22
Chapter 2. Sequences and Series
as desired.
Exercise 2.3.3. Let > 0 be arbitrary. We must show that there exists an N
such that n N implies |yn l| < . In terms of -neighborhoods (which are a
bit easier to use in this case), we must equivalently show yn
PLANAR KINETICS OF A RIGID BODY:
CONSERVATION OF ENERGY (Section 18.5)
Todays Objectives:
Students will be able to:
In-Class Activities:
a) Determine the potential
Check homework, if any
energy of conservative
Reading quiz
forces.
b) Apply the principle
PLANAR KINETICS OF A RIGID BODY: WORK AND
ENERGY (Sections 18.1-18.4)
Todays Objectives:
Students will be able to:
In-Class Activities:
a) Define the various ways a
Check homework, if any
force and couple do work.
b) Apply the principle of work Reading q
PLANAR KINETIC EQUATIONS OF MOTION:
TRANSLATION (Sections 17.2-17.3)
Todays Objectives:
In-Class Activities :
Students will be able to:
a) Apply the three equations of
Check homework, if any
motion for a rigid body in
Reading quiz
planar motion.
Applic
EQUATIONS OF MOTION: GENERAL PLANE MOTION
(Section 17.5)
Todays Objectives:
Students will be able to analyze
the planar kinetics of a rigid
In-Class Activities:
body undergoing general plane
Check homework, if any
motion.
Reading quiz
Applications
Equ
EQUATIONS OF MOTION: ROTATION ABOUT A
FIXED AXIS (Section 17.4)
Todays Objectives:
Students will be able to
analyze the planar kinetics of
a rigid body undergoing
rotational motion.
In-Class Activities:
Check homework, if any
Reading quiz
Applications
MASS MOMENT OF INERTIA (Section 17.1)
Todays Objectives:
Students will be able to
Determine the mass moment
of inertia of a rigid body or a
system of rigid bodies.
In-Class Activities:
Check homework, if any
Reading quiz
Applications
Mass moment of in
RELATIVE MOTION ANALYSIS: ACCELERATION
Todays Objectives: (Section 16.7)
Students will be able to:
a) Resolve the acceleration of a
point on a body into
components of translation and
rotation.
b) Determine the acceleration of a
point on a body by using a
RELATIVE MOTION ANALYSIS: VELOCITY (Section 16.5)
Todays Objectives:
Students will be able to:
a) Describe the velocity of a rigid
body in terms of translation
and rotation components.
b) Perform a relative-motion
velocity analysis of a point on
the body.
INSTANTANEOUS CENTER (IC) OF ZERO VELOCITY
(Section 16.6)
Todays Objectives:
Students will be able to:
In-Class Activities:
a) Locate the instantaneous
center (IC) of zero velocity. Check homework, if any
b) Use the IC to determine the Reading quiz
veloci
PLANAR KINETICS: IMPULSE AND MOMENTUM
(Sections 19.1-19.2)
Todays Objectives:
Students will be able to:
In-Class Activities:
a) Develop formulations for the
Check homework, if any
linear and angular
Reading quiz
momentum of a body.
b) Apply the principl
CONSERVATION OF MOMENTUM (Section 19.3)
Todays Objectives:
Students will be able to:
a) Understand the conditions
for conservation of linear and
angular momentum
b) Use the condition of
conservation of linear/
angular momentum
In-Class Activities:
Check
31
2.6. The Cauchy Criterion
2.6
The Cauchy Criterion
Exercise 2.6.1. (a) (1, 1/2, 1/3, 1/4, 1/5, 1/6, . . .)
(b) (1, 2, 3, 4, 5, 6, . . .)
(c) Impossible, if a sequence is Cauchy then Theorem 2.6.4 tells us that it
converges and Theorem 2.5.2 says that s
28
Chapter 2. Sequences and Series
then we can see that xn xn+1 is positive because x2n 2. Because we have
shown that (xn ) is decreasing and bounded below, we may set x = lim xn =
lim xn+1 . Taking limits across the recursive equation we find
x = lim xn+
2.4. The Monotone Convergence Theorem and a First Look atInfinite Series25
Exercise 2.3.12. (a) Intuitively speaking,
lim am,n
m,n!1
should be a number that is arbitrarily close to the values of am,n when m and
n are both large. However, with two index va
46
Chapter 3. Basic Topology of R
The sequence (an ) converges to x, so by Definition 2.2.3B, every -neighborhood
V (x) contains all but a finite number of the terms of (an ). Since (an ) is contained in A, this means that V (x)\A is non-empty and contain
37
2.7. Properties of Infinite Series
P
This means that eventually an < (L + 1)/n2 . We know that the series
1/n2
converges,
and by the Algebraic Limit Theorem for series (Theorem
P
P 2.7.1),
(L + 1)/n2 converges as well. Thus, by the Comparison Test
an m
34
Chapter 2. Sequences and Series
So, by virtue of the fact that (an ) ! 0, we can choose N so that m N implies
|am | . But this implies
|sn sm | = |am+1 am+2 + an | |am+1 | <
whenever n > m N , as desired.
(b) Let I1 be the closed interval [0, s1 ]. Th
40
Chapter 2. Sequences and Series
P1
Exercise 2.8.3. (a) As we have been doing, let bi = j=1 |aij | for all i 2 N.
P1
Our hypothesis tells us that there exists L 0 satisfying i=1 bi = L. Because
we are adding all non-negative terms, it follows that
tmn =
43
2.8. Double Summations and Products of Infinite Series
is bounded can be used to directly prove that it converges. We will use this
method. Let
1
1
X
X
|ai | = L and
|bj | = M.
i=1
j=1
For each fixed i 2 N, the Algebraic Limit Theorem allows us write
P
Chapter 2
Sequences and Series
2.1
Discussion: Rearrangements of Infinite Series
2.2
The Limit of a Sequence
Exercise 2.2.1. a) Let > 0 be arbitrary. We must show that there exists an
N 2 N such that n N implies | 6n21+1 0| < . Well,
1
1
=
0
6n2 + 1
6n2
RELATIVE MOTION ANALYSIS: ACCELERATION
Todays Objectives: (Section 16.7)
Students will be able to:
a) Resolve the acceleration of a
point on a body into
components of translation and
rotation.
b) Determine the acceleration of a
point on a body by using a
ABSOLUTE MOTION ANALYSIS (Section 16.4)
Todays Objective:
Students will be able to
determine the velocity and
acceleration of a rigid body
undergoing general plane
motion using an absolute
motion analysis.
In-Class Activities:
Check homework, if any
Rea
MOMENT OF A COUPLE (Section 4.6)
Todays Objectives:
Students will be able to
a) define a couple, and,
In-Class activities:
b) determine the moment of a couple.
Check homework, if any
Reading quiz
Applications
Moment of a Couple
Concept quiz
Group pr
MOMENT ABOUT AN AXIS (Section 4.5)
Todays Objectives:
Students will be able to determine the moment of a
force about an axis using
a) scalar analysis, and
In-Class Activities:
b) vector analysis.
Check Home work, if any
Reading quiz
Applications
Scala
MOMENT OF A FORCE (Section 4.1)
Todays Objectives :
Students will be able to:
a) understand and define moment,
and,
b) determine moments of a force in
2-D and 3-D cases.
Moment of a
force
In-Class Activities :
Check homework,
if any
Reading quiz
Applic
EQUILIBRIUM OF A PARTICLE IN 2-D
Todays Objectives:
Students will be able to :
a) Draw a free body diagram (FBD), and,
In-Class Activities:
b) Apply equations of equilibrium to solve Reading quiz
a 2-D problem.
Applications
What, why and how of a
FBD
E
DOT PRODUCT (Section 2.9)
Todays Objective:
Students will be able to use the dot product to
a) determine an angle between two vectors,
and,
b) determine the projection of a vector along a
In-Class Activities:
specified line.
Check homework
Reading quiz
POSITION & FORCE VECTORS (Sections 2.7 - 2.8)
Todays Objectives:
Students will be able to :
a) Represent a position vector in Cartesian
coordinate form, from given geometry.
b) Represent a force vector directed along In-Class Activities:
Check homework
a
3 D VECTORS (Section 2.5)
Todays Objectives:
Students will be able to :
a) Represent a 3-D vector in a
Cartesian coordinate system.
b) Find the magnitude and
coordinate angles of a 3-D vector
c) Add vectors (forces) in 3-D
space
In-Class Activities:
Read