Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Final Exam Solutions
Problem 1 Coupled oscillators
(k ' + k )/mB .
a) If mA = , then A = 0 and B =
'
b) If k = 0, then mA , mB and k form an isolated system
and the center of mass of th
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #5 Solutions
Problem 5.1 (French 6-12)1 Plucked string
A sketch of the string is shown.
a) Remember that the kinetic energy density of a wave y
2
1
y
dK
h
=
, and the potent
MIT 8.03 Fall 2004 Solutions to Problem Set 4
Problem 4.1 (French 7-12) Travelling pulse
(a) Since the pulse is travelling to the right, the piece of string on the right side of the peak is rising and
the piece on the left is falling. The transverse veloc
8.03 Fall 2005
Problem Set 3 Solutions
Solution 3.1: Take home experiment # 2
Coupled oscillators, resonance, and normal modes
1. How does the tension eect the time it takes for the energy to shift from one mass to the other?
As I, Igor Sylvester, increas
October 5, 2004 mini-Quiz #4 8.03
Your name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Circle your recitation R01 R02 R03 R04 R05
A string of length L is xed at both ends. The tension is T . A very small mass, m, is attached t
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #8 Solutions
Problem 8.1: Nature is in a hurry - Fermats Principle
A
a) See gure.
b) BO = B O and angle BO C = angle B O C .
So B O C is congruent to B O C .
Hence 2 = 3 (Co
MIT 8.03 Fall 2004 Solutions to Problem Set 6
Problem 6.1 Phase and group velocity in your bathtub
The dispersion relation for deep-water waves is given approximately by
2 = gk +
T3
k,
where = 2/.
(a) For very short wavelengths ( 1.7 cm), the k3 term domi
8.03 Fall 2004
Problem Set 2 Solutions
Solution 2.1: Take home experiment # 1 Inuence of mass on the damping of a
pendulum
I (Igor Sylvester) made my pendulum about 65 cm long. I measured the time of 20 complete oscillations of
the pendulum with the heavy
MIT 8.03 Fall 2004 Solutions to Problem Set 5
Problem 5.1 Piano galore
(a) The frequency of the n-th mode of a string is n = n /2 = n T /2L . Dierentiating with respect
to T gives
dn
dT
n
4L
=
=
n
2L
=
1
T
T
1
n
2T
1
2T
We know that n = 1, T = 250 N, C5 =
8.03 Fall 2004
Problem Set 9 Solutions
Solution 9.1: Nature is in a hurry - Fermats Principle
Part (a)
A
B
O
2
1
C
O
3
B
FIG. 1: Problem 9.1(a)
Shown in Fig. 1.
Part (b)
B O = B O and angle B O C = angle B O C . So
B O C is congruent to
2 = 3
1 = 3
1 = 2
September 28, 2004 mini-Quiz #3 8.03
Your name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R01 R02 R03 R04 R05
2 points
A constant current is owing through a resistor in the direction as indicated in the gure. In
B
going from A to B ,
8.03 Fall 2004
Problem Set 7 Solutions
Solution 7.1: Polarized radiation
Part (a)
For angle = /4 from the +y direction
E0
E/4 = cos(t k x)( + z )
y
2
E0
1
k E = x (y + z ) cos(t k x)
c2
E0
z
= cos(t k x)( y )
c2
(1)
B/4 =
(2)
For angle = /4 from the +y di
September 14, 2004 mini-Quiz #1 8.03
Your name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Circle your recitation R01 R02 R03 R04 R05
1. z = a + jb = Aej
a and b are real. Express A and in terms of a and b.
A = (a2 + b2 )0.5
Eu
MIT 8.03 Fall 2005 Solutions to Problem Set 8
Problem 8.1 - Doppler shifts of EM radiation a black-hole X-ray binary
The following figure shows the binary system at two different times.
m1
m2
r2 r1
observer
m2 m1
(a) Since the two orbits are circular, the
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #3 Solutions
Problem 3.1: (French 5-10)1 Coupled Oscillators using two springs
Let the displacement from the equilibrium posi
tions for masses m1 and m2 be x1 and x2 respec
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #2 Solutions
Problem 2.1: Driven oscillator with damping
a) An object of mass m is hung from a spring with spring constant 80 N/m. The resistive damping
force on the object
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #4 Solutions
b) The pulse shape is shown below. We can model the
pulse with a Gaussian function. That is, the pulse resembles
2
y ( ) = Ae where = x v t, A = 0.1 m and = 4 m
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #7 Solutions
Problem 7.1: Speed checked by radar
a) = c/f f = 3 108 /3 102 = 1010 Hz
b) Frequency received by moving car: f
f (1 + ) where = v/c is positive for approaching
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #6 Solutions
Problem 6.1 (Beke & Barrett 3.3)1 Electromagnetic plane waves
a) First note that B = By y . Hence, Bx = Bz = 0. We now proceed by applying Maxwells
equations to
Massachusetts Institute of Technology OpenCourseWare
8.03SC
Fall 2012
Problem Set #9 Solutions
Problem 9.1: (Beke & Barrett 8.1)1 Thin lm interference
The glass is too thick to produce thin lm interference. We will therefore only concentrate on
the air ga
MIT 8.03 Fall 2004 Solutions to Problem Set 11
Problem 11.1 - Take-home experiment #9 Transmission grating
Using a mini-maglite I (Igor Sylvester) was able to see the 0th, 1st and 2nd order spectra clearly. I observed that for both the 1st and 2nd order s
September 21, 2004 mini-Quiz #2 8.03
Your name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Circle your recitation R01 R02 R03 R04 R05
An object with a massm of 100 grams is attached between two massless springs,
each with a spr
8.03 Fall 2005
Problem Set 10 Solutions
Solution 10.1: (Beke & Barrett 8.1) Thin lm interference
a
b
c
1
n1
D
A
2
d
n2
r
C
1
n1
FIG. 1: Problem 10.1 Thin lm interference
The glass is too thick to produce thin lm interference. We will therefore only concen