 2 , and 3 kg, accelerate together across a horizontal table when a force of 10 N is applied as indicated in the drawing. The coefficients of friction are Ckinetic=0. 1, and Cstatic=0. 2 for all surfaces. Assume g = 10 m/sec 2. a. Draw all forces that act on each block in a free body diagram at right. b. Find the acceleration of the 3 kg block. 10 N 30 N 20 N

Problem 4 Name code A volume of 10 liters is filled with 2. 5 moles of helium (assumed to be an ideal gas) and is at 270 K. a. If the gas expands isothermally to a volume of 20 liters, how much work is done by the gas? b. If the gas expands adiabatically to 20 liters, what will be its final temperature? c. If the gas expands freely to 20 liters, what will be its final temperature and pressure?

Problem 5 Name code Pipe Wire a r b Cross section The figure (not to scale) shows an infinitely long, straight wire (carrying a uniform linear charge density =+10. 00 C/m) centered along the axis of an infinite pipe with inner radius a = 1. 00 m and outer radius b = 3. 00 m. The pipe (made of an insulating material) carries a uniform volume charge density = +1. 00 C/m 3 throughout its volume. a. Use Gauss’ Law to calculate E(r), the magnitude of the electric field at a perpendicular distance r meters from the wire, with a < r < b. b. At some distance r within the pipe (a < r < b), the magnitude of the electric field reaches a minimum. Find this distance r, in meters.

A 10. 0 F capacitor is fully charged by a battery, the battery is disconnected, and the charged capacitor Name code is quickly connected to a 200. 0 k series resistor. The first measurement of the current in the circuit, made at t = 4. 0 seconds, shows the current is 3. 0 10 -4 Amps. Problem 6 a. What was the potential difference across the capacitor at t = 0. 0 seconds? b. After a long time the current in the circuit reaches zero Amps. How much total energy has been dissipated in the resistor? Now imagine that the resistor has the value 400. 0 k (rather than 200. 0 k ). The rest of the given information is the same as above. c. For the case of the 400 k resistor, your new answer to part (a) would be: (circle one) Greater than Less than Equal to your original answer to part (a). d. For the case of the 400 k resistor, your new answer to part (b) would be: (circle one) Greater than your original answer to part (b). Less than Equal to

Problem 7 X X X Name code X b B into page X X a X Speed v X A uniform magnetic field B Tesla directed into the page fills the shaded region of the figure. The dark rectangle represents a single loop of wire with total resistance R ohms, and an external agent is moving the rectangle out of the field region at a constant speed to the right of v m/s. The length of the vertical sides of the rectangle is a meters, and the length of the horizontal sides is b meters. (a) Calculate the induced current in the rectangle at the instant shown in the figure. (b) Is the direction of the induced current in the rectangle clockwise or counterclockwise? (c) What force (magnitude and direction) does the magnetic field exert on the left side of the rectangle? (d) During the time the rectangle is exiting the field region (i. e. from the time its right side just exits the field region to the time its left side just exits), how much total energy is dissipated in the rectangle due to resistive heating? (e) Where does this energy come from?

Problem 8 Name code A Wheatstone bridge can be used to determine the value of an unknown resistance, Rx. The resistances of the resistors R 1 and R 2 are known. RS is adjustable. When adjusted so that the current read by the ammeter A completely vanishes, Rx RS a. what mathematical relationships must exist between the voltage drops across each resistor, Vs , Vx , V 1 , and V 2 For the ammeter to read 0? A R 1 R 2 Itotal b. Show under this setting, that Rx= R 2 R R 1 s c. When RS is adjusted too high (to produce the above condition), the current through the ammeter flows (circle one): UP DOWN.

o decaying at rest: o in flight: Problem 9 p 1 p 2 Name code K=150 Me. V o In its own rest frame, a neutral pion o (moc 2=135 Me. V) decays into 2 photons of equal energy and equal but opposite momentum. a. Why will the neutral pion not decay into a single photon with energy equal to the pion’s initial energy? A o traveling with kinetic energy, K = 150 Me. V, decays in flight to two photons. b. Find the pion’s speed before decay. Problem 9 continued

Problem 9 cont’d Name code c. For the o’s decay in flight, write an equation expressing the conservation of of energy. d. For the o’s decay in flight, write an equation expressing the conservation of of momentum. e. What are the highest and lowest photon energies E 1 and E 2 that could be observed in the lab from such decays?

Problem 10 a. The wave function 2 s has what possible quantum numbers? Name code n = ______ ℓ = ______ How many total 2 s m= ______ states are possible? ______ b. The wave function 3 d has what possible quantum numbers? n = ______ ℓ = ______ How many total 3 d m= ______ states are possible? ______ The 2 s state is described by the wave function c. Assume this function satisfies Schrodinger's equation. Collecting like terms in r , determine the energy eigenvalue E 2 of this state, and derive an expression for ao. E 2 = ao =

45 o Problem 11 nair = 1. 0 Name code A A monochromatic light ray, initially in air, strikes a glass block at an angle of 45 o as shown at left. a. What index of refraction, nglass, should the glass have if the ray behaves as shown when it reaches point A? b. If the index of refraction of the glass were larger than the value found in part (a), what would happen qualitatively to the light ray at point A? (Note: point A refers to the point where the ray strikes the left edge of the glass block. ) c. If the index of refraction of the glass were smaller than the value found in part (a), what would happen qualitatively to the light ray at point A? (Note: point A refers to the point where the ray strikes the left edge of the glass block. )

E p Problem 12 Name code A hydrogen atom at rest in an excited state of energy Ei decays to a final energy Ef by emitting a photon of energy E . a. Show that the recoil kinetic energy K of the atom is equal to E 2 /2 Mc 2. b. Show that when this recoil K is taken into account, the value of the photon’s frequency, , is reduced to approximately o ( 1 - E / 2 Mc 2) from the nominal o= E/h , where E = (Ei-Ef ). c. Using typical values, show that this change, / , is completely negligible. d. The gamma rays produced by nuclear decays when an excited nucleus (say, atomic number A 10) decays, are typically of the order of 1 Me. V in energy. How big is the correction in this case?