1. Find the equation of the line through points (1, 3) and (1, 4). (Write the equation
in slope-intercept form).
2. (a) What is the slope of the line dened by the equation
x + 2y = 6
(b) What is the y-intercept of that line?
(c) On the grid below, draw a
inversely 74 Chip formation
fundamentals Fig. 2.27 (a) Defining the
penetration distance l p of the lubricant
into the rear of the contact region and
(b) derived feed/speed regions of
complete and negligible penetration, for
oxygen Childs Part 1 28:3:2000
the machining process, it would be
desirable to be able to model both the
form of the variation of t with sn, in
terms of the coefficients m and m; and
also to understand what determines the
values of m and m. The most simple
friction model is to neglect
rake face heating is dominant and a
temperature in excess of 800C is
estimated at the cutting speed of 150
m/min. 3.1.2 Effects of pre-strain and
rake angle in machining copper In the
previous section, the machining of
annealed metals by a 6 rake angle to
temperatures. The body centred cubic
carbon and alloy steels form an
intermediate group. The behaviours of
these three different groups of alloys are
considered in Sections 3.1.3 to 3.1.5 of
this chapter, after sections in which the
machining of unalloyed
= s (2.25b) An Friction, lubrication
and wear 69 Fig. 2.24 (a) The sliding
contact of a chip on a tool; (b) schematic
dependence of A r /An on n/k Childs
Part 1 28:3:2000 2:37 pm Page 69 and its
size relative to k, t/k, is t Ar s =
(2.25c) k An k It is e
specific forces, up to 8 GPa for iron and
nickel and around 4 GPa for copper and
aluminium. These forces are some ten
times larger than the expected shear
flow stresses of these metals (Figure 3.1)
and arise from the very low shear plane
angles, between 5
roughnesses for inserts of these three
types found in the authors workshops. It
also records values of 10klocal/E*
calculated from data in Table 2.3, taking
klocal = mk, E for high speed steel tools
to be 210 GPa and for cemented
carbides to be 550 GPa. I
for oxygen. Although it is certain that
there can be no lubrication in the no
lubrication region of Figure 2.27, it is
not certain that there will be lubrication
in the possible lubrication region.
Whatever penetrates the channels must
also have time to r
to be on the chip. However, for large
values of s/klocal, both regions have
almost the same upper boundary, with c
(equation (2.26) approximately equal to
1. One would then expect s m
(2.28) klocal In those circumstances,
when m is measured to be < 1, th
rake face, and of the work past the flank,
causes the tool to wear away. Tool wear
will be considered in detail in Chapter 4.
Here, the purpose is briefly to review
knowledge of wear from other studies,
to create a standard to which tool wear
can be relat
(Figure 3.8 describes the machining of an
annealed brass. After cold-working, even
higher shear plane angles, and lower
specific forces are obtained.) These two
examples are ones where the reason for
the easier machining of the alloys
compared with the el
concept of the plasticity index (E*/k)Dq,
where k is the local shear stress of the
asperity, E* is an average Youngs
modulus for the Friction, lubrication and
wear 71 Childs Part 1 28:3:2000 2:37 pm
Page 71 asperity and tool material: 1/E*
= 1/Easperity +
speculation about departures from these
assumptions that could lead to larger
values of c and to m > 1. All proposals
require the shear strength of the
junction to be maintained while the
normal stress is unloaded. It is certain
that, for this to occur, t
If these equations were being derived
today, there would be discussion as to
whether the real contact pressure was H
(equivalent to 5k) or only to k (Section
2.4.1 and Appendix 3). However, such
discussion is pointless. It is found that
the K values so de
lightest loadings, when both an asperity
and the chip below it remain elastic
(range EE), displacements are taken up
by elastic compression. If the asperity
becomes plastic but the chip below it
remains elastic (range PE), plastically
displaced material f
For typical work plasticity index values of
about 20 (Table 2.4), this would place
the tool value at about 4, in the elastic
range of Figure 2.30. The mechanisms
available for tool wear are likely to be
fatigue and chemical reaction (oxidation)
with the a
However, it must be acknowledged that
understanding of this does not yet exist
in sufficient detail to be able
quantitatively to predict friction laws
from first principles. The following
sections may be omitted at a first
reading. Plastic asperities on a
717727. Childs, T. H. C. (1980a) Elastic
effects in metal cutting chip formation.
Int. J. Mech. Sci. 22, 457466. Childs, T.
H. C. (1980b) The sliding wear
mechanisms of metals, mainly steels.
Tribology International 13, 285293 .
Childs, T. H. C. (1988) Th
1. Here
m=
43
1
=
1 1
2
So our equation is
1
y 3 = (x 1)
2
1
1
y = x+ +3
2
2
1
7
y = x+
2
2
2. (a) We can re-write this equation as
1
y = x+3
2
so the slope is 1
2
(b) The y-intercept is 3.
(c) Heres a graph of the line:
3. (a) Heres the data using the li
1. You are playing a unique game of Rock-Scissors with a friend. If you both throw
rock, he pays you two dollars and if you both throw scissors you pay him two
dollars. If you throw rock and he throws scissors, he pays you a dollar, and if he
throws rock
1. Find the equation of the line through points (2, 2) and (4, 4). (Write the equation
in slope-intercept form).
2. (a) What is the slope of the line dened by the equation
4x 2y = 6
(b) What is the y-intercept of that line?
(c) On the grid below, draw a g
1. Suppose I have 4 solid nail polish colors, and 6 sparkly nail polish colors.
(a) How many ways can I color the nails on my left hand if I am allowed to repeat
colors?
(b) How many ways can I color the nails on my left hand if I am not allowed to
repeat
Math 3 HW1
Due Friday, Sept 7
Remember that your write-up should be polished and complete (and thus likely not
a rst draft). A Finite Mathematics student (not necessarily in our class) should be able
to read your write-up, understand the question, and und
1. Suppose I have 4 solid nail polish colors, and 6 sparkly nail polish colors.
(a) How many ways can I color the nails on my left hand if I am allowed to repeat
colors?
105
(b) How many ways can I color the nails on my left hand if I am not allowed to
re
Year
Data
GDP of Egypt (in billions of dollars)
2004
78.8452
2005
89.6857
2006
107.484
2007
130.4729
2008
162.8364
2009
188.9841
2010
218.9124
Country
Year
East Timor
Sierra
Leone
Somalia
Burundi
Malawi
GuineaBissau
Congo,
Democratic
Republic of
the
Tanza
Math 3 HW10
Due Friday, Dec 7
Remember that your write-up should be polished and complete (and thus likely not
a rst draft). A Finite Mathematics student (not necessarily in our class) should be able
to read your write-up, understand the question, and und
Math 3 (Finite Mathematics)
Monday
8/27
1.3 Linear Models
Wednesday
8/29
1.3, 1.4 Linear Regression
9/3
NO CLASS
9/5
2.1 Systems of Equations
9/10
2.2 More Gauss-Jordan
9/12
2.3 Applications of Linear
Systems
9/19
3.3 Matrix Inversion
9/26
Exam 1
10/3
4.1