Mathematics 55
Triple Integrals in Spherical Coordinates
Spherical Coordinate System
Let P (x, y, z) be a point in the 3D space and P 0 Examples.
be its projection onto the xy-plane.
in rectangular coordinates
1. 2, ,
4 3
2. (0, 2 3, 2) in spherical coo
5/10/2016
ES 12 Lecture 16
INTEGRATION OF
METHODS FOR RIGID
BODY KINETICS
FORCE-INERTIA, WORK-ENERGY, IMPULSE-MOMENTUM
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid Bodies Kine
2/10/2016
Lesson Objectives
At the end of the lecture, you are expected to:
Lecture 4
RELATIVE AND
DEPENDENT
MOTION
1. Distinguish relative motion from absolute motion,
2. Relate the absolute and relative velocities of two
particles, and
3. Relate the abs
8/16/2016
Uniform Rectilinear Motion
For a particle in uniform rectilinear motion, the
acceleration is zero and the velocity is constant.
ES 12 Dynamics of Rigid BodiesKinematics of Particles (by F.P. Beer and E.R.
Kinematics of Particles (by F.P. Beer an
_
ES 12 LEO 11
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4/26/2016
ES 12 Part 2: Lecture 13-15
KINETICS OF
PARTICLES AND
RIGID BODIES II
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid Bodies Kinematics of Particles (by F.P. Beer and E
3/16/2016
Introduction
ES 12 Part 2: Lecture 9-12
KINETICS OF
PARTICLES AND
RIGID BODIES I
Kinetics of
Particles
Force-Inertia
(Newtons 2nd
Law of Motion)
ImpulseMomentum
Work-Energy
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. J
3/9/2016
Lecture 8
GENERAL PLANE MOTION:
Lesson Objectives
At the end of the lecture, you are expected to:
POLE METHOD FOR
ACCELERATION
ANALYSIS
1. Relate the absolute and relative accelerations of various
points on a rigid body in plane motion, with the
3/8/2016
Lecture 7
GENERAL PLANE MOTION:
Lesson Objectives
At the end of the lecture, you are expected to:
INSTANTANEOUS
CENTER METHOD
FOR VELOCITY
ANALYSIS
1. Locate the center of rotation of a rigid body at a given
instant
2. Determine the angular veloc
4/14/2016
Introduction
ES 12 Part 2: Lecture 9-12
KINETICS OF
PARTICLES AND
RIGID BODIES I
Kinetics of
Rigid Bodies
Force-Inertia
(Newtons 2nd
Law of Motion)
ImpulseMomentum
Work-Energy
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R
Lecture 6
GENERAL PLANE MOTION:
POLE METHOD FOR
VELOCITY ANALYSIS
Engr. Maxell P. Lumbera
Kinematics of Rigid Bodies (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid Bodies Kinematics of Particles (by F.P. Beer and E.R.
Second
2/16/2016
Lesson Objectives
At the end of the lecture, you are expected to:
ES 12 Part 2: Lecture 5-8
KINEMATICS OF
RIGID BODIES
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid B
4/7/2016
Introduction
ES 12 Part 2: Lecture 9-12
KINETICS OF
PARTICLES AND
RIGID BODIES I
Kinetics of
Rigid Bodies
Force-Inertia
(Newtons 2nd
Law of Motion)
ImpulseMomentum
Work-Energy
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R.
Lecture 0
CLASS
POLICIES
ES 12 WBC1
ES 12 Dynamics of Rigid Bodies Kinematics of Particles (by F.P. Beer and E.R.
Second Semester AY 2015-2016
Johnston, Notes by J.W. Oler)
1
Class Policies
1. ATTENDANCE. Attendance is required!
2. Any form of intellectua
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3/31/2016
ES 12 Part 3: Lecture 9-12
KINETICS OF
PARTICLES AND
RIGID BODIES I
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid Bodies Kinematics of Particles (by F.P. Beer and E.R
5/5/2016
Introduction
ES 12 Part 2: Lecture 13-15
KINETICS OF
PARTICLES AND
RIGID BODIES II
Kinetics of
Rigid Bodies
Force-Inertia
(Newtons 2nd
Law of Motion)
ImpulseMomentum
Work-Energy
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.
Mathematics 55
1st Semester, A.Y. 2016-2017
THQ
Homework 4
Work independently. Do not consult anyone except your instructor. Provide neat and clear solutions.
Due: 18 October 2016 (Tuesday) at the start of the class. Use yellow pad for your work.
Do as in
Mathematics 55
1st Semester, A.Y. 2016-2017
THQ
Homework 3
Work independently. Do not consult anyone except your instructor. Provide neat and clear solutions.
Due: 04 October 2016 (Tuesday) at the start of the class. Use yellow pad for your work.
1. Let G
Mathematics 55
Exercises 8 - Triple Integrals I
Do as indicated.
1. Evaluate the following integrals.
ZZZ
(a)
(xy + z 2 ) dV where G = cfw_(x, y, z) : 0 x 2, 0 y 1, 0 z 3
ZG
ZZ
12xy 2 z 3 dV where G = [1, 2] [0, 3] [0, 2]
(b)
Z
G
0
Z
2e Z /3
y ln z tan x
Mathematics 55
Triple Integrals
each of length zk .
Recall.
b
Z
1. Single Integral:
Thus, the volume of each subbox is
f (x) dx
a
Vijk = xi yj zk .
Generally, this represents the net signed area of the
region bounded by the curve y = f (x) and the xaxis o
Name:
SN:
EEE 13 Quiz 2
CGI, Database Concepts, and SQL
Instructions: Write all your answers on the blank space provided.
1. Why do we assign a name attribute to <input/> tags? Why should name be unique
across multiple <input/>s?
-> We assign a name to <i
Mathematics 55
Exercises 10 - Vector Fields, FTLI
I. Curl and Divergence
1. Find the curl and divergence of F~ (x, y, z) = hex sin z, yex , z tan yi at the origin.
2. Given F~ (x, y, z) = hz y, x z, x yi, compute for div F~ and curl F~ .
3. Consider the v
Mathematics 55
Exercises 11 - Line Integrals, Greens Theorem
I. Line Integrals
2
Z
t
~
1. Evaluate
2y ds, where C is the curve described by R(t)
=
, t , t [0, 1].
2
C
Z
t3 2
~
~
~
~
2. Evaluate
F dR, where F (x, y, z) = h2x, 0, 2zi and C is the curve de
Mathematics 55
Exercises 9 - Triple Integrals II
I. Cylindrical Coordinates
1. Use cylindrical coordinates to find the volume of the solid in the first octant bounded by the planes
y = 2z, x = 0, z = 0 and the cylinder x2 + y 2 = 4.
2. Suppose that G is t
UNIT I
ENGINEERING ETHICS
Syllabus: Senses of 'Engineering Ethics' - variety of moral issues - types of inquiry - moral dilemmas - moral
autonomy - Kohlberg's theory - Gilligan's theory - consensus and controversy Models of Professional Roles theories abo
EEE 51 Assignment 2
2nd Semester SY 2016-2017
Due: 5pm Tuesday, February 14, 2017 (Rm 220)
Instructions: Write legibly. Show all solutions and state all assumptions. Write your full name, student number,
and section at the upper-right corner of each page.
MARK CEDRIC G. STA. ROMANA
20110109245
Unit L, 56 Valencia Townhouse, Valencia Street, Brgy.
Valencia, New Manila, Quezon City.
09175362260
[email protected]
WORK BACKGOUND
Company Name
Phitelco Information
Technology Services
Julia Vargas, Makati
4/26/2016
ES 12 Part 2: Lecture 13-15
KINETICS OF
PARTICLES AND
RIGID BODIES II
Engr. Maxell P. Lumbera
Kinematics of Particles (by F.P. Beer
and E.R. Johnston, Notes by J.W. Oler)
ES 12 Dynamics of Rigid Bodies Kinematics of Particles (by F.P. Beer and E
2/3/2016
Lesson Objectives
At the end of the lecture, you are expected to:
Lecture 2
CURVILINEAR
MOTION
1. Define tangential and normal components of
acceleration at each position,
2. Resolve the total acceleration into normal and tangential
components; o
#68, Roquid, Pauline Mae R.
2015-90339
Sept 6, 2017
Reaction Paper for Lorenzos Oil
Life has meaning only in struggle. Triumph or defeat is in the hands of God So, let us
celebrate the struggle! This quote that appeared at the very start of the movie desc
INTERNATIONAL FLAG FOOTBALL RULES
5 on 5 / non-contact
2015
Flag Football Rules 2015
Content
Flag Football Rules . 3
National Changes . 3
Diagram of Field . 4
Rule 1 - Game, Field, Ball and Equipment . 5
Rule 2 - Definitions . 6
Rule 3 - Periods
