Comprehensive Exam Session I Classical Mechanics Physics Department

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1 Letter Comprehensive Exam Session I Classical Mechanics Physics Department (April 2 nd, 2011) If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Show steps clearly. 1. Which of these statements about Mechanical Energy (E) are true: Circle all that are true. (Each correct answer = +1 each wrong answer = 1. Score will be normalized to 10 points) a) Consists of potential (V) and kinetic (T) energies. b) The T and V are never conserved separately. c) In free fall without air resistance T+V is a function of time. d) If E = T+V is conserved, the Lagrangian, L = T V is also conserved. e) If E = T+V is conserved then another quantity called the Gaussian q Form = exp [q (T+V) 2 / 4σ 2 ] is also conserved (q and σ are constants). f) For inelastic collisions E is sometimes conserved although T is never conserved in such cases. g) In free fall with air resistance E is conserved. h) If E is conserved the forces on a system are conservative and vice versa.

2 2. An existing frictionless roller coaster has a single hoop of radius R as shown in the figure. The height h was arranged so that the cars sent from the top at zero velocity just make the top of the hoop without falling. To make it more exciting the owner wants to add another larger hoop of radius 3/2 R, but he cannot raise the height due to regulations. He can however give the cars an initial velocity at the top. He asks you to figure out at what speed he needs to send the cars so that they will not fall down in the second hoop. What value of v o would you recommend? Derive an equation for v o in terms of R and find the numerical value if R = 10 m).

3 3. A chlorine molecule is moving at a velocity of 600 m/s when it absorbs a photon of wavelength 350 nm. The molecules dissociates into two chlorine atoms. One of the atoms is observed to be moving perpendicular to the direction of the initial molecule and at a speed of 1600 m/s. What is the binding energy of the chlorine molecule ignoring the momentum of the photon (7 points) express it in Joules and kj/mole? Is it reasonable to ignore the photon momentum (3 points)? Atomic mass of chlorine is 35. h Cl Cl Before v u Cl Cl After

4 4. A helix is made from a wire and a bead of mass m is threaded on the wire at the top. The bead slides down the wire. In cylindrical polar coordinates ρ = R (the constant radius of the helix) and z = λφ where λ is a constant. Use z as your generalized coordinate and write the Lagrangian for the bead. Find the Lagrange equation for the bead s vertical acceleration d 2 z/dt 2.. Solve the equation of motion for the bead starting at rest. (7 pts) What is the motion when R = 0? Explain (3 pts). R Bead H Helix R=Radius = # turns/l

5 5. A mass m is suspended by a massless thread. The upper end is wrapped around a cylinder of radius R and moment of inertia I that is free to rotate about its central axis which is fixed. Use the Lagrange method to find the equation of motion and then solve it to give the position of the mass as a function of time as it falls under gravity. I Axle fixed X X=generalized coordinate M

6 Letter Comprehensive Exam Session II E&M Physics Department (April 2 nd, 2011) If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Show steps clearly. Electricity and Magnetism: 1. A parallel plate capacitor, plate area A, radius a and separation s, is initially uncharged. A current I=I 0 exp( t/rc) flows to charge the capacitor. (a.) Find the charge on one plate as a function of time. s a I ANSWER: (b.) Find the electric displacement D(t) as a function of time. ANSWER:

7 2. The meter M in the circuit shown measures the total amount of charge that flows through it. The circuit is made of wires whose resistance is 10 Ω/meter. A magnetic field, B, is turned on such that the field inside the loop goes from zero to 0.5 tesla in 2 seconds, then remains constant. z (a.) What is the numerical value for the EMF that is induced while the magnetic field is changing? B 10 cm I 10 cm y 20 cm x M ANSWER: (b.) What total charge passes through the meter? ANSWER:

8 3. Two concentric cylindrical conductors, r 1 =0.01 m and r 2 =0.08 m, have charge densities 1 =40pC/m 2 and 2, such that D and E fields exist between the two cylinders but are zero elsewhere. Find the expression for D and E between the cylinders and 2. r 2 1 r 1 2 Answers: D= C/m 2 E= V/m 2 =

9 4. Find the magnetic flux (in Webers,W b ) crossing the portion of the plane = /4 defined by 0.01 < r < 0.05 m and 0< z <2 m (see figure). A current filament of 2.50 A along the z axis is in the z direction. 2 m z ds m 2.50 A 0.05 m /4 x = Wb

10 5. (a.) Find the equivalent capacitance in Figure 1. (show your work) A 1 A 2 V r1 r2 Figure 1 C eq = (b.) Find the equivalent capacitance in Figure 2.(show your work) d 2 d 1 V r2 r1 Figure 2 C eq =

11 Letter Comprehensive Exam Session III Modern Physics (Including Stat.Mech) Physics Department (April 2 nd, 2011) If you cannot solve the whole problem, write down all relevant equations and explain how you will approach the solution. Show steps clearly. 1. Which electronic configuration has a lower energy, [Ar]3d 4 4s 2 or [Ar]3d 5 4s 1? Identify this element and discuss Hund s rule for this particular case. (Note: The notation [Ar] represents the filled configuration for Ar.)

12 2. Determine if free electrons in a gold crystal can be modeled by Maxwell Boltzmann statistics at 0 C and ambient atmospheric pressure. Density of Au: 19.3 g cm 3, you should have all other constants and conversion factors on your formulae sheets and can, if needed, consult a periodic table of the elements. N.B. In your calculations, you do need to follow through thoroughly with the physical units in order to get full credit. You may start with Heisenberg s uncertainty principle )

13 3. When the idea of electron spin was introduced, the electron was thought to be a tiny charged sphere (today it is considered a point object with no extension in space). Find the equatorial speed under the assumption that the electron is a uniform sphere of radius 5x10 8 nm spinning about an axis through its center, as early theorists believed, and compare your result to the speed of light. (Note: The moment of inertia of a sphere is I = 2/5 mr 2 with m as mass and r as radius.)

14 Stat Mech. Problems: 1. Give the names of the 3 statistical distribution functions used to describe everything from molecules to electrons, the types of particles each applies to and any limitations on energy level occupancy.

15 2. For a 2 state system, separated by a positive energy difference E, the equilibrium number ratio is given by the equation below. Which state, 1 or 2, is lower in energy?

F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26

F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26 Physics 23 Exam 2 Spring 2010 Dr. Alward Page 1 1. A 250-N force is directed horizontally as shown to push a 29-kg box up an inclined plane at a constant speed. Determine the magnitude of the normal force,