L5(SHO) copy.ppt

Size: px
Start display at page:

Download "L5(SHO) copy.ppt"

Transcription

1 SIMPLE HARMONIC MOION: NEWON S LAW PRIOR READING: Main 1.1, 2.1 aylor 5.1,

2 r; r = L he siple pendulu Energy approach g t = t t ' = 0 # 0 " " 0 2 I d" ' ( ( 1# cos" ')) E # gl c d ' 2gL c 1" cos 2 L ax " 1" cos ' c ( ( )) t = # 0 2g L c d ' ( cos '" cos ) ax 1

3 L c = L he siple pendulu Energy approach t = # 0 2g L c d ' ( cos '" cos ) ax g 0 " g L angular frequency t = d ' # = 1 arcsin ' g ( L 2 ax " ' 2 ) $ 0 0 ax 0 t = arcsin " # arcsin " 0 " ax " # "# ax $ $ " ( t) = " ax sin (#t +$ ) 2

4 r; r = L he siple pendulu g = r " = I "" " F # =? Newton Known torque = Lgsin" ˆ# I "" = L 2 "" ˆ = " g L sin his is NO a restoring force proportional to displaceent (Hooke s law otion) in general, but IF we consider sall otion, I IS Expand the sin series sin = "

5 L g he siple pendulu in the liit of sall angular displaceents = " g L sin # = " g L or + g L = 0 Copare with x + k x = 0 What is (t) such that the above equation is obeyed? is a variable that describes position t is a paraeter that describes tie "dot" and "double dot" ean differentiate w.r.t. tie g, L are known constants, deterined by the syste. 4

6 L g REVIEW PENDULUM + g L = 0 (t) = Ce pt C, p are unknown (for now) constants, possibly coplex (t) = p 2 Ce pt = p 2 (t) Substitute: p 2 + g L = 0 p = ±i g L = ±i 0 p is now known (but C is not). Note that ω 0 is NO a new quantity It is just a rewriting of old ones - partly 5 shorthand, but also "ω" eans "frequency" to physicists

7 L g REVIEW PENDULUM WO possibilities. general solution is the su of the two and it ust be real (all angles are real). (t) = Ce i" 0t + C 'e #i" 0t If we force C' = C* (coplex conjugate of C), then x (t) is real, and there are only 2 constants, Re[C], and I[C]. A second order DEQ can deterine only 2 arbitrary constants. (t) = Ce i" 0t + C * e #i" 0t Siple haronic otion 6

8 L g (0) = C e i" C *ei" = C + C* = 2Re[C] 1 " Re[C] = 0 REVIEW PENDULUM 1 (t) = Ce i" 0t + C * e #i" 0t Re[C], I[C] chosen to fit initial conditions. Exaple: (0) = 0 rad and ddt(0) = 0.2 rad/sec (0) = i" 0 C e i" 0 0 i" C 0 *ei" rad / s = i" 0 C C * ( ) = i" 0 2iI[C] " I[C] = 0.2 2" 0 = 0.1 " 0 1 C = 0 + i " # $# cartesian = 0.1 e i " 2 ; C* =? "# 0 polar 7

9 L REVIEW PENDULUM (t) = Ce i" 0t + C * e i" 0t ; C = 0.1 e i # 2 " 0 g (t) = 0.1 e i 2 e i 0t e i 2 e i 0t 0 0 ( ) (t) = 0.1 e i ( 0t+ /2 ) + e i ( 0t+ /2) 0 ( ( )) (t) = 0.1 2cos 0 t + / 2 0 (t) = 0.2 " % cos $ 0 t + / 2' 0 # " & A 8

10 Reeber, all these are equivalent fors. All of the have a known ω 0 =(g/l) 1/2, and all have 2 ore undeterined constants that we find how? (t) = Acos (" 0 t + #) (t) = B p cos" 0 t + B q sin" 0 t (t) = C exp( i" 0 t) + C * exp (#i" 0 t) (t) = Re #$ D exp( i" 0 t) % & Do you reeber how the A, B, C, D constants are related? If not, go back and review until it becoes second nature 9

11 L he siple pendulu ("siple" here eans a point ass; your lab deals with a plane pendulu) = " g L g "(t) = " cos (# t + $ ) ax 0 " 0 = g L siple haronic otion ( potential confusion A siple pendulu does not always execute siple haronic otion ; it does so only in the liit of sall aplitude.) = 2" L g Period does not depend on ax, φ 10

12 k Free, undaped oscillators other exaples L k x r; r = L No friction x = kx I C q q = 1 LC q Coon notation for all " # g L + " 0 2 = 0 g

13 he following slides siply repeat the previous discussion, but now for a ass on a spring, and for a series LC circuit 12

14 REVIEW MASS ON IDEAL SPRING k F(x) = x F(x) = kx Newton Particular type of force., k known k x kx = x x + k x = 0 Linear, 2nd order differential equation What is x(t) such that the above equation is obeyed? x is a variable that describes position t is a paraeter that describes tie "dot" and "double dot" ean differentiate w.r.t. tie, k are known constants 13

15 REVIEW MASS ON IDEAL SPRING k x + k x = 0 k x x(t) = Ce pt C, p are unknown (for now) constants, possibly coplex x(t) = p 2 Ce pt = p 2 x(t) Substitute: p 2 x + k x = 0 p = ±i k = ±i 0 p is now known. Note that ω 0 is NO a new quantity It is just a rewriting of old ones - partly shorthand, but also ω 14 eans frequency to physicists

16 k x(t) = Acos ( 0 t + ") A, φ chosen to fit initial conditions: x(0) = x 0 and v(0) = v 0 k x x 0 = Acos v 0 = "# 0 Asin Square and add: Divide: x v 2 0 = ( 2 A2 cos 2 " + sin 2 ") = A 2 0 v 0 " 0 x 0 = tan# 15

17 x(t) = Acos ( 0 t + ") x(t) = x v 2 0 cos # k 2 0 t + arctan # "v 0 $ % $ % 0 x 0 & & ' ( ' ( x(t) = Acos cos" 0 t # Asin sin" 0 t x(t) = Aei 2 ei" 0t + Ae#i 2 e#i" 0t x(t) = Re Ae i e i" 0t #$ % & 2 arbitrary constants (A, φ) because 2nd order linear differential equation 16

18 Position: x(t) = Acos ( 0 t + ") A, φ are unknown constants - ust be deterined fro initial conditions ω 0, in principle, is known and is a characteristic of the physical syste Velocity: Acceleration: dx dt x(t) = "# 0 Asin # 0 t + $ ( ) d 2 x dt x(t) = "# Acos (# 0 t + $ ) = "# 0 2 x(t) his type of pure sinusoidal otion with a single frequency is called SIMPLE HARMONIC MOION 17

19 HE LC CIRCUI L V L + V C = 0 Kirchoff s law (not Newton this tie) I C q V L = L di dt = L d 2 q dt 2 = Lq V C = q C q = 1 LC q

2

2 2011 8 6 2011 5 7 [1] 1 2 i ii iii i 3 [2] 4 5 ii 6 7 iii 8 [3] 9 10 11 cf. Abstracts in English In terms of democracy, the patience and the kindness Tohoku people have shown will be dealt with as an exception.

More information

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin 2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ

More information

Euler Appendix cos, sin 2π t = 0 kx = 0, 2π x = 0 (wavelength)λ kλ = 2π, k = 2π/λ k (wavenumber) x = 0 ωt = 0, 2π t = 0 (period)t T = 2π/ω ω = 2πν (fr

Euler Appendix cos, sin 2π t = 0 kx = 0, 2π x = 0 (wavelength)λ kλ = 2π, k = 2π/λ k (wavenumber) x = 0 ωt = 0, 2π t = 0 (period)t T = 2π/ω ω = 2πν (fr This manuscript is modified on March 26, 2012 3 : 53 pm [1] 1 ( ) Figure 1: (longitudinal wave) (transverse wave). P 7km S 4km P S P S x t x u(x, t) t = t 0 = 0 f(x) f(x) = u(x, 0) v +x (Fig.2) ( ) δt

More information

25 II :30 16:00 (1),. Do not open this problem booklet until the start of the examination is announced. (2) 3.. Answer the following 3 proble

25 II :30 16:00 (1),. Do not open this problem booklet until the start of the examination is announced. (2) 3.. Answer the following 3 proble 25 II 25 2 6 13:30 16:00 (1),. Do not open this problem boolet until the start of the examination is announced. (2) 3.. Answer the following 3 problems. Use the designated answer sheet for each problem.

More information

最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 3 版 1 刷発行時のものです.

最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 3 版 1 刷発行時のものです. 最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6

More information

学習内容と日常生活との関連性の研究-第2部-第4章-1

学習内容と日常生活との関連性の研究-第2部-第4章-1 69 V A V + A V A 2A 2 http://www.jba-hp.jp/ http://www.kbn3.com/ http://www.usba.org/ 70 (1) (1996)35 7 pp.28-33 (2) (1994) 71 () 3 1 1 99 8 1 10 1 11.3 2.5 1 100 11.4 30.9 1 72 (1) http://www.stat.go.jp/data/zensho/1999/zuhyou/a906-6.xls

More information

Webster's New World Dictionary of the American Language, College Edition. N. Y. : The World Publishing Co., 1966. [WNWD) Webster 's Third New International Dictionary of the English Language-Unabridged.

More information

2 56

2 56 30 2006 pp. 55 67 1 Received October 31, 2006 Xu Weiwas an artist who was famous as a painter and a calligrapher and a drama writer in late Ming dynasty. But it is not known that he leaves poetry as a

More information

0 Speedy & Simple Kenji, Yoshio, and Goro are good at English. They have their ways of learning. Kenji often listens to English songs and tries to remember all the words. Yoshio reads one English book every

More information

国際恋愛で避けるべき7つの失敗と解決策

国際恋愛で避けるべき7つの失敗と解決策 7 http://lovecoachirene.com 1 7! 7! 1 NOT KNOWING WHAT YOU WANT 2 BEING A SUBMISSIVE WOMAN 3 NOT ALLOWING THE MAN TO BE YOUR HERO 4 WAITING FOR HIM TO LEAD 5 NOT SPEAKING YOUR MIND 6 PUTTING HIM ON A PEDESTAL

More information

WASEDA University Internship Guide http://www.waseda.jp/career/internship/ 1 2 3 For International Students International students who are interested in internships with Japanese corporations must be

More information

Title 社 会 化 教 育 における 公 民 的 資 質 : 法 教 育 における 憲 法 的 価 値 原 理 ( fulltext ) Author(s) 中 平, 一 義 Citation 学 校 教 育 学 研 究 論 集 (21): 113-126 Issue Date 2010-03 URL http://hdl.handle.net/2309/107543 Publisher 東 京

More information

123-099_Y05…X…`…‘…“†[…h…•

123-099_Y05…X…`…‘…“†[…h…• 1. 2 1993 2001 2 1 2 1 2 1 99 2009. 1982 250 251 1991 112 115 1988 75 2004 132 2006 73 3 100 3 4 1. 2. 3. 4. 5. 6.. 3.1 1991 2002 2004 3 4 101 2009 3 4 4 5 1 5 6 1 102 5 6 3.2 2 7 8 2 X Y Z Z X 103 2009

More information

4/15 No.

4/15 No. 4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem

More information

What s your name? Help me carry the baggage, please. politeness What s your name? Help me carry the baggage, please. iii

What s your name? Help me carry the baggage, please. politeness What s your name? Help me carry the baggage, please. iii What s your name? Help me carry the baggage, please. politeness What s your name? Help me carry the baggage, please. iii p. vi 2 50 2 2016 7 14 London, Russell Square iv iii vi Part 1 1 Part 2 13 Unit

More information

浜松医科大学紀要

浜松医科大学紀要 On the Statistical Bias Found in the Horse Racing Data (1) Akio NODA Mathematics Abstract: The purpose of the present paper is to report what type of statistical bias the author has found in the horse

More information

Public relations brochure of Higashikawa 4 2016 April No.750 CONTENTS 3 13 15 16 20 22 23 24 25 26 28 36 37 38 2

Public relations brochure of Higashikawa 4 2016 April No.750 CONTENTS 3 13 15 16 20 22 23 24 25 26 28 36 37 38 2 4 2016 April No.750 Public relations brochure of Higashikawa 4 2016 April No.750 CONTENTS 3 13 15 16 20 22 23 24 25 26 28 36 37 38 2 3 5 4 7 6 9 8 11 10 12 13 14 キトウシ 森 林 公 園 シーズン イン HIGASHIKAWA TOWN NEWS

More information

I

I I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

C. S2 X D. E.. (1) X S1 10 S2 X+S1 3 X+S S1S2 X+S1+S2 X S1 X+S S X+S2 X A. S1 2 a. b. c. d. e. 2

C. S2 X D. E.. (1) X S1 10 S2 X+S1 3 X+S S1S2 X+S1+S2 X S1 X+S S X+S2 X A. S1 2 a. b. c. d. e. 2 I. 200 2 II. ( 2001) 30 1992 Do X for S2 because S1(is not desirable) XS S2 A. S1 S2 B. S S2 S2 X 1 C. S2 X D. E.. (1) X 12 15 S1 10 S2 X+S1 3 X+S2 4 13 S1S2 X+S1+S2 X S1 X+S2. 2. 3.. S X+S2 X A. S1 2

More information

平成29年度英語力調査結果(中学3年生)の概要

平成29年度英語力調査結果(中学3年生)の概要 1 2 3 1 そう思う 2 どちらかといえば そう思う 3 どちらかといえば そう思わない 4 そう思わない 4 5 楽しめるようになりたい 6 1 そう思う 2 どちらかといえば そう思う 3 どちらかといえば そう思わない 4 そう思わない 7 1 そう思う 2 どちらかといえば そう思う 3 どちらかといえば そう思わない 4 そう思わない 8 1 そう思う 2 どちらかといえば そう思う

More information

AtCoder Regular Contest 073 Editorial Kohei Morita(yosupo) A: Shiritori if python3 a, b, c = input().split() if a[len(a)-1] == b[0] and b[len(

AtCoder Regular Contest 073 Editorial Kohei Morita(yosupo) A: Shiritori if python3 a, b, c = input().split() if a[len(a)-1] == b[0] and b[len( AtCoder Regular Contest 073 Editorial Kohei Morita(yosupo) 29 4 29 A: Shiritori if python3 a, b, c = input().split() if a[len(a)-1] == b[0] and b[len(b)-1] == c[0]: print( YES ) else: print( NO ) 1 B:

More information

ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k.

ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. K E N Z OU 8 9 8. F = kx x 3 678 ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. D = ±i dt = ±iωx,

More information

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e

More information

L1 What Can You Blood Type Tell Us? Part 1 Can you guess/ my blood type? Well,/ you re very serious person/ so/ I think/ your blood type is A. Wow!/ G

L1 What Can You Blood Type Tell Us? Part 1 Can you guess/ my blood type? Well,/ you re very serious person/ so/ I think/ your blood type is A. Wow!/ G L1 What Can You Blood Type Tell Us? Part 1 Can you guess/ my blood type? 当ててみて / 私の血液型を Well,/ you re very serious person/ so/ I think/ your blood type is A. えーと / あなたはとっても真面目な人 / だから / 私は ~ と思います / あなたの血液型は

More information

The Key Questions about Today's "Experience Loss": Focusing on Provision Issues Gerald ARGENTON These last years, the educational discourse has been focusing on the "experience loss" problem and its consequences.

More information

Z B- B- PHP - - [ ] PHP New York Times, December,,. The origins of the Japan-U.S. War and Adm. Isoroku Yamamoto Katsuhiko MATSUKAWA Abstract There are huge amount of studies concerning the origins

More information

2 10 The Bulletin of Meiji University of Integrative Medicine 1,2 II 1 Web PubMed elbow pain baseball elbow little leaguer s elbow acupun

2 10 The Bulletin of Meiji University of Integrative Medicine 1,2 II 1 Web PubMed elbow pain baseball elbow little leaguer s elbow acupun 10 1-14 2014 1 2 3 4 2 1 2 3 4 Web PubMed elbow pain baseball elbow little leaguer s elbow acupuncture electric acupuncture 2003 2012 10 39 32 Web PubMed Key words growth stage elbow pain baseball elbow

More information

〈論文〉興行データベースから「古典芸能」の定義を考える

〈論文〉興行データベースから「古典芸能」の定義を考える Abstract The long performance database of rakugo and kabuki was totaled, and it is found that few programs are repeated in both genres both have the frequency differential of performance. It is a question

More information

画像工学特論

画像工学特論 .? (x i, y i )? (x(t), y(t))? (x(t)) (X(ω)) Wiener-Khintchine 35/97 . : x(t) = X(ω)e jωt dω () π X(ω) = x(t)e jωt dt () X(ω) S(ω) = lim (3) ω S(ω)dω X(ω) : F of x : [X] [ = ] [x t] Power spectral density

More information

在日外国人高齢者福祉給付金制度の創設とその課題

在日外国人高齢者福祉給付金制度の創設とその課題 Establishment and Challenges of the Welfare Benefits System for Elderly Foreign Residents In the Case of Higashihiroshima City Naoe KAWAMOTO Graduate School of Integrated Arts and Sciences, Hiroshima University

More information

02-量子力学の復習

02-量子力学の復習 4/17 No. 1 4/17 No. 2 4/17 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = 2 2m 2 ψ(r,t)+v(r)ψ(r,t) ψ(r,t) Wave function ψ(r,t) = ϕ(r)e iωt steady state 2 2m 2 ϕ(r)+v(r)ϕ(r) = εϕ(r)

More information

Social ecology is based on looking rather than on analysis It is based on perception This I submit distinguishes it what is normally meant by a science. It is not only that it can not be reductionist.

More information

千葉県における温泉地の地域的展開

千葉県における温泉地の地域的展開 1) 1999 11 50 1948 23) 2 2519 9 3) 2006 4) 151 47 37 1.2 l 40 3.6 15 240 21 9.2 l 7. 210 1972 5) 1.9 l 5 1 0.2 l 6 1 1972 1.9 0.4 210 40-17- 292006 34 6 l/min.42 6) 2006 1 1 2006 42 60% 5060 4050 3040

More information

数学の基礎訓練I

数学の基礎訓練I I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

<4D F736F F D208BB38DDE5F F4390B394C52E646F6378>

<4D F736F F D208BB38DDE5F F4390B394C52E646F6378> Introduction [Track 1 13] Would you like to try our strawberry smoothie? No thank you 1. Hi. Would you like to try our parfait? Hi. Do you want to try our parfait? 1. No thanks. Can I get a #1(number one),

More information

A Contrastive Study of Japanese and Korean by Analyzing Mistranslation from Japanese into Korean Yukitoshi YUTANI Japanese, Korean, contrastive study, mistranslation, Japanese-Korean dictionary It is already

More information

第17回勉強会「英語の教え方教室」報告

第17回勉強会「英語の教え方教室」報告 -1- -2- -3- -4- -5- -6- -7- -8- -9- When I get older I will be stronger They'll call me freedom, just like a wavin' flag When I get older, I will be stronger They'll call me freedom just like a wavin'

More information

LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t)

LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t) 338 7 7.3 LCR 2.4.3 e ix LC AM 7.3.1 7.3.1.1 m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x k > 0 k 5.3.1.1 x = xt 7.3 339 m 2 x t 2 = k x 2 x t 2 = ω 2 0 x ω0 = k m ω 0 1.4.4.3 2 +α 14.9.3.1 5.3.2.1 2 x

More information

-1- -2- -1- A -1- -2- -3- -1- -2- -1- -2- -1- http://www.unicef.or.jp/kenri.syouyaku.htm -2- 1 2 http://www.stat.go.jp/index.htm http://portal.stat.go.jp/ 1871.8.28 1.4 11.8 42.7 19.3

More information

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier

1 Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier Fourier analog digital Fourier Fourier Fourier Fourier Fourier Fourier Green Fourier Fourier Fourier Fourier etc * 1 Fourier Fourier Fourier (DFT Fourier (FFT Heat Equation, Fourier Series, Fourier Transform, Discrete Fourier Transform, etc Yoshifumi TAKEDA 1 Abstract Suppose that u is

More information

fx-9860G Manager PLUS_J

fx-9860G Manager PLUS_J fx-9860g J fx-9860g Manager PLUS http://edu.casio.jp k 1 k III 2 3 1. 2. 4 3. 4. 5 1. 2. 3. 4. 5. 1. 6 7 k 8 k 9 k 10 k 11 k k k 12 k k k 1 2 3 4 5 6 1 2 3 4 5 6 13 k 1 2 3 1 2 3 1 2 3 1 2 3 14 k a j.+-(),m1

More information

005 1571 1630 17 1546 1601 16 1642 1727

005 1571 1630 17 1546 1601 16 1642 1727 I Takamitsu Sawa / 1561~1626 004 2010 / No.384 005 1571 1630 17 1546 1601 16 1642 1727 006 2010 / No.384 confirm refute verify significant 1902 1994 piecemeal engineering 1958 historicism 20 007 1990 90

More information

2013 Vol.1 Spring 2013 Vol.1 SPRING 03-3208-2248 C O N T E N T S 2013 03-3208-2248 2 3 4 7 Information 6 8 9 11 10 73 94 11 32 37 41 96 98 100 101 103 55 72 1 2 201345135016151330 3 1 2 URL: http://www.wul.waseda.ac.jp/clib/tel.03-3203-5581

More information

_05.indd

_05.indd : 82 1 2013 7 1 1 1 1 2 3 4 2 : 3 5 1 2 3 4 5 6 3 1 2 3 4 : 6 1 2 3 4 5 6 7 8 9 1 2 1998 The works, political, metaphysical, and chronological, of the late Sir James Steuart of Coltness, vol. I, London,

More information

08-Note2-web

08-Note2-web r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)

More information

Numerical Analysis II, Exam End Term Spring 2017

Numerical Analysis II, Exam End Term Spring 2017 H. Ammari W. Wu S. Yu Spring Term 2017 Numerical Analysis II ETH Zürich D-MATH End Term Spring 2017 Problem 1 Consider dx = f(t, x), t [0, T ] dt x(0) = x 0 R [28 Marks] with f C subject to the Lipschitz

More information

2 236

2 236 28 2004 pp. 235 245 1 Received November 5, 2004 In the eighth volume of Confessiones is a famous and critical passage in which Augustine describes the process leading to his conversion. It is a long and

More information

untitled

untitled GEEOO- -DISSAASSEERR PRREEVVEEI IOO SYYSSEEM - /6 - Spectru Analysis of Earthquae and Fourier ransfor *, he original of this docuent was fro the above text boo. - Strong-Motion Acceleration: SMA Iperial

More information

CONTENTS Public relations brochure of Higashikawa November No.745 Higashikawa 215 November 2

CONTENTS Public relations brochure of Higashikawa November No.745 Higashikawa 215 November 2 11 215 November No.745 CONTENTS 2 6 12 17 17 18 2 21 22 23 24 28 3 31 32 Public relations brochure of Higashikawa 11 215 November No.745 Higashikawa 215 November 2 816,18 832,686 8,326,862 196,93 43,573

More information

untitled

untitled SATO Kentaro Milk and its by-products are naturally nutritious food, and people in ancient Japan enjoyed tasting them as foods, drinks, or medicines. On the other hand, milk and its by-products were closely

More information

alternating current component and two transient components. Both transient components are direct currents at starting of the motor and are sinusoidal

alternating current component and two transient components. Both transient components are direct currents at starting of the motor and are sinusoidal Inrush Current of Induction Motor on Applying Electric Power by Takao Itoi Abstract The transient currents flow into the windings of the induction motors when electric sources are suddenly applied to the

More information

先端社会研究 ★5★号/4.山崎

先端社会研究 ★5★号/4.山崎 71 72 5 1 2005 7 8 47 14 2,379 2,440 1 2 3 2 73 4 3 1 4 1 5 1 5 8 3 2002 79 232 2 1999 249 265 74 5 3 5. 1 1 3. 1 1 2004 4. 1 23 2 75 52 5,000 2 500 250 250 125 3 1995 1998 76 5 1 2 1 100 2004 4 100 200

More information

R R S K K S K S K S K S K S Study of Samuhara Belief : Transformation from Protection against Injuries to Protection against Bullets WATANABE Kazuhiro Samuhara, which is a group of letters like unfamiliar

More information

Page 1 of 6 B (The World of Mathematics) November 20, 2006 Final Exam 2006 Division: ID#: Name: 1. p, q, r (Let p, q, r are propositions. ) (10pts) (a

Page 1 of 6 B (The World of Mathematics) November 20, 2006 Final Exam 2006 Division: ID#: Name: 1. p, q, r (Let p, q, r are propositions. ) (10pts) (a Page 1 of 6 B (The World of Mathematics) November 0, 006 Final Exam 006 Division: ID#: Name: 1. p, q, r (Let p, q, r are propositions. ) (a) (Decide whether the following holds by completing the truth

More information

1 ( ) 1 1 ( ) 1!! Shinji Akematsu 1

1 ( ) 1 1 ( ) 1!! Shinji Akematsu   1 30 + http://know-star.com/ 1 ( ) 1 1 ( ) 1!! Shinji Akematsu http://know-star.com/ info@know-star.com 1 2 (1) 1, 5, 13, 25,... 2 2 1,5,13,25,... 7 221 2 (2) (1) n (i) (ii) n 2 n = 2, 3, 4 n = 2 7, 1 +

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

More information

2015 8 65 87. J. Osaka Aoyama University. 2015, vol. 8, 65-87. 20 * Recollections of the Pacific War in the eyes of a school kid Hisao NAGAOKA Osaka Aoyama Gakuen Summary Seventy years have passed since

More information

P

P 03-3208-22482013 Vol.2 Summer & Autumn 2013 Vol.2 Summer & Autumn 90 527 P.156 611 91 C O N T E N T S 2013 03-3208-2248 2 3 4 6 Information 7 8 9 10 2 115 154 10 43 52 61 156 158 160 161 163 79 114 1 2

More information

™…

™… Review The Secret to Healthy Long Life Decrease in Oxidative and Mental Stress My motto is Health is not all. But nothing can be done without health. Health is the most important requisite for all human

More information

Motivation and Purpose There is no definition about whether seatbelt anchorage should be fixed or not. We tested the same test conditions except for t

Motivation and Purpose There is no definition about whether seatbelt anchorage should be fixed or not. We tested the same test conditions except for t Review of Seatbelt Anchorage and Dimensions of Test Bench Seat Cushion JASIC Motivation and Purpose There is no definition about whether seatbelt anchorage should be fixed or not. We tested the same test

More information

10 2000 11 11 48 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) CU-SeeMe NetMeeting Phoenix mini SeeMe Integrated Services Digital Network 64kbps 16kbps 128kbps 384kbps

More information

Title < 論文 > 公立学校における在日韓国 朝鮮人教育の位置に関する社会学的考察 : 大阪と京都における 民族学級 の事例から Author(s) 金, 兌恩 Citation 京都社会学年報 : KJS = Kyoto journal of so 14: 21-41 Issue Date 2006-12-25 URL http://hdl.handle.net/2433/192679 Right

More information

4 8 6 1 1 4 8 2001, 3 2 Marshall [1890]1920, 240 19 1 2001 2008 1990 1997 2007 2 Marshall [1890]1920

4 8 6 1 1 4 8 2001, 3 2 Marshall [1890]1920, 240 19 1 2001 2008 1990 1997 2007 2 Marshall [1890]1920 8 2011. 3 199 213 D. H. D. H. 1877-1953 The Evolution of Industry 1911 1904 1908 1919 1922 45 1906 Industrial Combination 2009 20 4 8 6 1 1 4 8 2001, 3 2 Marshall [1890]1920, 240 19 1 2001 2008 1990 1997

More information

Kyushu Communication Studies 第2号

Kyushu Communication Studies 第2号 Kyushu Communication Studies. 2004. 2:1-11 2004 How College Students Use and Perceive Pictographs in Cell Phone E-mail Messages IGARASHI Noriko (Niigata University of Health and Welfare) ITOI Emi (Bunkyo

More information

The Indirect Support to Faculty Advisers of die Individual Learning Support System for Underachieving Student The Indirect Support to Faculty Advisers of the Individual Learning Support System for Underachieving

More information

chap1.dvi

chap1.dvi 1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f

More information

126 学習院大学人文科学論集 ⅩⅩⅡ(2013) 1 2

126 学習院大学人文科学論集 ⅩⅩⅡ(2013) 1 2 125 126 学習院大学人文科学論集 ⅩⅩⅡ(2013) 1 2 127 うつほ物語 における言語認識 3 4 5 128 学習院大学人文科学論集 ⅩⅩⅡ(2013) 129 うつほ物語 における言語認識 130 学習院大学人文科学論集 ⅩⅩⅡ(2013) 6 131 うつほ物語 における言語認識 132 学習院大学人文科学論集 ⅩⅩⅡ(2013) 7 8 133 うつほ物語 における言語認識 134

More information

2012専門分科会_new_4.pptx

2012専門分科会_new_4.pptx d dt L L = 0 q i q i d dt L L = 0 r i i r i r r + Δr Δr δl = 0 dl dt = d dt i L L q i q i + q i i q i = q d L L i + q i i dt q i i q i = i L L q i L = 0, H = q q i L = E i q i i d dt L q q i i L = L(q

More information

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (

( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) ( 6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b

More information

DI DI

DI DI 7 UCHIDA, Suguru OKADA, Akira EGUCHI, Hiroshi 1 17 1827 4 6 37 8 12 1 7 2 3 8, 3 1 1,42 3 36.1 18 4 29.6 13 12 28 16 6 18 46 6 4 Vol.11 No.1 8 Spring 2 2.1 6 14 6 18 7 9 16.4 12.4 17 8 11.3 7.6 16 41 7

More information

駒田朋子.indd

駒田朋子.indd 2 2 44 6 6 6 6 2006 p. 5 2009 p. 6 49 12 2006 p. 6 2009 p. 9 2009 p. 6 2006 pp. 12 20 2005 2005 2 3 2005 An Integrated Approach to Intermediate Japanese 13 12 10 2005 8 p. 23 2005 2 50 p. 157 2 3 1 2010

More information

FAX-760CLT

FAX-760CLT FAX-760CLT ;; yy 1 f a n l p w s m t v y k u c j 09,. i 09 V X Q ( < N > O P Z R Q: W Y M S T U V 1 2 3 4 2 1 1 2 1 2 j 11 dd e i j i 1 ; 3 oo c o 1 2 3 4 5 6 j12 00 9 i 0 9 i 0 9 i 0 9 i oo

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

10 11 12 33.4 1 open / window / I / shall / the? 79.3 2 something / want / drink / I / to. 43.5 3 the way / you / tell / the library / would / to / me

10 11 12 33.4 1 open / window / I / shall / the? 79.3 2 something / want / drink / I / to. 43.5 3 the way / you / tell / the library / would / to / me -1- 10 11 12 33.4 1 open / window / I / shall / the? 79.3 2 something / want / drink / I / to. 43.5 3 the way / you / tell / the library / would / to / me? 28.7 4 Miyazaki / you / will / in / long / stay

More information

QMI_10.dvi

QMI_10.dvi 25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx

More information

QMI_09.dvi

QMI_09.dvi 25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h

More information

自分の天職をつかめ

自分の天職をつかめ Hiroshi Kawasaki / / 13 4 10 18 35 50 600 4 350 400 074 2011 autumn / No.389 5 5 I 1 4 1 11 90 20 22 22 352 325 27 81 9 3 7 370 2 400 377 23 83 12 3 2 410 3 415 391 24 82 9 3 6 470 4 389 362 27 78 9 5

More information

紀要1444_大扉&目次_初.indd

紀要1444_大扉&目次_初.indd 1. Hi, How are you? / What s up? / How s it going? A / Nice talking to you. 2. Oh really? / That s great! / That s A, B interesting! / Are you serious? / Sounds good. / You too! / That s too bad. 3. Sorry?

More information

L3 Japanese (90570) 2008

L3 Japanese (90570) 2008 90570-CDT-08-L3Japanese page 1 of 15 NCEA LEVEL 3: Japanese CD TRANSCRIPT 2008 90570: Listen to and understand complex spoken Japanese in less familiar contexts New Zealand Qualifications Authority: NCEA

More information

ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o

ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o dif engine 2017/12/08 Math Advent Calendar 2017(https://adventar.org/calendars/2380) 12/8 IST(Internal Set Theory; ) 1 1.1 (nonstandard analysis, NSA) ε ε (a) ε 0. (b) r > 0 ε < r. (a)(b) ε sin(x) d sin(x)

More information

1 ( 8:12) Eccles. 1:8 2 2

1 ( 8:12) Eccles. 1:8 2 2 1 http://www.hyuki.com/imit/ 1 1 ( 8:12) Eccles. 1:8 2 2 3 He to whom it becomes everything, who traces all things to it and who sees all things in it, may ease his heart and remain at peace with God.

More information

Cain & Abel

Cain & Abel Cain & Abel: False Religion vs. The Gospel Now Adam knew Eve his wife, and she conceived and bore Cain, saying, I have gotten a man with the help of the LORD. And again, she bore his brother Abel. Now

More information

Hospitality-mae.indd

Hospitality-mae.indd Hospitality on the Scene 15 Key Expressions Vocabulary Check PHASE 1 PHASE 2 Key Expressions A A Contents Unit 1 Transportation 2 Unit 2 At a Check-in Counter (hotel) 7 Unit 3 Facilities and Services (hotel)

More information

10-渡部芳栄.indd

10-渡部芳栄.indd COE GCOE GP ) b a b ) () ) () () ) ) .. () ) ) ) ) () ........... / / /.... 交付税額 / 経常費 : 右軸交付税額 /( 経常費 授業料 ): 右軸 . ) ()... /.. 自治体負担額 / 交付税額 : 右軸 ()......... / 自治体負担額 / 経常費 : 右軸 - No. - Vol. No. - IDE

More information

24 Depth scaling of binocular stereopsis by observer s own movements

24 Depth scaling of binocular stereopsis by observer s own movements 24 Depth scaling of binocular stereopsis by observer s own movements 1130313 2013 3 1 3D 3D 3D 2 2 i Abstract Depth scaling of binocular stereopsis by observer s own movements It will become more usual

More information

h23w1.dvi

h23w1.dvi 24 I 24 2 8 10:00 12:30 1),. Do not open this problem booklet until the start of the examination is announced. 2) 3.. Answer the following 3 problems. Use the designated answer sheet for each problem.

More information

ABSTRACT The "After War Phenomena" of the Japanese Literature after the War: Has It Really Come to an End? When we consider past theses concerning criticism and arguments about the theme of "Japanese Literature

More information

elemmay09.pub

elemmay09.pub Elementary Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Activity Bank Number Challenge Time:

More information

1 3 19 J.S. 7 1 2 1 1 1 3 1 1960 1928 2 3 10 1

1 3 19 J.S. 7 1 2 1 1 1 3 1 1960 1928 2 3 10 1 2006 4 2 47 3 1 3 3 25 26 2 1 3 19 J.S. 7 1 2 1 1 1 3 1 1960 1928 2 3 10 1 26 27... and when they have to answer opponents, only endeavour, by such arguments as they can command, to support the opposite

More information

50 23 10 26 11 13 1in 1 23 2 28 50 1 11 11 14 30 415 250 120 2

50 23 10 26 11 13 1in 1 23 2 28 50 1 11 11 14 30 415 250 120 2 第 6 号 2012 年 ( 平 成 24 年 ). 3. 31 後 期 インターナショナルオフィスの 活 動 23 3 FD SD 23 11 29 23 3 FD SD 1 20 1 50 23 10 26 11 13 1in 1 23 2 28 50 1 11 11 14 30 415 250 120 2 2011 9 18 2011 10 20 2011 10 25 2011 11 22 2011

More information

日本語教育紀要 7/pdf用 表紙

日本語教育紀要 7/pdf用 表紙 JF JF NC JF JF NC peer JF Can-do JF JF http : // jfstandard.jpjf Can-doCommon European Framework of Reference for Languages : learning, teaching,assessment CEFR AABBCC CEFR ABB A A B B B B Can-do CEFR

More information

Core Ethics Vol. a

Core Ethics Vol. a Core Ethics Vol. CP CP CP Core Ethics Vol. a - CP - - Core Ethics Vol. CP CP CP b CP CP CP e f a c c c c c c CP CP CP d CP ADL Core Ethics Vol. ADL ADL CP CP CP CP CP CP CP,,, d Core Ethics Vol. CP b GHQ

More information

【人】⑥入谷好樹先生【本文】/【人】⑥入谷好樹先生【本文】

【人】⑥入谷好樹先生【本文】/【人】⑥入谷好樹先生【本文】 Naranjo, C. Perls, F. Perls Perls Claudio Naranjo Naranjo Cattel, R. Perls, F. Simkin, J. Human Potential Movement Naranjo Naranjo P. Gestalt Naranjo actuality awareness responsibility Naranjo, C. Naranjo

More information

sein_sandwich2_FM_bounus_NYUKO.indd

sein_sandwich2_FM_bounus_NYUKO.indd Sandwich method bonus 24 At a store - Buying clothes Hello! You re looking for a shirt?!? Well, this shirt here is the latest style, and the price is really reasonable. David A. Thayne s 2 Special Methods

More information

早稲田大学現代政治経済研究所 ダブルトラック オークションの実験研究 宇都伸之早稲田大学上條良夫高知工科大学船木由喜彦早稲田大学 No.J1401 Working Paper Series Institute for Research in Contemporary Political and Ec

早稲田大学現代政治経済研究所 ダブルトラック オークションの実験研究 宇都伸之早稲田大学上條良夫高知工科大学船木由喜彦早稲田大学 No.J1401 Working Paper Series Institute for Research in Contemporary Political and Ec 早稲田大学現代政治経済研究所 ダブルトラック オークションの実験研究 宇都伸之早稲田大学上條良夫高知工科大学船木由喜彦早稲田大学 No.J1401 Working Paper Series Institute for Research in Contemporary Political and Economic Affairs Waseda University 169-8050 Tokyo,Japan

More information

53 ( 1 ) 120 Aries, P., 1975, Essais sur l'histoire de la mort en occident, Paris: Editions du Seuil., 1977, L' Homme devant la mort, Paris: Editions du Seuil. Becker, E., 1973, The Denial of

More information

きずなプロジェクト-表紙.indd

きずなプロジェクト-表紙.indd P6 P7 P12 P13 P20 P28 P76 P78 P80 P81 P88 P98 P138 P139 P140 P142 P144 P146 P148 #1 SHORT-TERM INVITATION GROUPS 2012 6 10 6 23 2012 7 17 14 2012 7 17 14 2012 7 8 7 21 2012 7 8 7 21 2012 8 7 8 18

More information