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1 ( ) / 32
2 : ( ) : ( ) :! ( ) / 32
3 ( ) / 32
4 ( ) ( ) ( : ) 1 2 (2014) 1 M2 2 ( ) / 32
5 ( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 1 M2 2 ( ) / 32
6 ( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 9/27( ) 1 M2 2 ( ) / 32
7 : (, ) ( ) ( ) / 32
8 (,, ) ( ) Entdeckungen über die Theorie des Klanges (, 1787) Die Akustik (, 1802) Neue Beyträge zur Akustik (, 1817) ( ) m n ( ) / 32
9 ( ) / 32
10 ( ) / 32
11 ( ) / 32
12 ( ) ( ) / 32
13 ( ) : ( ) ( ) / 32
14 1808 ( ) (1811, 1813, 1815) - (1850) - 2 ( u 4 ) t 2 = D u x u x 2 y u y 4 u = u(x, y, t) (x, y) t ( ) / 32
15 ( ) / 32
16 ( ) / 32
17 ( ) ( ) / 32
18 ( ) 1 ( ) 2 ( ) / 32
19 ( ) 1 ( ) 2 ( ) ( ) / 32
20 ( ) 1 ( ) 2 ( ) 20 ( 100 ) ( ) (1909 ) ( ) ( ) / 32
21 : 1 ( ) 2 ( ) 3 ( ) ( ) / 32
22 : 1 ( ) 2 ( ) 3 ( ) ( ) / 32
23 ( : ) ( ) (normal mode of vibration) ( ) / 32
24 : ( ) (, ) ( ) / 32
25 : ( ) (, ) ( ) T T = 2π l g (g = 9.8 [ms 2 ] ( ), l ) ( ) ( ) / 32
26 : ( ) ( ) / 32
27 : ( ) θ = A sin (ωt + ϕ), ω = 2π T ( ). A, ϕ T ( ω ) ( ) / 32
28 ( ) / 32
29 A 1 (x) = a 1 sin πx L, A 2(x) = a 2 sin 2πx L, A 3(x) = a 3 sin 3πx L, ( ) ( ) n A n (x) = a n sin nπx L. ( ) / 32
30 ( )? u n = A n (x)sin(ω n t + ϕ n ) = a n sin nπx L (ω n = nπc L ) ( nπc ) sin L t + ϕ n. (x A n (x) ) ( ) / 32
31 ( ) u = A 1 (x) sin (ω 1 t + ϕ 1 ) + A 2 (x) sin (ω 2 t + ϕ 2 ) + = n=1 A n (x) sin (ω n t + ϕ n ), A n (x) = a n sin nπx L. ( ) / 32
32 ( ) u = A 1 (x) sin (ω 1 t + ϕ 1 ) + A 2 (x) sin (ω 2 t + ϕ 2 ) + = A n (x) sin (ω n t + ϕ n ), n=1 A n (x) = a n sin nπx L. = A n (x) = a n sin nπx L ω n ( ) / 32
33 , ( ) / 32
34 : (the Tacoma Narrows Bridge) m ( ) / 32
35 ( [1]) ( ) / 32
36 ( ) ( ) / 32
37 ( ) / 32
38 ( [1]) ( WWW ) ( ) / 32
39 ( ) m = n m, n 1 m = n D 4 2 m = n D 4 3 m > n D 4 4 m > n D 4 5 m > n D 4 6 m > n D 4 7 m > n D 2 D n ( n, ). D 4 90 D ( ) / 32
40 ( ) m = n m, n 1 m = n D 4 2 m = n D 4 3 m > n D 4 4 m > n D 4 5 m > n D 4 6 m > n D 4 7 m > n D 2 D n ( n, ). D 4 90 D ( ) / 32
41 (Mary Désirée Waller, , ) Chladni Figures study in symmetry (1961) ( ) ( ) / 32
42 ( ) ( M2) ( [5], 2015/9/9) ( ) / 32
43 (oscillation, vibration) ( ) ( ;,, ) ( ) ( X ) ( ) / 32
44 ( ) 100 ( ) / 32
45 ( ) 100 ( ) / 32
46 Mary D. Waller, Vibrations of free square plates: part I. normal vibrating modes, Physical Society, Vol. 51 (1939), pp Mary D. Waller, Chladni figures a study in symmetry, G. Bell (1961). von Walter Ritz, Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern, Annalen der Physik Volume 333, Issue 4, pp (1909).,,, ,,, Chladni, 2015, ,. ( ) / 32
47 : 1 ( ) 200 (2006) ( ) ( ) ( ) / 32
48 ( ) / 32
49 : ( 0) x kx (k, ) 2 mα(t) = kx(t). x (t) = α(t) = k m x(t) = ω2 x(t) (ω = x (t) = ω 2 x(t) k m ). ( ) / 32
50 : x(t) = C 1 sin ωt + C 2 cos ωt ( ) x(t) = A sin(ωt + ϕ) x (t) = ω 2 x(t) T = 2π ω = 2π m k. ( ) ( ) / 32
51 : x(t) = C 1 sin ωt + C 2 cos ωt ( ) x(t) = A sin(ωt + ϕ) x (t) = ω 2 x(t) T = 2π ω = 2π m k. ( ) ( ) ( ) / 32
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63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
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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
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
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ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
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3 3.1 ( 1 m d2 x(t dt 2 = kx(t k = (3.1 d 2 x dt 2 = ω2 x, ω = x(t = 0, ẋ(0 = v 0 k m (3.2 x = v 0 ω sin ωt (ẋ = v 0 cos ωt (3.3 E = 1 2 mẋ2 + 1 2 kx2 = 1 2 mv2 0 cos 2 ωt + 1 2 k v2 0 ω 2 sin2 ωt = 1
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m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
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2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
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6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P
6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P
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9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
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2006 11 28 1. (1) ẋ = ax = x(t) =Ce at C C>0 a0 x(t) 0(t )!! 1 0.8 0.6 0.4 0.2 2 4 6 8 10-0.2 (1) a =2 C =1 1. (1) τ>0 (2) ẋ(t) = ax(t τ) 4 2 2 4 6 8 10-2 -4 (2) a =2 τ =1!! 1. (2) A. (2)
微分積分 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, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
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1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1
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http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ
1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
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