Line elements in the plane
REFERENCE: Hartle
In order to understand and describe general
relativity, a basic knowledge of differential
geometry is handy.
We begin with a look at
the Euclidean plane, which can be described
using Cartesian coordinates (x,y)
Line elements in the plane
PROBLEM SET
PROBLEM 1. Show that if x = r cos and y = r sin , then dx = dr cos r sin d and dy = dr sin + r cos d.
PROBLEM 2. Show that if the results of PROBLEM 1 are substituted into
dS2 = dx2 + dy2, then dS2 = dr2 + r2d2. What
Tides
PROBLEM SET
PROBLEM 1. To get an idea of the size and direction of the tidal force,
consider that the average earth-moon distance is R = 3.82108 m and the
average radius of the earth is r = 6.37106 m, so that R = 60r. If we
let r = 1, then R = 60. S
002-550
Fiberglass Boat Repair & Maintenance
epairing, restoring and prolonging the life of fiberglass boats with WEST SYSTEM Epoxy
Contents
1 Introduction
Understanding fiberglass boat construction and using W EST SYSTEM Epoxy for repair
2 Repairing Mino
Electric Power Annual 2011
Released: January 2012
Revised: October 2012
Next Update: November 2012
Table ES1. Summary Statistics for the United States, 1999 through 2010
Description
Net Generation (thousand
megawatthours)
Coal[1]
Petroleum[2]
Natural Gas[
Name_Per_
ALGEBRA 1 WEEKLY PACKET Week 2 (August 22 26, 2011)
Due:
August 29, 2011
Mrs. Hoang Yerba Buena High School
Classwork & Homework (50 points)
Depend on whether you are assigned to GROUP A or GROUP B, you will do the following problems under the g
User name: Linda Hoang Book: Intermediate Algebra: Functions & Authentic Applications, Fourth Edition Page: 30. No part of any book may be reproduced or transmitted by any
means without the publisher's prior permission. Use (other than qualified fair use)
YERBA BUENA HIGH SCHOOL
ALGEBRA 1
2011 2012
Mrs. HOANG
YERBA BUENA HIGH SCHOOL
ROOM #244
EMAIL: [email protected]
WEB: lindahoang.wikispaces.com
I. OBJECTIVES
1. Meet all Californias standards for ALGEBRA 1 (see cde.ca.gov under standards & frameworks)
2.
Stellar model 3: The polytrope model
REFERENCE: Bowers and Deeming
Suppose a star of mass M and radius R has its pressure proportional to
some arbitrary power of its density. Then
[Eq. 1]
P = K.
To analyze such a model we need the mass-continuity equation
Binary math
Question 1:
Counting practice: count from zero to thirty-one in binary, octal, and hexadecimal:
Question 2:
Add the following binary numbers:
Question 3:
If the numbers sixteen and nine are added in binary form, will the answer be any differen
Name_Per_
ALGEBRA 1 WEEKLY PACKET Week 1 (August 16 19, 2011)
Due:
August 22, 2011
Mrs. Hoang Yerba Buena High School
Classwork & Homework (50 points)
Below you will find a set of problems based on the basic math curriculum you have learned previously. We
Tides
REFERENCE: Barger & Olsson
Consider the gravitational forces acting on a mass element m on the
surface of the earth. The earth itself exerts a force to draw m into a
perfectly
spherical
configuration.
But
concurrent
with
that
spherically-symmetric f
Energy release in a supernova explosion
REFERENCE: None
When our sun finally runs out of nuclear fuel, the radiation pressure
that was opposing the sun's self-gravitational pressure will for the
most part cease. Gravity will then succeed in doing what it
Name_Per_
September 19, 2011
ALGEBRA 1 WEEKLY PACKET Week 5 (September 12 16, 2011)
Due:
CLASSWORK (50 points)
What we will do IN-CLASS:
Notes: statement of logics (hypothesis & conclusion); counterexamples (Mon)
Classwork: merge two signs; add/subtract i
Name_Per_
September 6, 2011
Due:
ALGEBRA 1 WEEKLY PACKET Week 3 (Aug. 29 Sept. 2, 2011)
Mrs. Hoang Yerba Buena High School
Classwork & Homework (50 points)
Depend on whether you are assigned to GROUP A or GROUP B, you will do the following problems under
Threshold mass of a neutron star
REFERENCE: None
Consider a star of mass M and radius R, and for simplicity assume that
the star is uniform in density , and that all parts rotate with the
same period. Since density and volume are given by
M
4
= V , and V
Stellar model 1: Constant density model
REFERENCE: Bowers and Deeming
The simplest form a star of mass M and radius R could take is that of a
constant density star in hydrostatic equilibrium.
To analyze such a
model we need the mass-continuity equation, a
Black holes - a classical look
REFERENCE: None
Let m be a mass in the near vicinity of the surface of M (of planetary
size). Recall that the potential energy stored in the configuration of
the two masses m and M is given by
[Eq. 1]
U = -GMm/r.
Since the t
The degenerate dwarf (white dwarf)
REFERENCE: Shu
At the end of the "normal" lifetime of a typical star, nuclear fuel will
run out, and the radiation pressure caused by the energy released by
fusion will decrease and eventually cease.
What will happen as
Stellar model 3: The Lane-Emden solutions
REFERENCE: Bowers and Deeming
In the last lesson we arrived at the second order differential LaneEmden equation of index n:
_1 d d
[Eq. 1]
d2
= -n,
2 d
with the boundary conditions
[Eq. 2]
= 1, = 0
d
= 0, = 0.
d
The degenerate dwarf (white dwarf)
PROBLEM SET
Consider the notes given at the end of the lecture.
they are:
(1)
When P 5/3 the exact formula Eq. 9 is
c = 5.99<> where <> 3M/4R3.
(2)
When P 5/3 the exact formula Eq. 9 is
Pc = 0.770GM2/R4.
(3)
The exact fo
Black holes - a classical look
PROBLEM SET
PROBLEM 1. Use the equation mv2/2 - mv02/2 + (-GMm/r) - (-GMm/R) = 0,
and the conditions v0 vesc, r = , and v = 0, to prove that
vesc =
2GM
R
1/2
.
PROBLEM 2. Suppose we define the Schwarzschild radius RSch to be
Stellar model 3:
PROBLEM SET
The polytrope model
1 r2 dP
.
G dr
PROBLEM 1.
Show that m(r) = -
PROBLEM 2.
dm
Show thatdr
PROBLEM 3.
1 d r2dP
Show thatG dr dr
PROBLEM 4.
Show that dP/dr = K-1d/dr.
1 d r2dP
= - G dr dr
.
= -(r)4r2.
d
1 d
r2K-2
= -4G. Which t
The equation of state of a degenerate gas
REFERENCE: Shu
We begin with three important equations: Two from classical physics, and
one from quantum physics.
[Eq. 1]
PV = NkT
3
[Eq. 2]
K = 2 kT
[Eq. 3]
(p)(r) > h
Eq. 1 is the perfect gas law of chemistry, w
A quick look at the sun
REFERENCE: None
Treating the sun as a self-gravitating mass M of radius R, its selfforce is just
[Eq. 1]
F GMM/R2.
Since pressure P = F/A and since A R2, from Eq. 1 we get
[Eq. 2]
P GM2/R4.
From the perfect gas law we have
[Eq. 3]
Energy release in a supernova explosion
REFERENCE: None
PROBLEM 1. What
pressure
gravitational collapse?
source
will
stop
the
sun's
ultimate
PROBLEM 2. What pressure source will stop a neutron star's ultimate
gravitational collapse? How many solar masses
Boolean Logic Review
Question 2:
Identify each of these logic gates by name, and complete their respective truth tables:
Question 2:
Convert the following logic gate circuit into a Boolean expression, writing Boolean sub-expressions next to each gate outp