Transcript ElementaryQualifierJan2003

Elementary Qualifier Examination
January 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:
g = 9.8 m/sec2
Planck constant h = 6.62610-34 J·sec
= 4.136 10-15 eV·sec
Speed of light, c = 3.00 108 m/sec
Permeability, 0 = 410-7 Tm/A
Relativistic kinematics
E = g moc2
E2 = p2c2+mo2c4
1
g 
v2
1- 2
c
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
Problem 1
m
Name code
M
R
Starting from rest, a small block of mass m slides down frictionlessly from the top
of a quarter circle path of radius R cut into the corner of a large block of mass M.
Initially at rest as well, the large block moves without friction across the table that
supports it. Find the velocity of each immediately after the small block leaves
the quarter circle path.
Problem 2
Name code

An amusement park ride consists if a vertical cylinder of radius R
that rotates about its vertical axis. Riders stand initially on a floor
which drops away once the ride starts and leaves them suspended,
pressed against the wall. Given the final rotational speed of the
cylinder, , derive the minimum coefficient of static friction to
ensure the riders’ safety.
Problem 3
Name code
ℓ
300-kg
A 300-kg block hangs at the end of a 100-kg 10-meter long horizontal beam.
A supporting cable is attached to the beam a distance ℓ out from the wall and
anchored to that wall 10 meters above the beam..
A. Draw the free body diagram of the long bar.
B. Find the value for ℓ such that the force exerted by the beam and block on the
wall has no vertical component.
Pa
a
Problem 4
Name code
P
T1
Pb
Pd
Pc
d
T2
b
c
Va
Vd
V
ab
bc
cd
da
Isothermal
Adiabatic
Isothermal
Adiabatic
Vb Vc
The pressure versus volume graph for 1 mole of an ideal monatomic gas undergoing
a cyclic thermodynamic process.
A. When a gas undergoes a sufficiently rapid change, it approaches the ideal
adiabatic process. In such a process (circle all that apply)
a. T = 0.
b. W = 0.
c. Q = 0.
d. U = 0
B.
Show that for the Carnot cycle graphed above Q2  - T2
Q1
T1
Problem 5
Name code
4
E
a.
Consider the circuit shown in the figure.
Calculate the equivalent resistance
between points A and B.
F
6
3
5
C
D
7
2
+
b.
Find the current through the 7 resistor.
A
B
6v
c.
If this circuit is on for 24 hours, find the electrical energy used in the 6 resistor.
Give your answer in kwh.
Problem 6
E=120V/m
Name code
a. Consider the uniform electric field between the parallel
plates A and B as shown in the figure at left. Calculate
the potential difference VA-VB.
P
5 cm
A
5 cm
B
b.
If an electron is released from rest at point P in the figure above, calculate the
kinetic energy it gains by the time it hits one of the plates. Express your
answer in eV.
c.
Which plate does the electron hit? (circle one)
A
B
Problem 7
Name code
v = 5103 m/sec
5 cm
An unknown particle undergoes circular motion
in the presence of a 0.510-4 T magnetic field
directed outward as shown in the figure at right.
Bout = 0.510-4 T
a.
Calculate its charge to mass (q/m) ratio. What is the sign of this charge?
b.
A proton is moving through this region parallel to the direction of the
magnetic field (directed out of the page) at a velocity v = 7.5104 m/sec.
Calculate the magnitude and direction of the force on the proton.
Problem 8
A
B
D
C
Name code
A copper wire (negligible resistance)
is bent into a circular shape of radius
0.5 m. A gap separates the ends B and
D. The radial section BC, fixed in
place, has a resistance of 3. A copper
bar AC sweeps clockwise at an angular
speed of 15 rad/sec. The loop lies in a
uniform magnetic field of 3.810-3 T.
The field is perpendicular to the loop
and points into the page. Find the
magnitude and direction of the current
induced in the loop ABC.
Problem 9
Name code
A muon, created by a collision between a primary cosmic ray and an atmospheric nuclei
at an altitude of 40 km, travels straight down with a speed ~0.99c. Given the mean
lifetime for a muon at rest is 2.2 sec, find the probability that it survives to reach the
earth’s surface
•
classically (i.e., without the effects of time dilation)
•
relativistically, taking into account its time dilation.
Problem 10
Name code
A. The wave function of a hydrogen atom depends on the 3 quantum numbers
n, ℓ, and m. Each is associated with a boundary condition on a 3 dimensional
geometric variable. Identify that variable for each:
n is associated with the boundary condition on ___.
ℓ is associated with the boundary condition on ___.
m is associated with the boundary condition on___.
B.
x, y, z
r, , 
The energy level of a free hydrogen atom depends solely on a single quantum
number, .  is actually which of the above?
_______
C. Each quantized energy level E depends on that quantum number in proportion
to (circle one)
a. 
d. 1/2
b.  2
e. e
c. 1/
f. e-
D. The 1st excited state, n=2 has what possible values of ℓ?
and for each ℓ, what possible values of m?
E.
For one of these states,
zr - Zr / 2 a
Ψ  e
cos
a0
0
Sketch the probability distribution P(r) for this wave function.
Solve for the normalization constant, .
Find the most probable radius for an electron in this state.
Questions A-F: An observer peers into a highly
reflective silvered, spherical Christmas ornament of
radius R as shown below. E marks the ornament’s
center. B and G are points on the surface. D, and F
are halfway between the center and surface.
Problem 11
Name code
R
___ A. The observer sees an ___ image of himself.
a. inverted, enlarged, and real
b. upright, enlarged, and real
c. upright, enlarged, and virtual
d. upright, reduced, and virtual
e. upright, reduced, and real
f. inverted, reduced, and virtual
___ B. The observer’s eye is approximately a distance R from the surface of the ornament.
An image of his eye forms approximately at point
a. A
b. B
c. C
d. D
e. E
f. F
C. Verify your answer by solving for the image’s location algebraically.
D. Graphically confirm your answer by drawing a neat ray diagram in the figure above
showing the formation of an image of any point of the face.
___ E. The image of distant objects behind him, appear somewhere between
a. AB
b. BC
c. CD
d. DE
e. EF
f. FG
___ F. Moving so close his nose touches the ornament, he sees the reflected image of his nose
a. roughly life-size, just behind the surface of the ornament.
b. reduced to a point at the center.
c. reduced to a point halfway between the center and surface.
d. magnified at a point beyond the center of the ornament.
Stopping potential, volts
Problem 12
3
Name code
2
1
Plot from:
Millikan, Phys.Rev.7,362 (1916)
0
30
40
50
43.91013 Hz
60
70
80
90
100
110
120
Frequency ( 1013 Hz)
According to Einstein’s interpretation of the photoelectric effect (Annalen der Physik,
Vol. 17; 1905 ) the slope of the line fit above should be given precisely by what simple
ratio of fundamental physical constants?
What is the photoelectric threshold wavelength, th for the metal Millikan used in the
experiment producing the plot above?
th =
What is the work function  for this metal?
Find the maximum speed of photoelectrons liberated from this metal by visible light
of wavelength 4000A.
Questions A-F: An observer peers into a highly
reflective silvered, spherical Christmas ornament of
radius R as shown below. E marks the ornament’s
center. B and G are points on the surface. D, and F
are halfway between the center and surface.
Problem 11
Name code
R
d
___ A. The observer sees an ___ image of himself.
a. inverted, enlarged, and real
b. upright, enlarged, and real
c. upright, enlarged, and virtual
d. upright, reduced, and virtual
e. upright, reduced, and real
f. inverted, reduced, and virtual
c
___ B. The observer’s eye is approximately a distance R from the surface of the ornament.
An image of his eye forms approximately at point
a. A
b. B
c. C
d. D
e. E
f. F
C. Verify your answer by solving for the image’s location algebraically.
-
2
R
=
1
R
+
1
i
D. Graphically confirm your answer by drawing a neat ray diagram in the figure above
showing the formation of an image of any point of the face.
___ E. The image of distant objects behind him, appear somewhere between
a. AB
b. BC
c. CD
d. DE
e. EF
f. FG
___ F. Moving so close his nose touches the ornament, he sees the reflected image of his nose
a. roughly life-size, just behind the surface of the ornament.
b. reduced to a point at the center.
c. reduced to a point halfway between the center and surface.
d. magnified at a point beyond the center of the ornament.