4/15/16
Shield: Herakles in Battle with
Amazons c. 700 BCE
Achilles and Penthesileia Amphora by Exekias, ca. 530 bce
Achilles Kills
Penthesileia,
Attic Kylix by
Penthesileia
Painter, c. 455
BCE
1
4/15/16
Amazons Preparing for Battle Attic red-figure
4/15/16
Theseus and Amazons Attic r-f squat lekythos425-400bce
Battle of Greeks and Amazons Temple at Bassae ca. 420bce
Antiope abducted by Theseus with Perithous
Attic amphora by Myson ca. 500bce
Greeks vs. Amazons: Replica of Pheidias Shield for Athena
4/15/16
Venus of
Willendorf
Paleolithic
28,000-
25,000 BCE
Naturhistorisches
Museum
Vienna
Venus of
Laussel
20,000-
18,000 BCE
Muse
dAquitaine
de la ville
de Bordeaux
Cycladic
Figurine
Neolithic
2800-2300
BCE
Cycladic
Figurine
Neolithic
3200-2800
BCE
Cycl
4/15/16
Scylla 5th c. bce from Melos
British Museum
Scylla Paestan red-figure krater 4th c. BCE J. Paul Getty Museum, Malibu
Skylla
1
4/15/16
Blinding of Polyphemos
Odysseus and Calypso red-
figure 5thc. bce
2
4/15/16
Statuette of young Spartan female athlete British Museum
Kouros
530 bce
Odysseus and Nausikaa. Athenian rf pyxis lid (Aison). Ca. 430
BCE. Boston.
1
4/15/16
Odysseus, Athena and
Nausicaa 475-25bce Etruria
Odysseus and Kirke. Athenian bf lekythos
4/15/16
Penelope, Roman
copy of a Greek statue
(original: Classical: 450 BCE)
[Vatican]
Telemachus and Penelope at the loom Attic red-figure
Skyphos Ca. 440BCE Chiusi, Italy
Telemachos and Penelope
Scenes from the Odyssey, Attic rf. vase
(Classical: 440
PHYSICS W3003: PHASE-PLANE PLOTS
1. Introduction and simple examples
1.1. The basic idea. Phase-plane plots are a very useful way of understanding qualitative
features of the solution of ordinary dierential equations, especially equations for a single
var
PHYSICS 3003: CONSERVATION OF ANGULAR MOMENTUM AND
MOTION OF RIGID BODIES
AJM
Because the discussion today in class was a little rushed I amplify my remarks here.
1. Generalities
Rigid Bodies: In a rigid body the position of every particle is xed if we sp
PHYSICS W3003: MATHEMATICAL BASICS
1. Differential equation basics
1.1. Example. Newtons equation for the motion of a particle,
d2 R(t)
= F(R, t)
dt2
is an example of a dierential equation, an equation relating one or more derivatives (in
this case the se
SOLUTIONS, MIDTERM 1, SPRING 2014
Problem 1
a) U (x) = 2k x. The plot of U is shown in the left panel
(b) all orbits are bound, there are no separatrices, unbound orbits or xed points.
2
3.0
2.5
1
2.0
0
1.5
1.0
1
0.5
2
2
4
6
8
0.0
10
0.5
1.0
1.5
2.0
Probl
PHY 3003 SPRING 2013
Solutions to Problem Set 4
1
(a) The energy is
m2
x + x2 + x2 + mgz
2
In polar coordinates referenced to the suspension point, with = meaning the mass is
directly below the suspension point, and noting that the cord length L is time-i
SOLUTIONS PROBLEM SET 6
W3003 SPRING 2014
(1) This oscillator is overdamped (natural frequency 0 = 1 and damping
general homogeneous solution xh is thus
xh (t) = A+ et/+ + At/
e
with
1
1
1
2
=
0 = 2 3
2
while a particular solution (as you can verify by s
PHY W3003 SPRING 2013
Midterm II solutions
(1a) Kinetic energy
(1)
KE =
m2
x + z2
2
but if z = kx2 then
(2)
z = 2kxx
so
(3)
KE =
m2
x + 4k 2 x2 x2
2
Potential energy = mgz = mgkx2 so
m2
(4)
L=
x + 4k 2 x2 x2 mgkx2
2
(1b)
(5)
m
d
x + 4k 2 x2 x + 2mgkx 4mk
Addition mistake!
I=5Ma^2/4
Thus w=(Sqrt[3]/10)(v/a)
Solution: problem 5, set 2
(a)
(i) Dene the y axis to the axis coming out of the plane of the gure so the component
of angular momentum in question is Ly . This is not conserved because the force is alo
SOLUTIONS PROBLEM SET 7
W3003 SPRING 2014
(1) (a) Using g = 10m/sec2 and h = 51 03 m we have v 2.3m/s. The distance across
the Pacic from say Tokyo to San Franciso 8000km so the time is about 36, 000
sec or about 10 hours.
(b) The numerator is the arc len
Carbon Cycle
Problem Set 11 Solutions
1) (We skipped this problem)
2)
CO32- (U/Ca)
CO -3-now
U / Canow
3.2 ppm
=
=
= 1 / 2 0.7
-CO 3 Miocene
U / CaMiocene
6.4 ppm
CO -3-now
CO -3-Miocene
= 121mol / kg
0.7
Ca + + now CO3 now = Ca + + Miocene CO3 Miocene
C
Carbon Cycle
Problem Set 11
Due: Thursday, April 24, 2008
1) If at some time in the past, the CO2 content of the atmosphere was three times lower
than todays and the earths tropical temperature was 8 colder than todays (i.e. 18 C
instead of 26C), roughly
Carbon Cycle
Homework 10 Solutions
1. If f is the fraction of C deposited as calcium carbonate, then (1-f) is the fraction
deposited as organic matter. We can then write the equation:
f (2) + (1 f )(28) = 5
Solving for f :
f = 0.77
So calcium carbonate de
Carbon Cycle
Homework 10
Due: Thursday, April 17, 2008
1. If you were to measure the 13C of organic matter and calcium carbonate in a given
rock layer and got the following values, what would be the proportions of organic
matter and calcium carbonate buri
Carbon Cycle
Homework 9 Solutions
1. Your answers will vary depending on the data and year you use. A correct answer
will take the total number of students enrolled in a year (for all undergraduates,
this number is somewhere between 7000 and 8000) and div
Carbon Cycle
Problem Set #9
Due: Thursday, April 10, 2008
For these problems you will need to use the following equation:
Residence time = (Total inventory)/(input or output rate)
where total inventory is the total amount in the reservoir, andinput or out
The Earths Carbon Cycle - Spring 2008
Problem Set 8 Solutions
1.
The boron isotopic composition of borate flattens out at lower pH than at present, making
11
B a less sensitive indicator of pH. Since lower ocean pH corresponds to higher pCO2, this
means