Common Exam - 2001

Department of PhysicsUniversity of UtahAugust 25, 2001

Examination booklets have been provided for recording your work and your

solutions. Please note that there is a separate booklet for each numbered question (i.e.,

use booklet #1 for problem #1, etc.).

To receive full credit, not only should the correct solutions be given, but a

sufficient number of steps should be given so that a faculty grader can follow your

reasoning. Define all algebraic symbols that you introduce. If you are short of time it

may be helpful to give a clear outline of the steps you intended to complete to reach a

solution. In some of the questions with multiple parts you will need the answer to an

earlier part in order to work a later part. If you fail to solve the earlier part you may

represent its answer with an algebraic symbol and proceed to give an algebraic answer to

the later part. This is a closed book exam: No notes, books, or other records should be

consulted. NO CALCULATORS MAY BE USED. The total of 250 points is divided

equally among the ten questions of the examination.

All work done on scratch paper should be NEATLY transferred to answer booklet.

SESSION 1

Problem 1 – Fundamental Mechanics

A mass m slides without friction on the track shown. It starts from rest at a height H. Frompoint 1 to point 2 the track is straight at an angle of 30° from the horizontal. The track is straightand vertical from point 4 to point 5. Between points 2 and 4 the track is a segment of a circle ofradius R.

(a) [5 pts.] What is the greatest speed the mass reaches while sliding?

(b) [5 pts.] Let be the momentum of the mass at point 3. What are the values of the

x ycomponents p and p ?(c) [5 pts.] What is the magnitude of the normal force on the mass at point 1?(d) [5 pts.] What is the magnitude of the normal force on the mass at point 5?(e) [5 pts.] What is the magnitude of the normal force on the mass at point 3?

Problem 2 – General Physics

(a) [4 pts.] Stars A and B have the same luminosity. (They radiate the same amount ofenergy per unit time.) Star A has a surface temperature of 2,000°K and has twice theradius of star B. What is the surface temperature of star B?

(b) [3 pts.] What is the order of magnitude of the wavelength of an x-ray?

(c) [4 pts.] An electron is accelerated into a screen, as in a television set. What order-of-magnitude voltage is necessary for the electron to create a nuclear particle in the collisionwith the screen?

(d) [4 pts.] A certain kind of Geiger counter has a probability of 50% of clicking in anygiven second. Four of these Geiger counters form a bank of detectors. What is theprobability that at least one of the counters clicks during any given second?

(e) [3 pts.] A 500 watt laser is used as an engine to propel a rocket ship. What is the thrust(force) provided by this engine?

(f) [4 pts.] A 30 kg cube, 1 meter on a side, floats half submerged in a fluid on the surfaceof the earth. What is the pressure in the fluid at a depth of 50 cm?

(g) [3 pts.] A certain camera has a lens of diameter 1 cm that has a focal distance of 5 cm. To order of magnitude, what is the resolution limit of this camera imposed by diffraction? (What is the size of the image on the film when a best-focused picture is taken of apoint?)

Problem 3 – Electricity and Magnetism

Consider a spherical capacitor consisting of two concentric spherical shells. The inner shell has

1 3outer radius r , while the outer shell has inner radius r . The space between the conductors is

1 2 1filled by two dielectrics. In the region r < r < r the dielectric has dielectric constant 6 and

1 2 3 2dielectric breakdown strength E . In the region r < r < r the dielectric has dielectric constant 6

2and dielectric breakdown strength E .

(a) [5 pts.] Suppose a charge Q is placed on the inner shell and -Q is placed on the outer

1 3shell. Find the electric field at an arbitrary r in the region r < r < r .

3 1(b) [5 pts.] Under the same conditions as in part (a), find the voltage V = |V(r ) ! V(r )|.(c) [5 pts.] What is the capacitance of the capacitor?(d) [5 pts.] What is the maximum energy that can be stored in the capacitor?

3 1(e) [5 pts.] What will be the voltage V = |V(r ) ! V(r )| at maximum energy?

Problem 4 – Modern Physics

Consider a planet with a nitrogen atmosphere of thickness 100 km. The atmosphere has aconstant density of .001 gm/cm . A proton of energy 10 eV impacts the atmosphere along a3 20

planetary radius (zenith angle = 0 deg).

(a) [5 pts.] If the proton-nitrogen cross section is 500 mb and is independent of protonenergy, what is the mean free path (collision length) between interactions in thisatmosphere?

(b) [5 pts.] What fraction of such protons will hit the planet's surface without interacting?(c) [5 pts.] A muon of the same energy enters the atmosphere with the same direction. If the

muon life-time is 2.2 × 10 sec and we can neglect muon-nitrogen interactions, what!6

fraction of such muons will reach the surface without decaying? What would thisfraction be if the muon energy was 10 eV?9

(d) [5 pts.] If the 10 eV muon-nitrogen interaction cross section is 10 :b, what is the20

probability that the muon will interact in the atmosphere before hitting the ground?(e) [5 pts.] If the muon energy is 10 eV, on average, and its cross section is the same as in9

(d), what is the probability that the muon will interact before it decays?

m(N) � 14.000 amum(proton) = 1.67 × 10 kg!27

1 eV = 1.6 × 10 J!19

muon mass � 105 meV1 mb = 10 m!31 2

Problem 5 – Quantum Mechanics

The Hamiltonian of a two-dimensional, isotropic harmonic oscillator with mass m and angularfrequency T has the form

(a) [12 pts.] By expressing as the sum of two similar Hamiltonians for one-

dimensional oscillators, show that the product of one-dimensional wavefunctions

solves the Schrödinger equation R(x,y) = ER(x,y), where E is the total energy.(b) [6 pts.] Show that the energy eigenvalue E is equal to (n + 1)ST, where n is zero or a

positive integer.(c) [7 pts.] Show that the degeneracy of the three lowest energy values are 1, 2, and 3,

respectively, and in general, the degeneracy of the n level is (n + 1) fold.th

Common Exam - 2001

Department of PhysicsUniversity of UtahAugust 25, 2001

Examination booklets have been provided for recording your work and your

solutions. Please note that there is a separate booklet for each numbered question (i.e.,

use booklet #1 for problem #1, etc.).

To receive full credit, not only should the correct solutions be given, but a

sufficient number of steps should be given so that a faculty grader can follow your

reasoning. Define all algebraic symbols that you introduce. If you are short of time it

may be helpful to give a clear outline of the steps you intended to complete to reach a

solution. In some of the questions with multiple parts you will need the answer to an

earlier part in order to work a later part. If you fail to solve the earlier part you may

represent its answer with an algebraic symbol and proceed to give an algebraic answer to

the later part. This is a closed book exam: No notes, books, or other records should be

consulted. NO CALCULATORS MAY BE USED. The total of 250 points is divided

equally among the ten questions of the examination.

All work done on scratch paper should be NEATLY transferred to answer booklet.

SESSION 2

Problem 6 – General Physics

3 3For the circuit shown, calculate the value of R such that the current in R in the direction shownis 1.0 Amp. (If you make calculational approximations, please make these clear.)

(a) [15 pts.] Write a complete set of equations describing the behavior of this circuit. Carefully, completely and precisely define the unknowns you use.

3(b) [10 pts.] Solve for the needed value of R .

1g = 4.00 V

2g = 3.00 V

1R = 10 S

2R = 2 S

Problem 7 – Thermodynamics

Consider a cylinder of cross-sectional area A and length L. It is filled with one mole of an idealgas. The gas has values for the specific heats Cp and Cv. One end of the cylinder has africtionless plug of mass M. The plug is free to move up or down and makes a good seal with thecylinder sides. The system is enclosed so no heat can flow to the surroundings. The outsidepressure is atmospheric. The system is at equilibrium at all times. The processes are reversible.

(a) [5 pts.] If the system is at equilibrium at temperature T, find the height of the plug abovethe bottom of the cylinder.

(b) [5 pts.] Suppose a small external force is applied to the top of the plug and then released. Describe the subsequent motion of the plug qualitatively.

(c) [5 pts.] What is the relation between the external force and the displacement of the plugat any given time?

(d) [5 pts.] What is the natural period of oscillation of the plug?(e) [5 pts.]Describe how you could use this set up to measure the ratio of specific heat at

constant pressure to the specific heat at constant volume.

Problem 8 – Lagrangian Mechanics

1Mass m slides on a horizontal frictionless surface and is attached to a fixed wall by a spring of

ospring constant k. The distance of the mass from the wall is x when the spring is at itsequilibrium length. The value of x in the figure is therefore the extension of the spring beyondits equilibrium length. A pendulum consists of a massless rod of length L with a pendulum bob

2 1of mass m . The pendulum is attached to a frictionless pivot on m . The angle the rod makes

2with the vertical is 2. The radius of m is small compared to L.

(a) [6 pts.] Write the Lagrangian for this system in terms of the coordinates x and 2. Do notassume that x and 2 are small.

(b) [5 pts.] Find the equations of motion of the system. [That is, find second orderdifferential equations satisfied by x(t) and 2(t).] Do not assume that x and 2 are small.

(c) [7 pts.] Find the normal mode frequencies governing motion with small x and 2.

1 2(d) [4 pts.] In the case that m = m / M, and , find the

(approximate) numerical values of the normal mode frequencies.

1 2(e) [3 pts.] What are the normal mode frequencies in the case m � m and k = 0?

Problem 9 – Electricity and Magnetism

Consider the set up shown, consisting of two parallel, resistanceless, conducting rails separatedby a distance w. A rod of resistance R and mass m slides along the frictionless rails. The track ishorizontal and there is a magnetic field B directed vertically downward, as shown. At time t = 0the rod is located at x = 0 and motionless. The battery has voltage V with polarity as shown. Att = 0 the switch, S, is closed. Assume the track is very long.

(a) [10 pts.] What is the maximum velocity achieved by the rod?(b) [10 pts.] What is the velocity of the rod at any time t?(c) [5 pts.] What is the condition for “very long” in terms of the parameters of the system

(m, R, V, w, B)?

Problem 10 – Quantum Mechanics

Suppose that, at t = 0, a particle in a one-dimensional infinite square potential well (width L) is in

1 2a superposition of the ground state Q and the first excited state Q :

1 2Q and Q are properly normalized. The energies of the ground state and the first excited state

1 1 2 2are E = ST and E = ST , respectively.

(a) [5 pts.] Verify that this wavefunction is correctly normalized.(b) [5 pts.] What is the expectation value for the energy of this particle?(c) [5 pts.] For t > 0, we must consider the time-dependent wavefunction of this particle.

Write down the time-dependent wavefunction for this particle.(d) [10 pts.] Find the probability distribution of this particle as a function of time.

Note: