Download ElementaryQualifierOct2003

Elementary Qualifier Examination
October 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:
e = 1.60 10-19 C
me = 511 keV/c2
Check the boxes below for the
10 problems you want graded
1 atm = 1.013105 N/m2
g = 9.8 m/sec2
mn = 1.6710-27 kg
1u = 1.66 10-27 kg
Compton wavelength, C = 2.43 10-12 m
Coulomb’s constant, k=1/(4o)=9.0109N·m2/c2
Planck constant h = 6.62610-34 J·sec
= 4.136 10-15 eV·sec
Speed of light, c = 3.00 108 m/sec
hc = 1240 eV·nm
Permeability, 0 = 410-7 Tm/A
Compton Scattering
- = C(1-cos)
Relativistic kinematics
E = g moc2
g 
1
v2
1- 2
c
Problem
Number
1
2
3
4
5
6
7
8
9
10
11
12
Total
Score
Problem 1
Name code
A neutron traveling with velocity v = 1107 m/sec collides head-on with a nucleus
(radius, r  10-14 m). Assuming the neutron decelerates uniformly between the outer
radius of the nucleus and its center where it comes to rest and is trapped:
a. estimate the magnitude of the average net force stopping the neutron.
b. find the time to complete this reaction.
Problem 2
Name code

L
m
R
Consider a “conic pendulum” : a mass, m, suspended by a cord of
length, L, at an angle  from the vertical as it traces a circular path
of radius R.
a. Find the acceleration due to gravity, g, in terms of L, , and
the pendulum’s period, T.
b. Given L = 1m,  = 45o, T = 1.69 sec, compute g.
Problem 3
Name code
For each bounce:
H
ℓ
ℓ
vi
vf
v h vh
A ball bounces down stairs, striking the center of each step, and bouncing each
time to the same height H above the step. The stair height equals its depth, ℓ,
and the coefficient of restitution e = -vf/vi is assumed given. vf and vi are the
vertical velocities just after and before a bounce (see diagram above).
A. Show that the vertical components of the ball’s velocity immediately before
and after each bounce can be written:
2 g
2 g
vi 
B.
1 - e2
v f  -e
1 - e2
Find an expression for the horizontal velocity vh in terms of only g, ℓ, and e.
Problem 4
Name code
Consider an ideal diatomic gas enclosed in an insulated chamber with a movable
piston. The values of the initial state variables are
P1 = 8 atm,
V1 = 4 m3
and
T1 = 400 K.
The final value of the pressure after an adiabatic expansion is
P2 = 1 atm.
Find V2 , T2 , W (the work done by the gas in expanding) and U (the change
in the gas’ internal energy). Recall that for an ideal diatomic gas
g
CP
 7 / 5.
CV
V2 =
T2 =
W =
U =
Problem 5
After the circuit shown in the figure at right has reached
the steady state, switch S1 is opened and S2 closed.
Name code
5F
Calculate:
•
the frequency of oscillation.
4mH
S2
R=2
•
S1
+
-
the energy in the circuit.
 = 80v
•
the maximum current.
Problem 6
Name code
Consider the four charges shown, at the corners of a square with side, a.
Calculate the energy in eV necessary to remove one of the charges to infinity.
a = 2.810-10 m
-e
+e
1
2
4
3
a
-e
a
+e
Problem 7
Name code
An electric field of 1.5 kV/m and a magnetic field of 0.40T act on a moving electron to
produce no net force.
Calculate the minimum speed of the electron.
 

Draw a diagram of the vectors E , B and v
Problem 8
Name code
Consider the circuit shown in the figure.
a. If the current in the straight wire is i, find the
the magnetic flux through the rectangular loop.
i
ℓ=
50cm
10 cm
5 cm
b.
If the current i decreases uniformly from 90A
to zero in 15 msec, calculate the magnitude and
direction of the induced current in the loop.
The resistance of the loop is 5m.
Problem 9
Name code
A particle has total energy 1.123 MeV and momentum 1.00 MeV/c.
a.
What is the particle’s (rest) mass?
b.
Find the total energy of this particle in a reference frame in which its momentum
is 2 MeV/c.
c.
Find the particle’s velocities in the first and second frames.
Problem 10
Name code
The 12C16O molecule absorbs infrared radiation of frequency 6.421013 Hz.
The atomic masses are MC = 12 u and MO = 16 u. Assuming that the system
is a harmonic oscillator, find:
a.
the ground-state vibrational energy of the CO molecule in eV.
b.
the molecular force constant.
c.
the classical amplitude of the ground-state vibrations.
Problem 11
Name code
An electron is described by the 1-dimensional wavefunction
 (x) 
where C is a constant.
0
if x  0
Cxe - x
if x  0
(a) Find the value of C that normalizes .
(b) For what value of x is the probability for finding the electron largest?
(c) Calculate the expectation value of x for this electron and comment
on any difference you find between it and the most likely position.
Problem 12
Name code
X-rays, produced in a cathode tube of voltage 62 kV, undergo
Compton scattering in the backward direction.
(a) What are the wavelengths of the incident and scattered X-rays?
(b) What is the momentum of the recoil electrons?
(c) What is the kinetic energy of the recoil electrons?