text:
# Recent questions tagged pigeonhole-principle

1
An alternative proof of Theorem $3$ based on the generalized pigeonhole principle is outlined in this exercise. The notation used is the same as that used in the proof in the text. Assume that $i_{k}\leq n\:\text{for}\: k = 1, 2,\dots ,n^{2} + 1.$ Use the ... $n + 1,$ then there must be a decreasing subsequence of this length.
2
Let $n_{1}, n_{2},\dots,n_{t}$ be positive integers. Show that if $n_{1} + n_{2} +\dots + n_{t} − t + 1$ objects are placed into $t$ boxes, then for some $i, i = 1, 2,\dots,t,$ the $i^{\text{th}}$ box contains at least $n_{i}$ objects.
3
Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n,$ the absolute value of the difference between $j x$ and the nearest integer to $j x$ is less than $1/n.$
4
There are $51$ houses on a street. Each house has an address between $1000\: \text{and}\: 1099,$ inclusive. Show that at least two houses have addresses that are consecutive integers.
5
Show that if $f$ is a function from $S\: \text{to}\: T,$ where $S\: \text{and}\: T$ are nonempty finite sets and $m = \left \lceil \mid S \mid / \mid T \mid \right \rceil ,$ then there are at least $m$ elements of $S$ mapped to the same value of $T.$ That is, show that there are distinct elements $s_{1}, s_{2},\dots,s_{m}\: \text{of}\: S$ such that $f (s_{1}) = f (s_{2}) =\dots = f (s_{m}).$
6
Is the statement in question $41$ true if $24$ is replaced by $2?$ $23?$ $25?$ $30?$
7
An arm wrestler is the champion for a period of $75$ hours. (Here, by an hour, we mean a period starting from an exact hour, such as $1\: \text{p.m.,}$ until the next hour.) The arm wrestler had at least one match an hour, but no more than $125$ total matches. Show that there is a period of consecutive hours during which the arm wrestler had exactly $24$ matches.
8
Prove that at a party where there are at least two people, there are two people who know the same number of other people there.
9
Find the least number of cables required to connect $100$ computers to $20$ printers to guarantee that $2$ every subset of $20$computers can directly access $20$ different printers. (Here, the assumptions about cables and computers are the same as in Example $9.$) Justify your answer.
10
Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers. Justify your answer.
11
A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers. [Hint: It is impossible to have a computer linked to none of the others and a computer linked to all the others.]
12
A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.
13
There are $38$ different time periods during which classes at a university can be scheduled. If there are $677$ different classes, how many different rooms will be needed?
14
Assuming that no one has more than $1,000,000$ hairs on the head of any person and that the population of New York City was $8,008,278\:\text{in}\: 2010,$ show there had to be at least nine people in NewYork City in $2010$ with the same number of hairs on their heads.
15
In the $17^{\text{th}}$ century, there were more than $800,000$ inhabitants of Paris. At the time, it was believed that no one had more than $200,000$ hairs on their head. Assuming these numbers are correct and that everyone has at least one hair ... generalized pigeonhole principle to show that there had to be at least five Parisians at that time with the same number of hairs on their heads.
16
Show that if there are $100,000,000$ wage earners in the United States who earn less than $1,000,000$ dollars (but at least a penny), then there are two who earned exactly the same amount of money, to the penny, last year.
17
Show that there are at least six people in California (population: $37$ million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume that everyone has three initials.
18
Show that if $m$ and $n$ are integers with $m \geq 2 \:\text{and}\: n \geq 2,$ then the Ramsey numbers $R(m, n)\:\text{and}\: R(n, m)$ are equal. $\text{(Recall that Ramsey numbers were discussed after Example}\: 13\: \text{in Section}\: 6.2.)$
19
Show that if $n$ is an integer with $n \geq 2,$ then the Ramsey number $R(2, n)$ equals $n.\text{(Recall that Ramsey numbers were discussed after Example}\: 13\:\text{in Section}\: 6.2.)$
20
Use question $27$ to show that among any group of $20$ people (where any two people are either friends or enemies), there are either four mutual friends or four mutual enemies.
21
Show that in a group of $10$ people (where any two people are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends.
22
Show that in a group of five people (where any two people are either friends or enemies), there are not necessarily three mutual friends or three mutual enemies.
23
Describe an algorithm in pseudocode for producing the largest increasing or decreasing subsequence of a sequence of distinct integers.
24
Suppose that $21$ girls and $21$ boys enter a mathematics competition. Furthermore, suppose that each entrant solves at most six questions, and for every boy-girl pair, there is at least one question that they both solved. Show that there is a question that was solved by at least three girls and at least three boys.
25
Show that whenever $25$ girls and $25$ boys are seated around a circular table there is always a person both of whose neighbors are boys.
26
Show that if there are $101$ people of different heights standing in a line, it is possible to find $11$ people in the order they are standing in the line with heights that are either increasing or decreasing.
Construct a sequence of $16$ positive integers that has no increasing or decreasing subsequence of five terms.
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
Suppose that every student in a discrete mathematics class of $25$ students is a freshman, a sophomore, or a junior. Show that there are at least nine freshmen, at least nine sophomores, or at least nine juniors in the class. Show that there are either at least three freshmen, at least $19$ sophomores, or at least five juniors in the class


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# Errata

The following corrections and other changes have been made in the DLMF, and are pending for the Handbook of Mathematical Functions. The Editors thank the users who have contributed to the accuracy of the DLMF Project by submitting reports of possible errors. For confirmed errors, the Editors have made the corrections listed here.

Printable errata .

## Version 1.0.6 (May 6, 2013)

Several minor improvements were made affecting display and layout; primarily tracking changes to the underlying LaTeXML system.

## Version 1.0.5 (October 1, 2012)

Subsection 1.2(i)

The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if is a positive integer. Previously was allowed to be zero.

Reported 2011-08-10 by Michael Somos.

Subsection 8.17(i)

The condition for the validity of (8.17.5) is that and are positive integers and . Previously, no conditions were stated.

Reported 2011-03-23 by Stephen Bourn.

Equation 10.20.14

Originally this coefficient was given incorrectly as . The other coefficients in this equation have not been changed.

Reported 2012-05-11 by Antony Lee.

Equation 13.16.4

The condition for the validity of this equation is . Originally it was given incorrectly as .

Subsection 14.2(ii)

Originally it was stated, incorrectly, that is real when and . This statement is true only for and .

Reported 2012-07-18 by Hans Volkmer and Howard Cohl.

Equation 21.3.4

Originally the vector was given incorrectly as .

Reported 2012-08-27 by Klaas Vantournhout.

Subsection 21.10(i)

The entire original content of this subsection has been replaced by a reference.

Figures 22.3.22 and 22.3.23

The captions for these figures have been corrected to read, in part, “as a function of ” (instead of ). Also, the resolution of the graph in Figure 22.3.22 was improved near .

Reported 2011-10-30 by Paul Abbott.

Equation 23.2.4

Originally the denominator was given incorrectly as .

Reported 2012-02-16 by James D. Walker.

Equation 24.4.26

This equation is true only for . Previously, was also allowed.

Equation 26.12.26

Originally this equation was given incorrectly as

Reported 2011-09-05 by Suresh Govindarajan.

Other Changes

## Version 1.0.4 (March 23, 2012)

Several minor improvements were made affecting display of math and graphics on the web site; the software index and help files were updated.

## Version 1.0.3 (Aug 29, 2011)

Equation 13.18.7

Originally the left-hand side was given correctly as ; the equation is true also for .

Other Changes

Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

## Version 1.0.2 (July 1, 2011)

Several minor improvements were made affecting display on the web site; the help files were revised.

## Version 1.0.1 (June 27, 2011)

Subsections 1.15(vi) and 1.15(vii)

The formulas in these subsections are valid only for . No conditions on were given originally.

Reported 2010-10-18 by Andreas Kurt Richter.

Figure 10.48.5

Originally the ordinate labels 2 and 4 in this figure were placed too high.

Reported 2010-11-08 by Wolfgang Ehrhardt.

Equation 14.19.2
.

Originally the argument to in this equation was incorrect (, rather than ), and the condition on was too weak (, rather than ). Also, the factor multiplying was rewritten to clarify the poles; originally it was .

Reported 2010-11-02 by Alvaro Valenzuela.

Equation 17.13.3

Originally the differential was identified incorrectly as ; the correct differential is .

Reported 2011-04-08.

Table 18.9.1

The coefficient for in the first row of this table originally omitted the parentheses and was given as , instead of .

0
0 1
2 0 1
1
4 −2 1
0
2 0
1 0

Reported 2010-09-16 by Kendall Atkinson.

Subsection 19.16(iii)

Originally it was implied that is an elliptic integral. It was clarified that is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, does not satisfy these conditions.

Reported 2010-11-23.

Table 22.5.4

Originally the limiting form for in the last line of this table was incorrect (, instead of ).

 1 1

Reported 2010-11-23.

Equation 22.16.14

Originally this equation appeared with the upper limit of integration as , rather than .

Reported 2010-07-08 by Charles Karney.

Equation 26.7.6

Originally this equation appeared with in the summation, instead of .

Reported 2010-11-07 by Layne Watson.

Equation 36.10.14

Originally this equation appeared with in the second term, rather than .

Reported 2010-04-02.

Other Changes

## Version 1.0.0 (May 7, 2010)

The Handbook of Mathematical Functions was published, and the Digital Library of Mathematical Functions was released.

format: plain


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## nissn Group Title find the fourier series of this one year ago one year ago

1. nissn Group Title

of this

2. AccessDenied Group Title

Is there any general form for a Fourier series of a function f(x)? I'm not particularly well-versed in the subject, although I'll try to help... :)

3. AccessDenied Group Title

My first observation of the problem is that f(x) goes through three cycles over the interval (-3pi, 3pi)...

4. nissn Group Title

yeah i know

5. nissn Group Title

the first thing I have to find is the fourier coefficient. I think it is 1/4?

6. nissn Group Title

$1/(2\pi)(\int\limits_{-\pi}^{0}0 dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

7. nissn Group Title

which is 1/4

8. AccessDenied Group Title

Yep, that appears correct to me.

9. nissn Group Title

$a _{n}=1/\pi \int\limits_{-\pi}^{0}0 dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

10. nissn Group Title

I am not sure if I am doing it right on the last one

11. AccessDenied Group Title

The resource I am checking indicate that the formula for a_n would be 1/pi * integral from -pi to pi of f(x) cos(nx) dx

12. AccessDenied Group Title

So... $$\displaystyle a_n = \frac{1}{\pi} \left( \int_{-\pi}^{0} 0 \; \textrm{d}x + \int_{0}^{\pi} \frac{1}{\pi} x \cos nx \; \textrm{d}x \right)$$

13. nissn Group Title

yeah so then it is correct

14. nissn Group Title

so then it is 1/pi?

15. nissn Group Title

sin (1/pi)

16. AccessDenied Group Title

So I am finding: $$\displaystyle a_n = \frac{\pi n \sin n \pi + \cos \pi n - 1}{\pi^2 n^2}$$

17. nissn Group Title

and then I must do a $b _{n}$

18. nissn Group Title

$b _{n}=1/\pi(\int\limits_{-\pi}^{0}0dx + \int\limits_{0}^{\pi}(1/\pi)x dx$

19. AccessDenied Group Title

My resource shows an additional sin(nx) there $$\displaystyle \frac{1}{\pi} \int_{0}^{\pi} \frac{1}{\pi} x \sin nx \; \textrm{d}x$$

20. nissn Group Title

oh it's the same I just forgot to write the nx

21. nissn Group Title

and that -cos (pi/pi)

22. AccessDenied Group Title

Sorry, I gotta go for school. I take the calculation of this integral to wolfram... Wolfram evaluates it out as: $$\displaystyle b_n = \frac{\sin \pi n - \pi n \cos \pi n}{\pi^2 n^2}$$ http://www.wolframalpha.com/input/?i=integral+from+0+to+pi+of+1%2Fpi%5E2+x+sin%28n+x%29+dx

23. nissn Group Title

okey. thank you

24. AccessDenied Group Title

You're welcome! :)

25. AccessDenied Group Title

and good luck, I think you're on the right track. :) I was using this as my resource: http://mathworld.wolfram.com/GeneralizedFourierSeries.html Just the end bit with the formula.

26. nissn Group Title

thank you :)


text:
# Decode this… before it's too late

Note: Story does not contain any clues

You are a detective trying to find out the whereabouts of a very dangerous criminal. You have captured his accomplice, who refuses to give you any information. Luckily, you have found three different notes from the criminal that the accomplice had with him.

First note:

First

Express yourself

Tell your friends and family how much they mean to you,

Tell them...before it's too late.

Under the ground, a being stirs

Bringer of death, he strikes

Reaps your soul

Under the ground you go

Terror strikes

End.

Second Note:

Second

All life must die

Ukuleles play the song of life

Am I alive or dead?

Funny how I don't know

By Arizona

Apache people live

Apache understand life and death like no other

Third note:

wba

hvsp

rshsh

qhwt

jsgg

ohhh

wqz

Hint #1:

Each note gives clues about how to solve the next one

After about 18 hours if this is not solved I will give another hint.

Partial (correct) answers will earn an upvote, person who contributed the most to the solution (or solved by themselves) will get accepted answer. Good luck.

• I would not worry about too much of my puzzle not being solved in 18 hours. – Matsmath Oct 20 '16 at 6:23
• Please read this about giving hints. – ABcDexter Oct 20 '16 at 7:36

## 2 Answers

Answer:

In your own attic.

Note 1:

The word "first" tells us to take the first letter of each sentence. These say ETTUBRUTE which indicates a Caesar cipher. (Thanks to @Rubio for solving this part.)

Note 2:

The second letter of each sentence is lkmuypp which with a Caesar shift of 16 becomes backoff.

Note 3:

Dropping the last letter of each line (taking the "back off") gives wbhvsrshsqhwjsgohhwq which rot-12s to inthedetectivesattic.

• Nice job, you got it! – suomynonA Oct 20 '16 at 15:54

Partial Answer
Note 1:

reading down the first letters (excluding the "First" line) gives
ETTBRUTE
which seems sufficiently similar to Et tu, Brute - the line attributed to Caesar - to not be a coincidence. Is there a "U" line missing?

In any case, this hints at a Caesar cipher for Note 2. While I'm not able to find anything in Note 2 that caesars to anything useful, I thought I'd toss this out there and see if it stirs anyone else's thinking.

• Yes, you figured out the first note, I fixed the typo also. – suomynonA Oct 20 '16 at 13:09


text:
# Proof in Parallelogram using vectors

• November 13th 2006, 09:41 AM
meismo
Proof in Parallelogram using vectors
Hello everyone,
I need to proof that in a parallelogram, the lines connecting one corner to the middle of its opposite sides split the diagonal of the parallelogram into three equal parts.

I've uploaded an image to clarify see attachment.

(ABCD is the parallelogram, BD is the diagonal, and AE and AF are the lines splitting the diagonal. The rest are lines I thought might be useful, but haven't found out how.

I need to do this proof using vectors. Since it's a parallelogram I know that AB=DC,BC=AD, BD||EF||IJ.
So I need to proof that DH=HG=GB.
So far all my tries lead nowhere and I'm stuck. Any help would be appreciated.

meismo
• November 14th 2006, 09:37 AM
TriKri
Draw the line AC, it will cross DB at the middle because of the symetry of the parallelogram. Consider the two triangles BEA and ECA, their baces is the same (along the line BC) and so is their heights, hence also their area.

Now consider the two triangles BEG and ECG, by the same reason they have equal area as well. That means the two triangles GCA and GBA also has the same area, cause GCA + ECG = ECA, and GBA + BEG = BEA.

Now, also AC is split on the middle by BD, because of the symetry. If the midpoint (where they cross) is M, that means also triangles AMG and CMG has the same area, hence half the area of ACG. And since ACG and ABG has the same area, AMG has half the area of ABG. If we say their bases is along the line the line BM, GMA has the same height as BGA, hence the half base, hence GM is half the distance of BG and one third of BM.

If you do the same thing on the other half of the parallelogram, you'll se HM is one third of DM. And by adding two and two together you'll se DH equals HG equals GB.

/Kristofer
• November 14th 2006, 11:06 AM
earboth
Quote:

Originally Posted by meismo
Hello everyone,
I need to proof that in a parallelogram, the lines connecting one corner to the middle of its opposite sides split the diagonal of the parallelogram into three equal parts.
...

Hello, meismo,

if I understand your problem right you have to use vectors to prove the property:

a) As you have mentioned the vector $\overrightarrow{AB} = \overrightarrow{DC}$

b) You can prove by the property of the mid-parallel in a triangle that $\overrightarrow{AG} = \frac{2}{3}\overrightarrow{AE}$

You get 3 vectors:
$\overrightarrow{DH} = -\overrightarrow{AD}+\frac{2}{3}(\overrightarrow{AD }+\frac{1}{2}\overrightarrow{AB})$ = $-\frac{1}{3}\overrightarrow{AD}+\frac{1}{3}\overrig htarrow{AB}$

$\overrightarrow{BG} = -\overrightarrow{AB}+\frac{2}{3}(\overrightarrow{AB }+\frac{1}{2}\overrightarrow{AD})$ = $-\frac{1}{3}\overrightarrow{AB}+\frac{1}{3}\overrig htarrow{AD}$

$\overrightarrow{HG} = -\frac{2}{3}(\overrightarrow{AD}+\frac{1}{2}\overri ghtarrow{AB}) +\frac{2}{3}(\overrightarrow{AB}+\frac{1}{2}\overr ightarrow{AD})$ = $\frac{1}{3}\overrightarrow{AB}-\frac{1}{3}\overrightarrow{AD}$

So you can see that all three vectors have the same length and they only differ in their direction.

EB


text:
# 21 CFR § 177.1520 - Olefin polymers.

§ 177.1520 Olefin polymers.

The olefin polymers listed in paragraph (a) of this section may be safely used as articles or components of articles intended for use in contact with food, subject to the provisions of this section.

(a) For the purpose of this section, olefin polymers are basic polymers manufactured as described in this paragraph, so as to meet the specifications prescribed in paragraph (c) of this section, when tested by the methods described in paragraph (d) of this section.

(1)

(i) Polypropylene consists of basic polymers manufactured by the catalytic polymerization of propylene.

(ii) Propylene homopolymer consists of basic polymers manufactured by the catalytic polymerization of propylene with a metallocene catalyst.

(2)

(i) Polyethylene consists of basic polymers manufactured by the catalytic polymerization of ethylene.

(ii) Fumaric acid-grafted polyethylene (CAS Reg. No. 26877-81-6) consists of basic polymers manufactured by the catalytic polymerization of ethylene followed by reaction with fumaric acid in the absence of free radical initiators. Such polymers shall contain grafted fumaric acid at levels not to exceed 2 percent by weight of the finished polymer.

(3) Olefin basic copolymers consist of basic copolymers manufactured by the catalytic copolymerization of:

(i) Two or more of the 1-alkenes having 2 to 8 carbon atoms. Such olefin basic copolymers contain not less than 96 weight-percent of polymer units derived from ethylene and/or propylene, except that:

(a)(1) Olefin basic copolymers manufactured by the catalytic copolymerization of ethylene and hexene-1 or ethylene and octene-1 shall contain not less than 90 weight-percent of polymer units derived from ethylene;

(2) Olefin basic copolymers manufactured by the catalytic copolymerization of ethylene and hexene-1 shall contain not less than 80 but not more than 90 weight percent of polymer units derived from ethylene.

(3) Olefin basic copolymers manufactured by the catalytic copolymerization of ethylene and pentene-1 shall contain not less than 90 weight-percent of polymer units derived from ethylene.

(4) Olefin basic copolymers manufactured by the catalytic polymerization of ethylene and octene-1 shall contain not less than 50 weight-percent of polymer units derived from ethylene.

(b) Olefin basic copolymers manufactured by the catalytic copolymerization of ethylene and 4-methylpentene-1 shall contain not less than 89 weight-percent of polymer units derived from ethylene;

(c)(1) Olefin basic copolymers manufactured by the catalytic copolymerization of two or more of the monomers ethylene, propylene, butene-1, 2-methylpropene-1, and 2,4,4-trimethylpentene-1 shall contain not less than 85 weight-percent of polymer units derived from ethylene and/or propylene;

(2) Olefin basic copolymers manufactured by the catalytic copolymerization of propylene and butene-1 shall contain greater than 15 but not greater than 35 weight percent of polymer units derived from butene-1 with the remainder being propylene.

(d) Olefin basic terpolymers manufactured by the catalytic copolymerization of ethylene, hexene-1, and either propylene or butene-1, shall contain not less than 85 weight percent polymer units derived from ethylene.

(e) Olefin basic copolymers manufactured by the catalytic polymerization of ethylene and octene-1, or ethylene, octene-1, and either hexene-1, butene-1, propylene, or 4-methylpentene-1 shall contain not less than 80 weight percent of polymer units derived from ethylene.

(ii) 4-Methylpentene-1 and 1-alkenes having from 6 to 18 carbon atoms. Such olefin basic copolymers shall contain not less than 95 molar percent of polymer units derived from 4-methylpentene-1, except that copolymers manufactured with 1-alkenes having from 12 to 18 carbon atoms shall contain not less than 97 molar percent of polymer units derived from 4-methylpentene-1; or

(iii) Ethylene and propylene that may contain as modifiers not more than 5 weight-percent of total polymer units derived by copolymerization with one or more of the following monomers:

5-Ethylidine-2-norbornene.
5-Methylene-2-norbornene.

(iv) Ethylene and propylene that may contain as a modifier not more than 4.5 weight percent of total polymer units derived by copolymerization with 1,4-hexadiene.

(v) Ethylene and butene-1 copolymers (CAS Reg. No. 25087-34-7) that shall contain not less than 80 weight percent of polymer units derived from ethylene.

(vi) Olefin basic copolymers (CAS Reg. No. 61615-63-2) manufactured by the catalytic copolymerization of ethylene and propylene with 1,4-hexadiene, followed by reaction with fumaric acid in the absence of free radical initiators. Such polymers shall contain not more than 4.5 percent of polymer units deriving from 1,4-hexadiene by weight of total polymer prior to reaction with fumaric acid and not more than 2.2 percent of grafted fumaric acid by weight of the finished polymer.

(vii) Ethylene and 2-norbornene (CAS Reg. No. 26007-43-2) copolymers that shall contain not less than 30 and not more than 70 mole percent of polymer units derived from 2-norbornene.

(4) Poly(methylpentene) consists of basic polymers manufactured by the catalytic polymerization of 4-methylpentene-1.

(5) Polyethylene graft copolymers consist of polyethylene complying with item 2.2 of paragraph (c) of this section which subsequently has 3a,4,7,7a-tetrahydromethyl-4,7-methanoisobenzofuran-1,3-dione grafted onto it at a level not to exceed 1.7 percent by weight of the finished copolymer.

(6) Ethylene-maleic anhydride copolymers (CAS Reg. No. 9006-26-2) containing no more than 2 percent by weight of copolymer units derived from maleic anhydride.

(b) The basic olefin polymers identified in paragraph (a) of this section may contain optional adjuvant substances required in the production of such basic olefin polymers. The optional adjuvant substances required in the production of the basic olefin polymers or finished food-contact articles may include substances permitted for such use by applicable regulations in parts 170 through 189 of this chapter, substances generally recognized as safe in food and food packaging, substances used in accordance with a prior sanction or approval, and the following:

Substance Limitations
Aromatic petroleum hydrocarbon resin, hydrogenated (CAS Reg. No. 88526-47-0), produced by the catalytic polymerization of aromatic-substituted olefins from distillates of cracked petroleum stocks with a boiling point no greater than 220 °C (428 °F), and the subsequent catalytic hydrogenation of the resulting aromatic petroleum hydrocarbon resin, having a minimum softening point of 110 °C (230 °F), as determined by ASTM Method E 28-67 (Reapproved 1982), “Standard Test Method for Softening Point by Ring-and-Ball Apparatus,” and a minimum aniline point of 107 °C (225 °F), as determined by ASTM Method D 611-82, “Standard Test Methods for Aniline Point and Mixed Aniline Point of Petroleum Products and Hydrocarbon Solvents,” both of which are incorporated by reference in accordance with 5 U.S.C. 552(a) and 1 CFR part 51. Copies are available from the American Society for Testing and Materials, 100 Barr Harbor Dr., West Conshohocken, Philadelphia, PA 19428-2959, or from the Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html. For use only as an adjuvant at levels not to exceed 25 percent by weight in blends with polypropylene complying with paragraph (c), item 1.1 of this section. The finished polymer may be used in contact with food Types I, II, IV-B, VI-A through VI-C, VII-B, and VIII identified in table 1 of § 176.170(c) of this chapter and under conditions of use B through H described in table 2 of § 176.170(c) of this chapter; and with food Types III, IV-A, V, VII-A, and IX identified in table 1 of § 176.170(c) of this chapter and under conditions of use D through G described in table 2 of § 176.170(c) of this chapter.
Colorants used in accordance with § 178.3297 of this chapter
2,5-Dimethyl-2,5-di(tert-butylperoxy)hexane (CAS Reg. No. 78-63-7) For use as an initiator in the production of propylene homopolymer complying with § 177.1520(c), item 1.1 and olefin copolymers complying with § 177.1520(c), items 3.1 and 3.2 and containing not less than 75 weight percent of polymer units derived from propylene, provided that the maximum concentration of tert-butyl alcohol in the polymer does not exceed 100 parts per million, as determined by a method titled “Determination of tert-Butyl Alcohol in Polypropylene,” which is incorporated by reference. Copies are available from the Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or available for inspection at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html.
Methyl methacrylate/butyl acrylate-grafted polypropylene copolymer containing methyl methacrylate/butyl acrylate-grafted polypropylene (CAS Reg. No. 121510-09-6), methyl methacrylate/butyl acrylate copolymer (CAS Reg. No. 25852-37-3), methyl methacrylate homopolymer (CAS Reg. No. 9011-14-7), and polypropylene (CAS Reg. No. 9003-07-0), resulting from the reaction of a mixture of methyl methacrylate and butyl acrylate with polypropylene. The finished product contains no more than 55 percent by weight of polymer units derived from methyl methacrylate and butyl acrylate as determined by a method entitled, “Determination of the Total Acrylic in PP-MMA/BA Polymers,” which is incorporated by reference in accordance with 5 U.S.C. 552(a) and 1 CFR part 51. Copies are available from the Office of Food Additive Safety, Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the Food and Drug Administration's Main Library, 10903 New Hampshire Ave., Bldg. 2, Third Floor, Silver Spring, MD 20993, 301-796-2039, or at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html For use only at levels not to exceed 6 percent by weight of olefin polymers complying with paragraph (c) of this section, items 1.1, 3.1a, 3.2a, and 3.2b, where the copolymers complying with items 3.1a, 3.2a, and 3.2b contain not less than 85 weight-percent of polymer units derived from propylene.
Petroleum hydrocarbon resins (cyclopentadiene-type), hydrogenated (CAS Reg. No. 68132-00-3) produced by the thermal polymerization of dicyclopentadiene and cyclodiene codimers (consisting of a mixture of cyclopentadiene, methyl cyclopentadiene, and C4-C5 acyclic dienes), followed by hydrogenation and having a ring-and-ball softening point of 119 °C minimum as determined by ASTM Method E 28-67 (Reapproved 1982), “Standard Test Method for Softening Point by Ring-and-Ball Apparatus,” and a minimum viscosity of 3,000 centipoise, measured at 160 °C, as determined by ASTM Method D 3236-88, “Standard Test Method for Apparent Viscosity of Hot Melt Adhesives and Coating Materials,” both of which are incorporated by reference in accordance with 5 U.S.C. 552(a) and 1 CFR part 51. Copies are available from the American Society for Testing and Materials, 100 Barr Harbor Dr., West Conshohocken, Philadelphia, PA 19428-2959, or from the Center For Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html. For use only as an adjuvant at levels not to exceed 30 percent by weight in blends with: (1) Polypropylene complying with paragraph (c), item 1.1 of this section, or (2) a copolymer of propylene and ethylene containing not less than 94 weight percent propylene and complying with paragraph (c), item 3.2 of this section. The average thickness of the food-contact film is not to exceed 0.1 millimeter (0.004 inch). The finished polymer may be used in contact with (1) Food types I, II, IV-B, VI-A, VI-B, VII-B, and VIII identified in table 1 of § 176.170(c) of this chapter and under conditions of use C through G described in table 2 of § 176.170(c) of this chapter; and (2) food types III, IV-A, V, VI-C, VII-A, and IX identified in table 1 of § 176.170(c) of this chapter and under conditions of use D through G described in table 2 of § 176.170(c) of this chapter.
Polymethylsilsesquioxane (CAS Reg. No. 68554-70-1) For use only as a surface lubricant or anti-blocking agent in films.
Poly(vinylidene fluoride) homopolymer (CAS Reg. No. 24937-79-9), having a melt viscosity of 6 to 37 kilopoise at a shear rate of 100−1 seconds at 232 °C as determined by ASTM Method D 3835-79 (Reapproved 1983), “Standard Test Method for Rheological Properties of Thermoplastics with a Capillary Rheometer” using a capillary of 15:1 L/D, which is incorporated by reference in accordance with 5 U.S.C. 552(a) and 1 CFR part 51. Copies are available from the Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html. For use only as a processing aid in the production of olefin polymers complying with paragraph (c) of this section at levels not to exceed 1.0 percent by weight of the polymer. The finished polymers may be used only under the conditions described in § 176.170(c) of this chapter, table 2, under conditions of use B though H.
Polyoxyethylene-grafted polydimethylsiloxane (CAS Reg. No. 68937-54-2) For use as an extrusion aid in the production of extruded olefin polymers that comply with § 177.1520(c) at levels not to exceed 0.3 percent by weight of the polymer. The finished polymer is used in contact with foods under conditions of use B through H described in table 2 of § 176.170 of this chapter.
Triisopropanolamine (CAS Reg. No. 122-20-3) For use as a Zeigler-Natta-type catalyst deactivator and antioxidant in the production of olefin polymers complying with § 177.1520(c), items 2.1, 2.2, and 2.3, and having a minimum density of 0.94 grams per cubic centimeter, and copolymers complying with § 177.1520(c), items 3.1 and 3.2, for use in contact with all foods under the following conditions of use: (a) films with a maximum thickness of 0.102 millimeter (0.004 inch) may be used under conditions A through H defined in table 2 of § 176.170(c) of this chapter; and (b) articles with thickness greater than 0.102 millimeter (0.004 inch) may be used under conditions C through G defined in table 2 of § 176.170(c) of this chapter.
Trimethylpyridine and dimethylpyridine mixture having percent by weight composition as follows: 2,4,6-trimethylpyridine (CAS Reg. No. 108-75-8), not less than 60 percent; 2,3,6-trimethylpyridine (CAS Reg. No. 1462-84-6), not more than 27 percent; 3,5-dimethylpyridine (CAS Reg. No. 591-22-0), not more than 12 percent; and other dimethylpyridines, not more than 6 percent For use only as an adjuvant substance in the production of propylene homopolymers complying with items 1.1, 1.2, and 1.3, and propylene copolymers complying with items 3.1, and 3.2 of paragraph (c) of this section provided that the adjuvant is used at a level not to exceed 20 parts per million by weight of the olefin polymers.
Vinylidene fluoride-hexafluoropropene copolymer (CAS Reg. No. 9011-17-0) having a fluorine content of 65 to 71 percent and a Mooney viscosity of at least 28, as determined by a method entitled “Mooney Viscosity,” which is incorporated by reference in accordance with 5 U.S.C. 552(a). Copies are available from the Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html. For use only as an extrusion aid in the production of extruded olefin polymers at levels not to exceed 0.2 percent by weight of the polymer. The finished polymers may be used only under the conditions described in § 176.170(c) of this chapter, table 2, under conditions of use B through H.
Vinylidene fluoride-hexafluoropropene copolymer (CAS Reg. No. 9011-17-0), having a vinylidene fluoride content of not less than 87 percent but less than 100 percent by weight and a melt viscosity of 12 to 27 kilopoise at a shear rate of 100−1 seconds at 232 °C as determined by ASTM Method D 3835-79 (Reapproved 1983), “Standard Test Method for Rheological Properties of Thermoplastics with a Capillary Rheometer” using a capillary of 15:1 L/D, which is incorporated by reference in accordance with 5 U.S.C. 552(a) and 1 CFR part 51. Copies are available from the Center for Food Safety and Applied Nutrition (HFS-200), Food and Drug Administration, 5001 Campus Dr., College Park, MD 20740, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html. For use only as a processing aid in the production of olefin polymers complying with paragraph (c) of this section at levels not to exceed 1.0 percent by weight of the polymer. The finished polymers may be used only under the conditions described in § 176.170(c) of this chapter, table 2, under conditions of use B though H.

(c) Specifications:

Olefin polymers Density Melting Point (MP) or softening point (SP) (Degrees Centigrade) - Maximum extractable fraction (expressed as percent by weight of the polymer) in N-hexane at specified temperatures Maximum soluble fraction (expressed as percent by weight of polymer) in xylene at specified temperatures
1.1a. Polypropylene described in paragraph (a)(1)(i) of this section 0.880-0.913 MP: 160°-180 °C 6.4 pct at reflux temperature 9.8 pct at 25 °C
1.1b. Propylene homopolymer described in paragraph (a)(1)(ii) of this section 0.880-0.913- MP: 150°-180 °C 6.4 pct at reflux temperature 9.8 pct at 25 °C
1.2. Polypropylene, noncrystalline; for use only to plasticize polyethylene described under items 2.1 and 2.2 of this table, provided that such plasticized polymers meet the maximum extractable fraction and maximum soluble fraction specifications prescribed for such basic polyethylene 0.80-0.88
1.3. Polypropylene, noncrystalline, for use only: To plasticize polypropylene described by item 1.1 of this table, provided that such plasticized polymers meet the maximum extractable fraction and maximum soluble fraction specifications prescribed for such basic polypropylene, and further provided that such plasticized polypropylene contacts food only of the types identified in § 176.170(c) of this chapter, table 1, under Types I, II, IV-B, VI-B, VII-B, and VIII; and for use at levels not to exceed 50 pct by weight of any mixture employed as a food-contact coating provided such coatings contact food only of the types identified in § 176.170(c) of this chapter, table 1, under Types I, II, IV-B, VI-B, VII-B, and VIII 0.80-0.88 SP:115°-138 °C
2.1. Polyethylene for use in articles that contact food except for articles used for packing or holding food during cooking 0.85-1.00 5.5 pct at 50 °C 11.3 pct at 25 °C
2.2. Polyethylene for use in articles used for packing or holding food during cooking 0.85-1.00 2.6 pct at 50 °C Do.
2.3. Polyethylene for use only as component of food-contact coatings at levels up to and including 50 percent by weight of any mixture employed as a food-contact coating 0.85-1.00 53 pct at 50 °C 75 pct at 25 °C
2.4. Olefin polymers described in paragraph (a)(2)(ii) of this section, having a melt flow index not to exceed 17 grams/per 10 minutes as determined by the method described in paragraph (d)(7) of this section, for use in blends with other polymers at levels not to exceed 20 percent by weight of total polymer, subject to the limitation that when contacting food of types III, IV-A, V, VI-C, VII-A, and IX identified in § 176.170(c) of this chapter, Table 1, the polymers shall be used only under conditions of use C, D, E, F, and G, described in § 176.170(c) of this chapter, Table 2.
3.1a. Olefin copolymers described in paragraph (a)(3)(i) of this section for use in articles that contact food except for articles used for packing or holding food during cooking; except olefin copolymers described in paragraph (a)(3)(i)(a)(3) of this section and listed in item 3.1c of this table and olefin copolymers described in paragraph (a)(3)(i)(e) of this section and listed in item 3.1b of this table 0.85-1.00 5.5 pct at 50 °C 30 pct at 25 °C
3.1b. Olefin copolymers described in paragraph (a)(3)(i)(e) of this section for use in contact with food only under conditions of use D, E, F, G, and H described in § 176.170(c) of this chapter, table 2 0.9-1.00 Do Do.
3.1c. Olefin copolymers described in paragraph (a)(3)(i)(a)(3) of this section for use in contact with food only under conditions of use B, C, D, E, F, G, and H described in § 176.170(c) of this chapter, table 2; except that such copolymers when used in contact with food of the types identified in § 176.170(c), table 1, under types III, IVA, V, VIIA, and IX, shall be used only under conditions of use D, E, F, and G described in § 176.170(c) of this chapter, table 2 Not less than 0.92
3.2a. Olefin copolymers described in paragraph (a)(3)(i) of this section for use in articles used for packing or holding food during cooking; except olefin copolymers described in paragraph (a)(3)(i)(c)(2) of this section and listed in item 3.2b of this table; except that olefin copolymers containing 89 to 95 percent ethylene with the remainder being 4-methyl-pentene-1 contacting food Types III, IVA, V, VIIA, and IX identified in § 176.170(c) of this chapter, table 1, shall not exceed 0.051 millimeter (mm) (0.002 inch (in)) in thickness when used under conditions of use A and shall not exceed 0.102 mm (0.004 in) in thickness when used under conditions of use B, C, D, E, and H described in § 176.170(c) of this chapter, table 2. Additionally, olefin copolymers described in (a)(3)(i)(a)(2) of this section may be used only under conditions of use B, C, D, E, F, G, and H described in § 176.170(c) of this chapter, table 2, in contact with all food types identified in § 176.170(c) of this chapter, table 1 0.85-1.00 2.6 pct at 50 °C Do.
3.2b. Olefin copolymers described in paragraph (a)(3)(i)(c)(2) of this section have a melt flow index no greater than 10 grams per 10 minutes as determined by the method described in paragraph (d)(7) of this section, and the thickness of the finished polymer contacting food shall not exceed 0.025 mm (0.001 in). Additionally, optional adjuvants permitted for use in olefin copolymers complying with item 3.2a of this table may be used in the production of this copolymer Do.
3.2c. Olefin copolymers described in paragraph (a)(3)(i)(a)(4) of this section have a melt flow index no greater than 50 grams per 10 minutes as determined by the method described in paragraph (d)(7) of this section. Articles manufactured using these polymers may be used with all types of food under conditions of use C through H as described in table 2 of § 176.170(c) of this chapter 0.85-0.92
3.3a. Olefin copolymers described in paragraph (a)(3)(ii) of this section and manufactured with 1-alkenes having from 6 to 10 carbon atoms
3.3b. Olefin copolymers described in paragraph (a)(3)(ii) of this section, provided that such olefin polymers have a melt temperature of 220 °C to 250 °C (428 °F to 482 °F) as determined by the method described in paragraph (d)(8) of this section and minimum intrinsic viscosity of 1.0 as determined in paragraph (d)(9) of this section.
3.4. Olefin copolymers, primarily non-crystalline, described in par. (a)(3)(iii) of this section provided that such olefin polymers have a minimum viscosity average molecular weight of 120,000 as determined by the method described in par. (d)(5) of this section and a minimum Mooney viscosity of 35 as determined by the method described in par. (d)(6) of this section, and further provided that such olefin copolymers contact food only of the types identified in § 176.170(c) of this chapter, table 1, under Types I, II, III, IV-B, VI, VII, VIII, and IX 0.85-0.90
3.5. Olefin copolymers, primarily non-crystalline, described in paragraph (a)(3)(iv) of this section, provided that such olefin polymers have a minimum viscosity average molecular weight of 95,600 as determined by the method described in paragraph (d)(5) of this section, and further provided that such olefin polymers are used only in blends with olefin polymers described under items 1.1, 2.1, and 2.2 of this table at a maximum level of 25 pct by weight, and provided that such olefin copolymers contact food only of the types identified in § 176.170 (c) of this chapter, table 1, under Types I, II, IV-B, VI, VII-B, and VIII at temperatures not exceeding 190 °F 0.85-0.90
3.6. Olefin copolymers described in paragraph (a)(3)(v) of this section for use in blends with olefin polymer resins have a melt flow index no greater than 5 grams/10 minutes as determined by the method described in paragraph (d)(7) of this section and the thickness of the finished blends shall not exceed 0.1 millimeter (0.004 inch). The ethylene/butene-1 copolymer may be used subject to the following conditions: (1) For use at a level not to exceed 20 weight percent in polypropylene as described under item 1.1 of this table. (2) For use at a level not to exceed 40 weight percent in polyethylene as described under items 2.1 and 2.2 of this table. (3) For use at a level not to exceed 40 weight percent in olefin copolymers as described under items 3.1 and 3.2 of this table Not less than 0.88
3.7. Ethylene/propylene copolymers, meeting the identity described in paragraph (a)(3)(i) of this section, containing not less than 80 mole-percent of polymer units derived from ethylene and having a minimum viscosity average molecular weight of 95,000 as determined by the method described in paragraph (d)(5) of this section, and a minimum Mooney viscosity of 13 as determined by the method described in paragraph (d)(6) of this section. Ethylene/propylene copolymers described in this item 3.7 are to be used only in blends with other olefin polymers complying with this section, at levels not to exceed 30 percent by weight of the total polymer blend, and in contact with food only of types identified in § 176.170(c) of this chapter, Table 1, under Types I, II, III, IV-B, VI, VII, VIII, and IX. Additionally, optional adjuvants permitted for use in olefin copolymers complying with item 3.4 of this table may be used in the production of this copolymer Not less than 0.86
3.8. Olefin polymers described in paragraph (a)(3)(vi) of this section, having a melt flow index not to exceed 9.2 grams per 10 minutes as determined by the method described in paragraph (d)(7) of this section, for use in blends with other polymers at levels not to exceed 8 percent by weight of total polymer, subject to the limitation that when contacting food of types III, IV-A, V, VI-C, VII-A, and IX, identified in § 176.170(c) of this chapter, Table 1, the polymers shall be used only under conditions of use C, D, E, F, and G, described in § 176.170(c) of this chapter, Table 2.
3.9. Olefin copolymers described in paragraph (a)(3)(vii) of this section may only be used in contact with dry foods, Type VIII, as identified in § 176.170(c) of this chapter, Table 1 Not less than 1.0
4. Poly(methylpentene) 0.82-0.85 MP: 235°-250 °C 6.6 pct at reflux temperature 7.5 pct at 25 °C
5. Polyethylene copolymer described in paragraph (a)(5) of this section and having a melt index not to exceed 2, for use, either alone or in blends with other olefin polymers, subject to the limitation that when contacting foods of types III, IV-A, V, VI-C, VII-A, VIII, and IX identified in § 176.170(c) of this chapter, table 1, the thickness of the film (in mils) containing the polyethylene graft copolymer times the concentration of the polyethylene graft copolymer shall not exceed a value of 2 Not less than 0.94 0.45 pct at 15 °C 1.8 pct at 25 °C
6. Ethylene-maleic anhydride copolymers described in paragraph (a)(6) of this section for use as the adhesive component in multilaminate structures, or as the sealant layer in flexible packaging, in contact with food at temperatures not exceeding 49 °C (120 °F) 0.92 or greater 1.36 pct at 50 °C 2.28 pct at 25 °C

(d) The analytical methods for determining whether olefin polymers conform to the specifications prescribed in this section are as follows, and are applicable to the basic polymer in film form not exceeding 4 mils in thickness. The film to be tested shall be cut into approximately 1-inch squares by any convenient method that avoids contamination by dust, dirt, or grease (Note: Do not touch samples with bare fingers - use forceps to hold or transfer samples).

(1) Density. Density shall be determined by ASTM method D1505-68 (Reapproved 1979), “Standard Test Method for Density of Plastics by the Density-Gradient Technique,” which is incorporated by reference. Copies may be obtained from the American Society for Testing Materials, 100 Barr Harbor Dr., West Conshohocken, Philadelphia, PA 19428-2959, or may be examined at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html.

(2) Melting point or softening point -

(i) Melting point. The melting point shall be determined by ASTM method D2117-82, “Standard Test Method for Melting Point of Semicrystalline Polymers by the Hot Stage Microscopy Method,” which is incorporated by reference. The availability of this incorporation by reference is given in paragraph (d)(1) of this section.

(ii) Softening point. The softening point shall be determined by ASTM method E28-67 (Reapproved 1982), “Standard Test Method for Softening Point by Ring-and-Ball Apparatus,” which is incorporated by reference. The availability of this incorporation by reference is given in paragraph (d)(1) of this section.

(3) Maximum extractable fraction in n-hexane -

(i) Olefin copolymers described in paragraph (a)(3)(ii) of this section, polypropylene, and poly(methylpentene). A sample is refluxed in the solvent for 2 hours and filtered at the boiling point. The filtrate is evaporated and the total residue weighed as a measure of the solvent extractable fraction.

(a) Apparatus.

(1) Erlenmeyer flasks, 250-milliliter, with ground joint.

(2) Condensers, Allihn, 400-millimeter jacket, with ground joint.

(3) Funnels, ribbed 75-millimeter diameter, stem cut to 40 millimeters.

(4) Funnels, Buchner type, with coarse-porosity fritted disc, 60-millimeter diameter.

(5) Bell jar for vacuum filtration into beaker.

(b) Reagent. n-Hexane, commercial grade, specific gravity 0.663-0.667 (20 °C/20 °C), boiling range 66 °C-69 °C, or equivalent.

(c) Procedure. Weigh 1 gram of sample accurately and place in a 250-milliliter Erlenmeyer flask containing two or three boiling stones. Add 100 milliliters of solvent, attach the flask to the condenser (use no grease), and reflux the mixture for 2 hours. Remove the flask from the heat, disconnect the condenser, and filter rapidly, while still hot, through a small wad of glass wool packed in a short-stem funnel into a tared 150-millimeter beaker. Rinse the flask and filter with two 10-milliliter portions of the hot solvent, and add the rinsings to the filtrate. Evaporate the filtrate on a stream bath with the aid of a stream of nitrogen. Dry the residue in a vacuum oven at 110 °C for 2 hours, cool in a desiccator, and weigh to the nearest 0.0001 gram. Determine the blank on 120 milliliters of solvent evaporated in a tared 150-milliliter beaker. Correct the sample residue for this blank if significant. Calculation:

$\frac{\text{Grams of residue}}{\text{Grams of sample}×100}=\text{Percent extractable with}n\text{-hexane}$

(ii) Olefin copolymers described in paragraph (a)(3)(i) of this section and polyethylene. A preweighed sample is extracted at 50 °C for 2 hours and filtered. The filtrate is evaporated and the total residue weighed as a measure of the solvent extractable fraction. Alternatively, the sample is reweighed after the extraction period to give a measure of the solvent extractable fraction. The maximum n-hexane-extractable fraction may be determined by the methods set forth in paragraphs (d)(3)(ii)(a) through (d)(3)(ii)(i) of this section.

(a) Extraction apparatus. Two-liter, straight-walled, Pyrex (or equivalent) resin kettles, fitted with three-hole ground-glass covers are most convenient for this purpose. The cover is fitted with a thermometer, a gas-tight stirrer driven by an air motor or explosion-proof electric motor, and a reflux condenser. The kettle is fitted with an electric heating mantle of appropriate size and shape, which is controlled by a variable-voltage transformer.

(b) Evaporating apparatus. Rapid evaporation of large volumes of solvent requires special precautions to prevent contamination by dust. This is facilitated by a special “gas” cover consisting of an inverted flat Pyrex crystallizing dish of an appropriate size (190 millimeters × 100 millimeters) to fit a 1-liter beaker. Through the center of the dish are sealed an inlet tube for preheated, oxygen-free nitrogen, and an outlet tube located 1 inch off center. Nitrogen is fed from the supply source through a coil of 1/4-inch stainless steel tubing immersed in the same steam bath used to supply heat for solvent evaporation. All connections are made with flexible tetrafluoroethylene tubing.

(c) Reagents - (1) n-Hexane. Spectrograde n-hexane.

(2) Nitrogen. High-purity dry nitrogen containing less than 10 parts per million of oxygen.

(d) Procedure. Transfer 2.5 grams (accurately weighed to nearest 0.001 gram) of the polymer to the resin kettle. Add 1 liter of solvent and clamp top in position. Start water flowing through jacket of the reflux condenser and apply air pressure to the stirring motor to produce vigorous agitation. Turn on heating jacket with transformer set at a predetermined voltage to bring the temperature of the contents to 50 °C within 20-25 minutes. As the thermometer reading approaches 45 °C-47 °C, reduce the voltage to the predetermined setting that will just maintain the temperature at 50 °C. Do not overshoot the prescribed temperature. Should this occur discard the test and start afresh. Exactly 2 hours after the solvent temperature has reached 50 °C, disconnect the heater, remove the resin kettle from the heating jacket, and decant the solvent, while still warm, through a coarse filter paper placed on top of a fritted-glass funnel, collecting the filtrate in a tared, glass-stoppered Erlenmeyer flask of 1-liter capacity. Determine the weight of the filtrate recovered to the nearest gram. Recovery should be at least 90 percent of the original solvent. Losses due to evaporation during heating and filtering have been found not to exceed 10 percent. Transfer about half of the solvent filtrate to a 1-liter beaker placed on an opening in the steam bath and immediately cover with the special “gas” cover, the inlet tube of which has been attached with flexible tetrafluoroethylene tubing to a source of high-purity nitrogen in series with a stainless steel heating coil immersed directly in the body of the steam bath. Maintain a positive flow of warm nitrogen gas throughout the evaporation of the solvent, adding the remainder of the filtrate from the Erlenmeyer flask as the evaporation proceeds. When the volume of the solvent has been reduced to about 50 milliliters, transfer the concentrated liquid to a previously tared weighing dish of suitable size. Wash the beaker twice with 20-30 milliliter portions of warm solvent, adding the washings to the weighing dish while continuing to evaporate the remainder of the solvent under the gas cover with its flow of warm nitrogen directed toward the center of the dish. In the event that an insoluble residue that cannot be removed with warm solvent remains in the beaker, it may be necessary to heat with a small amount of a higher boiling solvent such as benzene or toluene, transferring these washings to the weighing dish before final evaporation to dryness. Transfer the weighing dish with its residue to a vacuum desiccator, and allow it to remain overnight (at least 12 hours), after which the net weight of the dry residue is determined to the nearest 0.0001 gram. Correct the result for any solvent blank equivalent to the nonvolatile matter determined to be contained in the amount of solvents used in the test.

(e) Extraction apparatus for alternate method. Two-liter extraction vessel, such as a resin kettle or round bottom flask, fitted with an Allihn condenser (size C), a 45/50 male joint with a Teflon sleeve, and a Teflon coated stir bar. Water bath maintained at 49.5 °C ±0.5 °C containing a submersible magnetic stirrer motor with power supply. Other suitable means of maintaining temperature control, such as electric heating mantles, may be used provided that the temperature range can be strictly maintained.

(f) Sample basket (Optional). A perforated stainless steel cylindrical basket that is approximately 1.5 inches in diameter, 1.6 inches high, and has perforations of 0.125 inches in diameter for 33 holes/in 2, or 40 percent open area. The basket should pass freely through the 45/50 female joint of the extraction flask. A No. 6-32 stainless steel eye-bolt is attached to the lid for positioning the basket in the extraction vessel. The positioning rod, approximately 18 inches long and made from 1/16 inch outside diameter 316 stainless steel welding rod or equivalent and hooked at both ends, is used to position the basket in the extraction apparatus.

(g) Vacuum oven. Capable of maintaining 80 °C ±5 °C and a minimum of 635 millimeters of mercury pressure.

(h) Reagents. n-Hexane, reagent or spectrograde, aromatic free (less than 1 milligram per liter), minimum 85 percent n-hexane. This reagent may be reused until it contains a maximum of 1.5 grams polyolefin extractables or has been used for 12 determinations.

(i) Procedure. Assemble the extraction vessel, condenser, and magnetic stir bar. Add n-hexane (1 liter) to the extraction vessel and clamp the assembly into a water bath set at 49.5 °C ±0.5 °C. Start the water flowing through the jacket of the reflux condenser. Adjust the air flow through the stirring motor to give a smooth and uniform stir rate. Allow the n-hexane to preheat for 1 hour to bring the temperature to 49.5 °C±0.5 °C. Temperature is a critical factor in this analysis and it must not vary more than 1 °C. If the temperature exceeds these limits, the test must be discontinued and restarted. Blown, compression molded, or extrusion cast films can be tested. Ideally, the film should be prepared by the same process as will be used with the production resin. Using gloves and metal tweezers to avoid sample contamination, cut about 2.7 grams of the prepared film (4 mils or less in thickness) into about 1-inch squares using clean sharp scissors. Proceed with Option 1 or 2.

Option 1. Using tweezers and noting the number of film pieces, transfer 2.5 grams (accurately weighed to 0.1 milligram) of polymer to the extraction vessel. Extract the film sample for 2 hours. Allow the vessel to cool and filter the contents through a fritted porcelain funnel. Wash the film pieces with fresh n-hexane, aspirate to dryness, and transfer, using tweezers, to a beaker. Recount the film pieces to verify that none were lost during the transfer. Place the beaker in the vacuum oven for 2 hours at 80 °C ±5 °C. After 2 hours, remove and place in a desiccator to cool to room temperature (about 1 hour). After cooling, reweigh the film pieces to the nearest 0.1 milligram. Calculate the percent hexane-extractables content from the weight loss of the original sample. Multiply the result by 0.935 and compare with extraction limits in paragraph (c) of this section. Repeat the above procedure for successive samples.

Option 2. Transfer 2.5±0.05 grams of the prepared 1-inch film sections into a tared sample basket and accurately weigh to the nearest 0.1 milligram. Carefully raise the condenser until the hook on the positioning rod is above the neck of the 2-liter extraction vessel. The basket should be totally below the level of n-hexane solvent. Extract the sample resin film for 2 hours and then raise the basket above the solvent level to drain momentarily. Remove the basket and rinse the contents by immersing several times in fresh n-hexane. Allow the basket to dry between rinsings. Remove the excess solvent by briefly blowing the basket with a stream of nitrogen or dry air. Place the basket in the vacuum oven for 2 hours at 80 °C ±5 °C. After 2 hours, remove and place in a desiccator to cool to room temperature (about 1 hour). After cooling, reweigh the basket to the nearest 0.1 milligram. Calculate the percent hexane extractables content from the weight loss of the original sample. Multiply the result by 0.935 and compare with extraction limits in paragraph (c) of this section. Repeat the above procedure for successive samples. The same solvent charge should remain clear and can be used for at least 12 determinations. Applications of solvent reuse should be confirmed for each resin type before use.

(4) Maximum soluble fraction in xylene -

(i) Olefin copolymers described in paragraph

(a)(3)(ii) of this section, polypropylene, and poly(methylpen-tene). A sample is dissolved completely in xylene by heating and stirring in a bottle with little free space. The solution is allowed to cool without stirring, whereupon the insoluble portion precipitates and is filtered off; the total solids content of the filtrate is then determined as a measure of the soluble fraction.

(a) Apparatus.

(1) Pyrex (or equivalent) reagent bottle, 125-milliliter, glass-stoppered.

(2) Heating mantle of size for 150-milliliter beaker (or suitable aluminum block to fit the 125-milliter bottle described in paragraph (d)(4)(i)(a)(1) of this section.

(3) Magnetic stirrer for use under the heating mantle (combination magnetic stirrer and hotplate may be used if aluminum block is used in place of heating mantle).

(4) Variable-voltage transformer, 7.5 amperes.

(5) Tetrafluoroethylene-resin-coated stirring bar, 1-inch long.

(6) Constant temperature water bath maintained at 25 °C±0.5 °C.

(7) Aluminum dishes, 18 millimeters × 60 millimeters, disposable.

(8) Funnel, Buchner type, with coarse-porosity fritted disc, 30-60 millimeter diameter.

(b) Reagent. Xylene with antioxidant. Dissolve 0.020 gram of phenyl-β- naphthylamine in 1 liter of industrial grade xylene having specific gravity 0.856-0.867 (20 °C/20 °C) and boiling range 123 °C-160 °C.

(c) Procedure. Weigh 1 to 2 grams of sample to the nearest 0.001 gram and place in a 125-milliliter Pyrex reagent bottle containing a 1-inch long tetrafluoroethylene-resin-coated stirring bar. Add 100 milliliters of solvent, set the stopper in lightly, and place the bottle in the heating mantle or aluminum block maintained at a temperature of 120 °C, and stir with a magnetic stirrer until the sample is completely dissolved. Remove the bottle from the heat and allow it to cool 1 hour in the air, without stirring. Then place the bottle in a water bath maintained at 25 °C ±0.5 °C, and allow to stand 1 hour without stirring. Next, remove the bottle from the water bath, shake, and pour part of the contents into the coarse-porosity fritted-glass funnel. Apply suction, and draw 30-40 milliliters of filtrate through, adding more slurry to the funnel, and catching the filtrate in a large test tube. (If the slurry is hard to filter, add 10 grams of diatomaceous earth filter aid to the bottle and shake vigorously just prior to the filtration.) Pipet a suitable aliquot (preferably 20 milliliters) of the filtrate into a tared aluminum disposable dish. Place the dish on a steam bath covered with a fresh sheet of aluminum foil and invert a short-stemmed 4-inch funnel over the dish. Pass nitrogen (heated if desired) down through the funnel at a rate sufficient to just ripple the surface of the solvent. When the liquid has evaporated, place the dish in a vacuum oven at 140 °C and less than 50 millimeters mercury pressure for 2 hours. Cool in a desiccator and weigh. (Note: If the residue value seems high, redry in the vacuum oven for one-half hour to ensure complete removal of all xylene solvent.) Calculation:

$\frac{\text{Grams of residue}}{\text{Grams of sample}}×\frac{100\text{milliliters}}{\text{volume of aliquot in milliliters}}×100=\text{Percent soluble in xylene}$

(ii) Olefin copolymers described in paragraph

(a)(3)(i) of this section and polyethylene. A sample is extracted in xylene at reflux temperature for 2 hours and filtered. The filtrate is evaporated and the total residue weighed as a measure of soluble fraction.

(a) Apparatus -

(1) Extraction apparatus. Two-liter, straight-walled Pyrex (or equivalent) resin kettles, fitted with ground-glass covers, are most convenient for this purpose. The cover is equipped with a thermometer and an efficient reflux condenser. The kettle is fitted with an electric heating mantle of appropriate size and shape which is controlled by a variable-voltage transformer.

(2) Constant temperature water bath. It must be large enough to permit immersion of the extraction kettle and set to maintain 25 °C ±0.1 °C.

(3) Evaporating apparatus. Gas cover consisting of a flat Pyrex crystallizing dish (190 millimeters × 100 millimeters) inverted to fit over a 1-liter beaker with 8-millimeter gas inlet tube sealed through center and an outlet tube 1 inch off center. The beaker with gas cover is inserted in an electric heating mantle equipped with a variable-voltage transformer. The outlet tube is attached to an efficient condenser mounted on a receiving flask for solvent recovery and having an outlet for connection to an aspirator pump. The heating mantle (with the beaker) is mounted on a magnetic stirring device. An infrared heat lamp is mounted vertically 3-4 inches above the gas cover to prevent condensation of the solvent inside the cover. Make all connections with flexible tetrafluoroethylene tubing.

(b) Reagents -

(1) Xylene. American Chemical Society reagent grade that has been redistilled through a fractionating column to reduce the nonvolatile residue.

(2) Nitrogen. High-purity dry nitrogen containing less than 10 4 parts per million oxygen.

(c) Procedure. Transfer 5 grams ±0.001 gram of sample to the resin kettle, add 1,000 milliliters (840 grams) of xylene, and clamp top in position after inserting a piece of glass rod to prevent bumping during reflux. Start water flowing through the jacket of the reflux condenser and apply full voltage (115 volts) to the heating mantle. When the xylene starts to boil, reduce the voltage to a level just sufficient to maintain reflux. After refluxing for at least 2 hours, disconnect the power source to the mantle, remove the kettle, and allow to cool in air until the temperature of the contents drops to 50 °C, after which the kettle may be rapidly cooled to 25 °C-30 °C by immersing in a cold water bath. Transfer the kettle to a constant temperature bath set to maintain 25 °C ±0.1 °C, and allow to equilibrate for a least 1 hour (may be left overnight if convenient). Break up any precipitated polymers that may have formed, and decant the xylene solution successively through a fast filter paper and then through a fritted-glass filter into a tared 1-liter Erlenmeyer flask, collecting only the first 450 milliliters - 500 milliliters of filtrate (any attempt to collect more of the xylene solution usually results in clogging the filter and risking losses). Reweigh the Erlenmeyer flask and calculate the weight of the filtrate obtained to the nearest 0.1 gram. Transfer the filtrate, quantitatively, from the Erlenmeyer flask to the 1-liter beaker, insert the beaker in its heating mantle, add a glass-coated magnetic stirring bar, and mount the gas cover in place, connecting the inlet tube to the nitrogen source and the outlet to the condenser of the receiving flask. Start a flow of nitrogen (2 to 3 liters per minute) into the gas cover and connect an aspirator to the receiver using a free-flow rate equivalent to 6-7 liters of air per minute. With the infrared lamp on, adjust the voltage to the heating mantle to give a distillation rate of 12-13 milliliters per minute when the magnetic stirrer is revolving just fast enough to promote good boiling. When the volume of solvent in the beaker has been reduced to 30-50 milliliters, transfer the concentrated extractive to a suitable weighing dish that has been previously tared (dry). Rinse the beaker twice with 10-20 milliliter portions of fresh xylene, adding the rinsings to the weighing dish. Evaporate the remainder of the xylene on an electric hotplate set at low heat under the gas cover with a stream of nitrogen directed toward the center of the dish. Avoid any charring of the residue. Transfer the weighing dish to a vacuum desiccator at room temperature and allow to remain under reduced pressure for at least 12 hours (overnight), after which determine the net weight of the residue to the nearest 0.0001 gram. Correct the result for nonvolatile solvent blank obtained by evaporating the equivalent amount of xylene under identical conditions. Calculate the weight of residue originally present in the total weight of solvent (840 grams), using the appropriate factor based on the weight of filtrate evaporated.

(5) Viscosity average molecular weight olefin copolymers described in paragraphs

(a)(3)(iii) and (iv) of this section. The viscosity average molecular weight shall be determined from the kinematic viscosity (using ASTM method D445-74, “Test for Kinematic Viscosity of Transparent and Opaque Liquids” (Revised 1974), which is incorporated by reference; copies are available from American Society for Testing and Materials (ASTM), 100 Barr Harbor Dr., West Conshohocken, Philadelphia, PA 19428-2959, or available for inspection at the National Archives and Records Administration (NARA). For information on the availability of this material at NARA, call 202-741-6030, or go to: http://www.archives.gov/federal_register/code_of_federal_regulations/ibr_locations.html.) of solutions of the copolymers in solvents and at temperatures as follows:

(i) Olefin polymers described in paragraph (a)(3)(iii) of this section in decahydronaphthalene at 135 °C.

(ii) Olefin polymers described in paragraph (a)(3)(iv) of this section in tetrachloroethylene at 30 °C.

(6) Mooney viscosity - olefin copolymers described in paragraph (a)(3)(iii) of this section. Mooney viscosity is determined by ASTM method D1646-81, “Standard Test Method for Rubber - Viscosity and Vulcanization Characteristics (Mooney Viscometer),” which is incorporated by reference (the availability of this incorporation by reference is given in paragraph (d)(1) of this section), using the large rotor at a temperature of 100 °C, except that a temperature of 127 °C shall be used for those copolymers whose Mooney viscosity cannot be determined at 100 °C. The apparatus containing the sample is warmed for 1 minute, run for 8 minutes, and viscosity measurements are then made.

(7) Melt flow index. The melt flow index of olefin polymers described below shall be determined by ASTM method D-1238-82, “Standard Test Method for Flow Rates of Thermoplastics by Extrusion Plastometer,” which is incorporated by reference in accordance with 5 U.S.C. 552(a). The availability of this incorporation by reference is given in paragraph (d)(1) of this section. The olefin polymers and test conditions and procedures are as follows:

List of polymers Conditions/procedures
Olefin copolymers described in paragraph (a)(3)(i)(c)(2) of this section Condition L, procedure A.
Olefin copolymers described in paragraph (a)(3)(v) of this section Condition E, procedure A.
Olefin polymers described in paragraph (a)(2)(ii) of this section Condition E, procedure A.
Olefin polymers described in paragraph (a)(3)(vi) of this section Condition E, procedure A.

(8) Melting peak temperature. The melt temperature of the olefin polymers described in paragraph (a)(3)(ii) of this section shall be determined by ASTM method D 3418-82, “Standard Test Method for Transition Temperatures of Polymers by Thermal Analysis,” which is incorporated by reference in accordance with 5 U.S.C. 552(a). The availability of this incorporation by reference is given in paragraph (d)(1) of this section.

(9) Intrinsic viscosity. The intrinsic viscosity of the olefin polymers described in paragraph (a)(3)(ii) of this section shall be determined by ASTM method D 1601-78, “Standard Test Method for Dilute Solution Viscosity of Ethylene Polymers,” which is incorporated by reference in accordance with 5 U.S.C. 552(a). The availability of this incorporation by reference is given in paragraph (d)(1) of this section.

(e) Olefin copolymers described in paragraph (a)(3)(i) of this section and polyethylene, alone or in combination, may be subjected to irradiation bombardment from a source not to exceed 2.3 million volts intensity to cause molecular crosslinking of the polymers to impart desired properties, such as increased strength and increased ability to shrink when exposed to heat.

(f) The olefin polymers identified in and complying with this section, when used as components of the food-contact surface of any article that is the subject of a regulation in parts 174, 175, 176, 177, 178, and § 179.45 of this chapter, shall comply with any specifications and limitations prescribed by such regulation for the article in the finished form in which it is to contact food.

(g) The provisions of this section are not applicable to olefin polymers identified in § 175.105(c)(5) of this chapter and used in food-packaging adhesives complying with § 175.105 of this chapter.

[42 FR 14572, Mar. 15, 1977]
Editorial Note:
For Federal Register citations affecting § 177.1520, see the List of CFR Sections Affected, which appears in the Finding Aids section of the printed volume and at www.govinfo.gov.


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# Use logarithmic differentiation to find the derivative of $y = (1 +\frac1x)^{2x}$

Can someone guide me through solving this problem? $$\dfrac{\mathrm d}{\mathrm dx}\left(1 +\dfrac1x\right)^{2x}$$

Let $$y=\left(1 +\dfrac1x\right)^{2x}$$ then $$\ln y=2x\ln\left(1 +\dfrac1x\right).$$ Thus \begin{align} \frac{d}{dx}\ln y&=\frac{d}{dx}2x\ln\left(1 +\frac1x\right)\\ \frac1y\frac{dy}{dx}&=2\ln\left(1 +\frac1x\right)+\frac{2x}{1 +\frac1x}\cdot\left(-\frac1{x^2}\right)\\ &=2\ln\left(1 +\frac1x\right)-\frac{2}{x +1}\\ \frac{dy}{dx}&=y\left(2\ln\left(1 +\frac1x\right)-\frac{2}{x +1}\right)\\ &=\left(1 +\dfrac1x\right)^{2x}\left(2\ln\left(1 +\frac1x\right)-\frac{2}{x +1}\right).\\ \end{align}
$(u^v)^\prime = (e^{v\ln u})^\prime = e^{v\ln u}\cdot(v \ln u)^\prime=u^v\cdot(v^\prime\ln u+v\cdot\frac{u^\prime}{u})$
Take logs on both sides and differentiate, \begin{align} y &=\left(1+\frac{1}{x}\right)^{2x}\\ \ln y &=2x \ln \left(1+\frac{1}{x}\right)\\ \frac{\frac{dy}{dx}}{ y} &=2 \ln \left(1+\frac{1}{x}\right)-2 x\frac{\frac{1}{x^2}}{1+\frac{1}{x}} \\ \frac{dy}{dx} &=2 y\ln \left(1+\frac{1}{x}\right)-2 yx\frac{\frac{1}{x^2}}{1+\frac{1}{x}} \end{align} and you can substitute in for $y$ and simplify.
Taking the log of both sides results in $$\ln y = 2x\ln\left(1+\frac{1}{x}\right)$$ Differentiating this leads to $$\frac{1}{y}\frac{dy}{dx}=2\ln\left(1+\frac{1}{x}\right)+2x\left(\frac{-1}{x^2}\frac{x}{x+1}\right)=2\ln\left(1+\frac{1}{x}\right)-\frac{2}{x+1}$$ where, for the right hand side, we used both the product rule and the fact that $\large\frac{d}{dx}\log f(x)=\frac{f'(x)}{f(x)}$
This leads to $$\frac{dy}{dx}=2y\left[\ln\left(1+\frac{1}{x}\right)-\frac{1}{x+1}\right]$$ Expressing $y$ in terms of $x$ on the right hand side of the equation leads to:- $$\frac{dy}{dx}=2\left(1+\frac{1}{x}\right)^{2x}\left[\ln\left(1+\frac{1}{x}\right)-\frac{1}{x+1}\right]$$


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# Simplify Each of the Following Expressions - CBSE Class 9 - Mathematics

ConceptOperations on Real Numbers

#### Question

Simplify of the following expression :-

(3+sqrt3)(2+sqrt2)

#### Solution

(3+sqrt3)(2+sqrt2)

(3+sqrt3)(2+sqrt2) = 3(2+sqrt2)+sqrt3(2+sqrt2)
Left Distributive law of multiplication over addition

= (3)(2)+3sqrt2+(sqrt3(2))+(sqrt3)(sqrt2)

= 6+3sqrt2+2sqrt3+sqrt((3)(2))            thereforesqrtasqrtb=sqrt(ab)

= 6+3sqrt2+2sqrt3+sqrt6

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Mathematics Class 9 (2018 to Current)
Chapter 1: Number Systems
Ex. 1.50 | Q: 2.1 | Page no. 24

#### Video TutorialsVIEW ALL [1]

Solution Simplify Each of the Following Expressions Concept: Operations on Real Numbers.
S


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# How do I finish this off?

1. Sep 28, 2009

### neutron star

I think I'm close but I don't know what the next step is.

1. The problem statement, all variables and given/known data
A ball is dropped off a cliff on a planet with an exact gravity of $$10m/s^2$$, 1.5 seconds later a lead ball is thrown straight down with an initial speed of $$20 m/s$$ If the two balls hit the base of the cliff at the same time, find the height of the cliff.

2. Relevant equations
$$X$$f=$$X$$o+$$V$$o$$t$$ -$$1/2gt^2$$

3. The attempt at a solution$$-d=-1/2gt^2$$
$$t^2=2d/g$$
$$t=\sqrt{2d/g}$$

$$-d=0-20(\sqrt{2d/g}-1.5)-1/2g(\sqrt{2d/g}-1.5)^2$$

Last edited: Sep 28, 2009
2. Sep 28, 2009

### Delphi51

Wow, it took me a while to see that your last expression is d = Vi*t + .5at^2 with t replaced by (t-1.5). So that is your distance for the second ball at time t from when the first ball is dropped. To finish the job, you equate that distance to a similar expression for the first ball's distance.

If I may offer a more general tip, it would be to begin with
ball 1 distance = ball 2 distance
.5*a*t^2 = Vi*(t-1.5) + .5a(t-1.5)^2
so you have the big picture clearly stated, and then it is just detail work to finish it.
Also, you might take "down" to be positive in this problem to reduce the number of troublesome minus signs. I see it isn't really a quadratic equation - the t^2 term cancels a step or two further on.


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# Oscillations of a cylinder inside a cylinder [closed]

Please read the whole thing I'm asking for a concept not the problem itself, but I have to show the problem to explain myself

Find the period of the small oscillations of a cylinder of radius r that rolls without slide inside a cylindrical surface of radius R.

I tried going with the conservation of mechanic energy $$E_m$$ which is equal to $$E_m=E_k+E_p+E_r$$ being $$k, p, r$$ kinetic, potential and rotational (energy). We know that $$E_k=\frac{1}{2}mv^2=\frac{1}{2}m(\frac{d\theta}{dt}(R-r))^2$$, being $$\theta$$ the angle of the small cilinder from the center of the big cilinder with respect to the vertical, and $$E_p=-mg(R-r)\cos(\theta)$$ but for $$E_k=\frac{1}{2}I\omega_{cm}^2$$ I had no idea on how to continue, since $$\omega_{cm}$$ here refers to the rotation of the small cylinder with respect to it's center of mass, not the rotation of the small cylinder with respect to the big cylinder so we can't do $$\omega_{cm}=\frac{d\theta}{dt}$$

Doubt. So I looked the solution and it said that the velocity of the small cylinder with respect to the big cylinder was equal to the velocity of the points inside the small cylinder with respect to the center of masses of the small cylinder, why is that? That makes no sense to me. The procedure they make afterwards is $$v_{cm}=v_{O}\Rightarrow \omega_{cm}=\frac{R-r}{r}\omega_0=\frac{R-r}{r}\frac{d\theta}{dt}$$ being $$\omega_0$$ the rotation of the small cylinder with respect to the center of the big cylinder.

• you need to use the roll-without-slip condition to connect the motion of the center of mass of the system to the motion of the moving cylinderl Nov 25, 2022 at 13:58
• Is bigger cylinder also moving with smaller one? Nov 25, 2022 at 14:01
• I'd ask that the close voter reconsider. It seems to me there is a legitimate conceptual issue being asked here, note the final paragraph to see it. Nov 25, 2022 at 14:03
• @KshitijKumar no, the big cylinder is not moving
– user348222
Nov 25, 2022 at 14:09
• This is the rolling condition $~ \left( R-r\right) \theta =\varphi r~$ where $\varphi$ is the rotation of the small Zylinder, the generalized coordinate
– Eli
Nov 25, 2022 at 14:41

Well honestly I couldn't undertsand the paragraph part of what are you trying to say but mathematically equation is correct, let me explain my way.

There are two types of rotations of small cylinder, one on its own axis, and one about the big cylinder's axis. Let us take the case of a sliding cylinder, not a rolling, one. We can write:

$$v_{cm}=\omega (R-r)$$

where $$\omega$$ is rotation about big cylinder's axis, and $$v_{cm}$$ is velocity of small cylinder's center of mass. Now here due to pure rolling of small cylinder $$v_{cm}=\omega_{cm}r$$. Here $$\omega_{cm}$$ is about the small cylinder's axis. This will yield the same result that you got.

Hope it helps, ask any doubts.

Edit:

The pure rolling means that bottom-most point of sphere or its point of contact with ground remains at rest with repect to ground. Two motions are going on simultaneosuly: Translational and rotational, which is summed up in the above image and leads us to: $$v_{cm}=\omega_{cm}r$$

• Oh I think I've understood the statement "doesn't slip" wrongly all this time, does that statement implies $v_{cm}=\omega_{own\ axis}r$?
– user348222
Nov 25, 2022 at 14:32
• Because I might have always thought that $v_{cm}=\omega_{cm}r$ which is wrong because I thought $\omega_{cm}$ was with respect to it's own axis itself as well as $v_{cm}$ but it turns out it isn't, right? It turns out $v$ in that equation refers to the velocity of the center of masses itself, not the points of the small cylinder
– user348222
Nov 25, 2022 at 14:36
• @Ivy Added details appropriately, hope it helps. Nov 25, 2022 at 14:44
• It helps but I want my doubt answered that I asked above your comment, still unsure what's with respect to what. And as a second question, what's the name of that book? To see if I can buy it, it looks good.
– user348222
Nov 25, 2022 at 14:47
• @Ivy couldn't understand what are trying to say. Image i got from net Nov 25, 2022 at 15:19

starting with the rolling condition

$$r\,\varphi=(R-r)\theta\tag 1$$

where $$~\varphi~$$ is the rotation of the small cylinder (generalized coordinate)

the position vector to the center of the small cylinder is

$$\vec R_c=\begin{bmatrix} x \\ y \\ \end{bmatrix}=(R-r)\begin{bmatrix} \sin(\theta(\varphi)) \\ \cos(\theta(\varphi)) \\ \end{bmatrix}\quad\Rightarrow\\ T_k=\frac m2 (\dot x^2+\dot y^2)$$ the potential energy is $$~U=-m\,g\,y~$$ and the kinetic energy of the small cylinder $$T_r=\frac 12\,I_c\,\dot\varphi^2=\frac 14\,m\,r^2\,\dot\varphi^2$$

The Langragian

$$L=T_k+T_r-U=L(\varphi~,\dot\varphi)$$

with EL and $$~\sin(a\,\varphi)=a\,\varphi$$

$$\omega^2=\frac 23\frac{g}{R-r}$$


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# Covalent Bonds

Show/Hide Sub-topics (Matter | O Level)

Covalent bonds are usually formed between atoms of non-metals.

• There is a sharing of electrons between the two atoms in order for them to achieve the stable noble gas configuration.
• Each atom contribute the same number of electrons for the bond
• If the two atoms share one pair of electrons, it is a single bond
• Share two pairs of electrons $\rightarrow$ double bond
• Share three pairs of electrons $\rightarrow$ triple bond

Before we go into more details about covalent bonds, we will need to distinguish between molecules and compounds.

• A molecule is a group of two or more atoms chemically bonded together. Examples: Hydrogen molecules ($\text{H}_{2}$), Oxygen molecules ($\text{O}_{2}$)
• A compound is a group of atoms created by chemically combining two or more elements together. Examples: Water ($\text{H}_{2}\text{O}$), carbon dioxide ($\text{CO}_{2}$)

### Some examples to illustrate covalent bonding in molecules and compounds

Covalent bond in hydrogen molecules:

• Electron configuration of hydrogen is 1
• Number of valence electron: 1
• Each hydrogen atom will contribute 1 electron for sharing.
• This will result in both atoms sharing 1 pair of electrons, hence achieving the stable noble gas configuration. (Because they will now have 2 valence electrons $\rightarrow$ similar to helium)
• A single bond has formed between the two hydrogen atoms

Covalent bond in oxygen molecules:

• Electron configuration of oxygen is 2.6
• Number of valence electron is 6
• Each oxygen atoms will contribute 2 electrons for sharing
• This will result in both atoms sharing 2 pairs of electrons. Number of valence electrons will then be 8.
• Stable noble gas configuration is achieved
• A double bond has formed between the two oxygen atoms

Covalent bond in nitrogen molecules:

• Electron configuration of nitrogen is 2.5
• Number of valence electron is 5
• Each nitrogen atoms will contribute 3 electrons for sharing
• This will result in both atoms sharing 3 pairs of electrons. Number of valence electrons will then be 8.
• Stable noble gas configuration is achieved
• A triple bond has formed between the two nitrogen atoms

Covalent bond in water:

• Electron configuration of oxygen is 2.6 while electron configuration of hydrogen is 1
• Each hydrogen atom can share 1 electron
• Oxygen need 2 electrons to achieve the stable noble gas configuration
• Hence, two hydrogen atoms will pair with one oxygen atom
• The oxygen atom will form 2 covalent bonds – 1 for each hydrogen atom.
• Two single bonds has formed between the two hydrogen atoms and one oxygen atom

### Properties of covalent compounds

• Low melting and boiling points
• Covalent compounds do not conduct electricity
• Covalent compounds dissolve in organic solvent but do not dissolve in water

Reason for low melting and boiling points: The molecules are held together by very weak Van der Waals’ forces of attraction. Hence, very little energy is required to overcome the weak intermolecular forces of attraction.

Reason for non-conductance of electricity: Covalent compounds do not carry charges. Hence, there are no ions to carry electricity


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Generalized continued fraction

In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity. They are useful in the theory of infinite summation of series.

A generalized continued fraction is an expression such as:

$x = \frac{b_1}{a_1\pm\frac{b_2}{a_2\pm\frac{b_3}{a_3+\,\cdots}}}$

where all symbols are integers. A convenient notation is

$\frac{b_1}{a_1\pm}\, \frac{b_2}{a_2\pm}\, \frac{b_3}{a_3\pm}\ldots$

The successive convergents are formed in a similar way to those of continued fractions. If all $\pm$ signs are positive,

$x_1=\frac{b_1}{a_1}\qquad x_2=\frac{a_2b_1}{a_2a_1+b_2}\qquad x_3=\frac{a_3a_2b_1+b_3b_1}{a_3(a_2a_1+b_2)+b_2a_1}$

If we write xn = pn / qn, then

$p_{n+1}=a_{n+1}p_n+b_{n+1}p_{n-1},\qquad q_{n+1}=a_{n+1}q_n+b_{n+1}q_{n-1}$

(if the signs are negative, replace "+" with "-" in the above formula).

If the positive sign is chosen, then (as for ordinary continued fractions) all convergents of odd order are greater than x but uniformly decrease; and all convergents of even order are less than x but uniformly increase.

Thus odd convergents tend to a limit, and even convergents tend to a limit. If the limits are not equal, the continued fraction is said to be oscillating. To determine whether the limits are equal, define

$s_n= \frac{a_na_{n+1}}{b_{n+1}}.$

Then if $\exists\epsilon>0$ and integer n0 such that n > n0 implies sn > ε, then the limits are equal and the continued fraction has a definite value.

 Contents

Generalized continued fractions and series

The series

$\frac{1}{u_1}+ \frac{1}{u_2}+ \frac{1}{u_3}+ \cdots+ \frac{1}{u_n}$

is equal to the continued fraction

$\frac{1}{u_1-}\, \frac{u_1^2}{u_1+u_2-}\, \frac{u_2^2}{u_2+u_3-}\cdots \frac{u_{n-1}^2}{u_{n-1}+u_n}.$

The series

$\frac{1}{a_0}+\frac{x}{a_0a_1}+\frac{x^2}{a_0a_1a_2}+ \cdots +\frac{x^n}{a_0a_1a_2\ldots a_n}$

is equal to

$\frac{1}{a_0-}\, \frac{a_0x}{a_1+x-}\, \frac{a_1x}{a_2+x-}\, \cdots \frac{a_{n-1}x}{a_n-x}$

Examples

$\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots= \frac{x}{1+}\, \frac{1^2x}{2-x+}\, \frac{2^2x}{3-2x+}\, \frac{3^2x}{4-3x+}\ldots$
$\exp(x)=1+x+\frac{x^2}{2!}+\ldots= 1+\frac{x}{1-}\, \frac{x}{x+2-}\, \frac{2x}{x+3-}\, \frac{3x}{x+4-}\, \ldots$
$\exp(x)=\frac{1}{1-}\, \frac{z}{1+}\, \frac{z}{2-}\, \frac{z}{3+}\, \frac{z}{2-}\, \frac{z}{5+}\, \frac{z}{2-}\ldots\qquad\forall z\in C$

Higher dimensions

Another meaning for generalized continued fraction would be a generalisation to higher dimensions. For example, there is a close relationship between the continued fraction for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Therefore one can ask for something relating to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomials in several real numbers, take the logarithmic form and consider how small it can be.

There have been numerous attempts, in fact, to construct a generalised theory. Two notable ones are those of Georges Poitou and George Szekeres .

References

• William B. Jones and W.J. Thron, "Continued Fractions Analytic Theory and Applications", Addison-Wesley, 1980. (Covers both analytic theory and history).
• Lisa Lorentzen and Haakon Waadeland, "Continued Fractions with Applications", North Holland, 1992. (Covers primarily analytic theory and some arithmetic theory).
• Oskar Perron, B.G. Teubner, "Die Lehre Von Den Kettenbruchen" Band I, II, 1954.
• George Szekeres, "Multidimensional Continued Fractions." G.Ann. Univ. Sci. Budapest Eotvos Sect. Math. 13, 113-140, 1970.
• H.S. Wall, "Analytic Theory of Continued Fractions", Chelsea, 1973.
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