Download Common Exam - 2001 Department of Physics University of Utah August 25, 2001

Common Exam - 2001
Department of Physics
University of Utah
August 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. From
point 1 to point 2 the track is straight at an angle of 30° from the horizontal. The track is straight
and vertical from point 4 to point 5. Between points 2 and 4 the track is a segment of a circle of
radius R.
(a)
(b)
(c)
(d)
(e)
[5 pts.] What is the greatest speed the mass reaches while sliding?
[5 pts.] Let be the momentum of the mass at point 3. What are the values of the
components px and py?
[5 pts.] What is the magnitude of the normal force on the mass at point 1?
[5 pts.] What is the magnitude of the normal force on the mass at point 5?
[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 of
energy per unit time.) Star A has a surface temperature of 2,000°K and has twice the
radius 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-ofmagnitude voltage is necessary for the electron to create a nuclear particle in the collision
with the screen?
(d)
[4 pts.] A certain kind of Geiger counter has a probability of 50% of clicking in any
given second. Four of these Geiger counters form a bank of detectors. What is the
probability 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 surface
of 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 a
point?)
Problem 3 – Electricity and Magnetism
Consider a spherical capacitor consisting of two concentric spherical shells. The inner shell has
outer radius r1, while the outer shell has inner radius r3. The space between the conductors is
filled by two dielectrics. In the region r1 < r < r2 the dielectric has dielectric constant 61 and
dielectric breakdown strength E1. In the region r2 < r < r3 the dielectric has dielectric constant 62
and dielectric breakdown strength E2.
(a)
(b)
(c)
(d)
(e)
[5 pts.] Suppose a charge Q is placed on the inner shell and -Q is placed on the outer
shell. Find the electric field at an arbitrary r in the region r1 < r < r3.
[5 pts.] Under the same conditions as in part (a), find the voltage V = |V(r3) ! V(r1)|.
[5 pts.] What is the capacitance of the capacitor?
[5 pts.] What is the maximum energy that can be stored in the capacitor?
[5 pts.] What will be the voltage V = |V(r3) ! V(r1)| at maximum energy?
Problem 4 – Modern Physics
Consider a planet with a nitrogen atmosphere of thickness 100 km. The atmosphere has a
constant density of .001 gm/cm3. A proton of energy 1020 eV impacts the atmosphere along a
planetary radius (zenith angle = 0 deg).
(a)
(b)
(c)
(d)
(e)
[5 pts.] If the proton-nitrogen cross section is 500 mb and is independent of proton
energy, what is the mean free path (collision length) between interactions in this
atmosphere?
[5 pts.] What fraction of such protons will hit the planet's surface without interacting?
[5 pts.] A muon of the same energy enters the atmosphere with the same direction. If the
muon life-time is 2.2 × 10!6 sec and we can neglect muon-nitrogen interactions, what
fraction of such muons will reach the surface without decaying? What would this
fraction be if the muon energy was 109 eV?
[5 pts.] If the 1020 eV muon-nitrogen interaction cross section is 10 :b, what is the
probability that the muon will interact in the atmosphere before hitting the ground?
[5 pts.] If the muon energy is 109 eV, on average, and its cross section is the same as in
(d), what is the probability that the muon will interact before it decays?
m(N) – 14.000 amu
m(proton) = 1.67 × 10!27 kg
1 eV = 1.6 × 10!19 J
muon mass – 105 meV
1 mb = 10!31 m2
Problem 5 – Quantum Mechanics
The Hamiltonian of a two-dimensional, isotropic harmonic oscillator with mass m and angular
frequency 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
(b)
(c)
solves the Schrödinger equation R(x,y) = ER(x,y), where E is the total energy.
[6 pts.] Show that the energy eigenvalue E is equal to (n + 1)ST, where n is zero or a
positive integer.
[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 nth level is (n + 1) fold.
Common Exam - 2001
Department of Physics
University of Utah
August 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
For the circuit shown, calculate the value of R3 such that the current in R3 in the direction shown
is 1.0 Amp. (If you make calculational approximations, please make these clear.)
(a)
(b)
[15 pts.] Write a complete set of equations describing the behavior of this circuit.
Carefully, completely and precisely define the unknowns you use.
[10 pts.] Solve for the needed value of R3.
g1 = 4.00 V
g2 = 3.00 V
R1 = 10 S
R2 = 2 S
Problem 7 – Thermodynamics
Consider a cylinder of cross-sectional area A and length L. It is filled with one mole of an ideal
gas. The gas has values for the specific heats Cp and Cv. One end of the cylinder has a
frictionless plug of mass M. The plug is free to move up or down and makes a good seal with the
cylinder sides. The system is enclosed so no heat can flow to the surroundings. The outside
pressure is atmospheric. The system is at equilibrium at all times. The processes are reversible.
(a)
(b)
(c)
(d)
(e)
[5 pts.] If the system is at equilibrium at temperature T, find the height of the plug above
the bottom of the cylinder.
[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.
[5 pts.] What is the relation between the external force and the displacement of the plug
at any given time?
[5 pts.] What is the natural period of oscillation of the plug?
[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
Mass m1 slides on a horizontal frictionless surface and is attached to a fixed wall by a spring of
spring constant k. The distance of the mass from the wall is xo when the spring is at its
equilibrium length. The value of x in the figure is therefore the extension of the spring beyond
its equilibrium length. A pendulum consists of a massless rod of length L with a pendulum bob
of mass m2. The pendulum is attached to a frictionless pivot on m1. The angle the rod makes
with the vertical is 2. The radius of m2 is small compared to L.
(c)
[6 pts.] Write the Lagrangian for this system in terms of the coordinates x and 2. Do not
assume that x and 2 are small.
[5 pts.] Find the equations of motion of the system. [That is, find second order
differential equations satisfied by x(t) and 2(t).] Do not assume that x and 2 are small.
[7 pts.] Find the normal mode frequencies governing motion with small x and 2.
(d)
[4 pts.] In the case that m1 = m2 / M, and
(e)
(approximate) numerical values of the normal mode frequencies.
[3 pts.] What are the normal mode frequencies in the case m1 … m2 and k = 0?
(a)
(b)
, find the
Problem 9 – Electricity and Magnetism
Consider the set up shown, consisting of two parallel, resistanceless, conducting rails separated
by a distance w. A rod of resistance R and mass m slides along the frictionless rails. The track is
horizontal and there is a magnetic field B directed vertically downward, as shown. At time t = 0
the rod is located at x = 0 and motionless. The battery has voltage V with polarity as shown. At
t = 0 the switch, S, is closed. Assume the track is very long.
(a)
(b)
(c)
[10 pts.] What is the maximum velocity achieved by the rod?
[10 pts.] What is the velocity of the rod at any time t?
[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
a superposition of the ground state Q1 and the first excited state Q2:
Q1 and Q2 are properly normalized. The energies of the ground state and the first excited state
are E1 = ST1 and E2 = ST2 , respectively.
(a)
(b)
(c)
(d)
[5 pts.] Verify that this wavefunction is correctly normalized.
[5 pts.] What is the expectation value for the energy of this particle?
[5 pts.] For t > 0, we must consider the time-dependent wavefunction of this particle.
Write down the time-dependent wavefunction for this particle.
[10 pts.] Find the probability distribution of this particle as a function of time.
Note: