Transcript ElementaryQualifierOct2006

Elementary Qualifier Examination
October 9, 2006
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 ħ = h/(2) = 6.583  10-16 eV·sec = 1.05457  10-34 J·sec
1 eV = 1.602189  10-19 J
 
Gauss’ Law:  E  dA  q / 
Ampere’s Law:
0

 
Biot-Savart Law: dB  0 Ids  rˆ
4 r 2
Speed of light, c = 3.00 10 8 m/sec
Permeability, 0 = 410-7 Tm/A
q of electron e = 1.6010-19 C
me = 9.109  10-31 kg = 0.511 MeV/c2
m = 1.883  10-28 kg = 105.6 MeV/c2
m0 = 2.407  10-28 kg = 135.0 MeV/c2
Atomic weight N = 14.00674 u
1u = 1.660 10-27 kg = 931.5 MeV/c2
E
di / dt
Harmonic Oscillator   k 
Inductance L  -
T  2 
Relativistic kinematics
E = g moc2
E2 = p2c2 + mo2c4
1
g 
v2
1- 2
c
 
 B  ds  0 I
Check the boxes below for the
10 problems you want graded
Problem
Number
1
2
3
4
5
6
7
8
9
10
11
12
Total
Score
m1
T1
m2
T2
m3
Problem 1
T3
Name code
As shown above, three connected blocks are pulled to the right on a horizontal frictionless
table by a force of magnitude T3 = 65.0 N. If m1 = 12.0 kg, m2 = 24 kg, and m3 = 31 kg,
calculate:
a. the magnitude of the system’s acceleration
b. the tension, T1
c. the tension, T2
A hollow sphere of mass, M and outer radius a, and with
an inner concentric cavity of radius b is shown in cross
section below right.
Problem 2
Name code
a. Sketch a curve of the magnitude of the gravitational
force F from the sphere on a mass m located a distance
r from the center of the sphere, as a function of r over
the range 0  r  . Show any formulas used in
determining the curve you draw.
Consider r = 0, b, a, and  in particular.
M
r
m
b
a
F
b
a
r
continued
Problem 2 continued
Name code
b.
Sketch the corresponding curve for the potential energy U(r)
Show any formulas used in determining the curve you draw.
Consider r = 0, b, a, and  in particular.
U(r)
b
a
r
k
k
m
Problem 3
Name code
Two springs are joined and connected to a block of mass m = 0.245 kg (see the
figure above) that is set oscillating over a frictionless floor. The springs each
have spring constant k =6430 N/m.
a. Find an expression giving the total force acting on the mass when the two springs are
elongated to give a total elongation x = x1 + x2. From this derive the effective
spring constant of this system of two springs.
b. What is the frequency of the oscillations?
Problem4
1.00 mole of an ideal monatomic gas is taken through
the cycle shown in the graph at right. Assume that
p = 2p0, V = 2V0, p0 = 1.01  105 Pa, V0 = 0.0225 m3.
b
Pressure
Calculate:
(a) the work done during the cycle, and
(b) the energy added as heat during the stroke abc.
Name code
c
V0,p0
a
Volume
V,p
d
A
B
+ + + + + + + + + + + +
L
Problem 5
a
P
Name code
In the figure above a very thin nonconducting rod of length L carries a uniform linear positive
charge density. The total charge of the rod is q. The left end of the rod is at position A and
its right end at position B. Take the potential to be zero at infinity, and consider a point P
on the line containing the rod, as indicated in the Figure. The distance between P and B is a.
a. Which direction has the electric field at point P? Carefully explain your answer.
b. Calculate the magnitude E of the electric field at point P in terms of L, q, and a.
c. Calculate the electric potential VP at point P in terms of L, q, and a.
continued
A
Problem 5
continued
B
+ + + + + + + + + + + +
L
a
P
Name
code
For parts d. and e. assume the rod is L = 30 cm long and that point P is at a distance
a = 30 cm from the rod. (You should not have used these values in parts a., b., and c.)
d. Imagine a point charge with negative charge -q (i.e., equal in magnitude to the total charge
of the rod). Where on the rod would we have to position this point charge to make the
potential at point P zero? Is this position closer to A, closer to B, or exactly midway
between A and B? Explain.
e. Where would the point charge have to be positioned to make the electric field at point
P zero? Is this position closer to A, closer to B, or exactly midway between A and B?
Explain.
In a simple model of the hydrogen atom (H), the electron
orbits around the stationary proton in a circular orbit of
radius a0  0.53 Å
Problem 6
Name code
a. Assuming the only force acting on the electron is the electrostatic force due to the
stationary proton, show that the orbital period of the electron (the time it needs to
revolve around the proton once) equals 1.5 10-16 s.
b. Based on part a., calculate the time-averaged current i in the electron’s circular orbit.
c. Calculate the magnitude B of the magnetic field at the location of the proton that is
produced by this time-averaged current of the electron. Note: if you were unable to
find the answer for part b., you may assume in this part c. that the current is i  3.0 mA
(which is not the real answer for part b.).
Problem 7
R = 47 W
Name code
E = 48 V
+
L=
37 mH
S
Circuit with DC battery, resistor
and inductor. S is a switch.
A solenoid with a self-inductance of L = 37 mH and a resistor with resistance R = 47 W
are connected in series to a 48-volt DC battery as shown in the figure above. Initially,
there is no current in the circuit. The switch S is closed at time t = 0.
a. Calculate the current in the circuit immediately after the switch was closed. Explain.
b. Calculate the current a very long time after the switch was closed. Explain.
c. Calculate the amount of energy in the magnetic field of the solenoid a very long time
after the switch was closed.
d. Was there a time at which the potentials across the inductor and the resistor were equal?
If yes, when was that? If no, explain.
e. At what time was the power delivered from the battery to the solenoid greatest?
What was this peak power?
Problem 7 continued
Name code
plastic insulation
(A)
Problem 8
copper sheath
dielectric
Name code
copper core
(B)
copper sheath
copper
core
(A) Structure of a coaxial cable.
(B) Coaxial cable as a cylindrical capacitor
ra
rb
L
dielectric
The coaxial cable shown in Fig. (A) consists of a copper core (round wire), a dielectric
insulator, and a cylindrical outer conductor (“copper sheath”), all coaxial with each other.
For protection, the cable also has a plastic sheath. Figure (B) shows a piece of this cable
of length L and with the plastic sheath removed. Consider this as a cylindrical capacitor:
imagine there is a linear charge density + on the core and a linear charge density of - on
the sheath, as indicated. These charges set up the radial field indicated by arrows in the gap
between the two conductors. The core has a diameter of 0.812 mm. The dielectric has an
outer diameter of 3.70 mm, and consists of cellular polyethylene (dielectric constant 2.25).
Calculate the capacitance of this cable, in pF/ft (picofarad per foot). Note: 1 foot = 30.5 cm.
Problem 8 continued
Name code
Problem 9
Two converging lenses, separated by 26 centimeters,
bring to focus the image of an object set a distance of
24 cm before the 1st one as shown. The first lens has
a focal length of 8-cm, the second lens 15-cm.
Name code
26cm
f1
f1
f2
f2
___ The image produced by the first lens is
a. real.
b. virtual.
___ The image produced by the first lens is
a. upright.
b. inverted.
___ The final image seen through both lenses is a.upright & real. b.upright & virtual.
c.inverted & real. d. inverted & virtual.
What is the ratio (magnification factor) of the final image size to the object’s actual size?
Show all work.
A negative pion decays, at rest, to a muon and uncharged neutrino:
 -   - 
Problem 10
Name code
The muon and neutrino move relativistically. Assume the neutrino is massless.
a. Find the momentum (in units of MeV/c) of the emitted muon.
b. Find the speed of the emitted muon.
c. Find the energy of the neutrino (in units of MeV).
d. A pion is a spin-0 particle (it has no intrinsic angular momentum, but a muon
is a fermion (spin-½ particle). What must the spin of the neutrino be?
Below are energy shell levels of nucleons including the
spin-orbit splitting of energy levels. Nucleons (neutrons
and protons) are spin-½ fermions, so fill these energy
levels just like electrons fill atomic orbitals. In exactly
the same way, the letters s, p, d, f label each orbital’s
angular momentum (  = 0, 1, 2, 3).
Problem 11
Name code
Fill in the boxes below, giving in the 2nd column the total angular momentum j for
each level (see the “1s” and “1p”examples at bottom of the column). In the 3 rd
column give the number of the degenerate states that fill each shell.
2p
1f
2p
2p
1f
1f
2s
1d
Large energy gaps (see the grayed,
dashed lines) show particularly
stable configurations for nuclei
when a nucleon total exactly matches
one of several “magic numbers.”
2s
1d
1d
This chart shows the 1st four magic
numbers for nucleons are:
1p
1p1/2
1p3/2
1s
1s1/2
principal
quantum
level
coupled
with
spin
2
Degeneracy
(number of
nucleons that fill
each energy level)
a. Express the rotational kinetic energy of a system
classically in terms of its angular momentum L
and its moment of inertia I.
Problem 12
Name code
b. Give the rotational energy EJ of a system quantum
mechanically in terms of its moment of inertia I and
total angular momentum quantum number J.
From spectroscopic observations, the energy difference between the ground state and
the second excited rotational state of the N2 molecule is known to be 14.9  10-4 eV.
c. Deduce the moment of inertia of the molecule.
d. Deduce the center- to-center distance between the N atoms.