http://www.ike-dyn.ritsumei.ac.jp/ hyoo/dynamics.html 1 (i) (ii) 2 (i) (ii) (*) [1] 2 1 3 1.1................................ 3 1.2..................................... 3 2 4 2.1.................................... 4 2.1.1 G(x(t), y(t)).................... 4 2.1.2............................. 4 2.2................................ 4 2.2.1............................. 4 2.2.2............................. 4 2.3....................... 4 3 5 3.1 [1]....................................... 5 3.2 [1]....................................... 6 3.3 [1]....................................... 7 3.4 [1]....................................... 8 3.5 [3]....................................... 9 1
3.6 [3]....................................... 10 3.7 [2]....................................... 11 3.8 [2]....................................... 12 3.9 [5]....................................... 13 3.10 [5]....................................... 14 3.11 [5] [ ]................................... 15 3.12 [3]....................................... 16 3.13 [3]....................................... 17 3.14 [3] [ ]................................... 18 3.15 [4] [ ]................................... 19 3.16 [2]....................................... 20 3.17 [2] [ ]................................... 21 3.18 [6]....................................... 22 3.19 [7]....................................... 23 3.20 [8]....................................... 24 3.21 [8]............................. 25 3.22 [8]............................. 26 [1] [2] 1a-1b R.W.Serway [3] I,II [4] [5] [6] [7] [8] W.T.Thomson 2
1 1.1 ( ) ( ) I 0 θ(t) = τ0 (1.1) I 0 1 [1] θ(t) 2 τ 0 3 [1]p62-64 1.2 θ t Z θ(t) θ(0) ( ) = Z θ(t) I 0 θ(t)dθ = τ 0 dθ (1.2) = = Z t 0 Z t I 0 θ(t) θdt θ(0) Ω d 1 0 dt 1 2 I 0{ θ(t)} 2 2 I 0( θ(t)) 2 t 0 æ dt (1.3) E R (t) 1 2 I 0{ θ(t)} 2 (1.4) E R (t) E R (0) Z θ(t) θ(0) τ 0 dθ (1.5) [Memo] [1] 2-9(p79) 1 2 θ(t) θ(t) 3 τ N 3
2 2.1 2.1.1 G(x(t), y(t)) Mẍ(t) = F x Mÿ(t) = F y (2.1) 2.1.2 I G θ(t) = τg (2.2) I G θ(t) τ G 2.2 2.2.1 E K Kinetic Energy ~ V (t) = (ẋ(t), ẏ(t)), W (t) = θ(t). E K (t) = 1 2 M Ø ØØ ~ V (t) Ø ØØ 2 + 1 2 I GW 2 (t) (2.3) 2.2.2 E P Potential Energy M Mgh 2.3 d dt (E K + E P ) = 0 (2.4) 4
3 3.1 [1] [ 2-3]p65- I r m N 0 ( ) N 0 = 0 I = 1 2 Mr2 M Ans. θ(t) 1 = r g 1 + M 2m 5
3.2 [1] [ 2-6](p73-) M r I f 0 x Ẍ f 0 ( ) I = 1 2 Mr2 f 0 Ans. Ẍ(t) = 2 3 M 6
3.3 [1] [ 2-7](p75-) M r I α Ẍ ( ) I = 1 2 Mr2 Ans. Ẍ(t) = 2 3 g sin α 7
3.4 [1] [ 2-8](p77-) M r I ( ) I = 1 2 Mr2 Ans. Ẍ(t) = 2 3 g 8
3.5 [3] M R I MR 2 1 2 MR2 ( ) h s 2gh Ans. V = 1 + I Mr 2 9
3.6 [3] M R V 0 I = 1 2 MR2 Ans. h = 3V 0 2 4g 10
3.7 [2] L M O I 0 = 1 3 ML2 Ans. W = r 3g L 11
3.8 [2] R M L ω = 12gR/L O I 0 = 1 12 ML2 12
3.9 [5] Ans. ω = r 3g 2l 13
3.10 [5] I 0 = ml 2 Ans. ω = a r k l m 14
3.11 [5] [ ] 13.2 I 0 = 1 2 Mr2 M s k Ans. ω = m + 1 2 M 15
3.12 [3] 16
3.13 [3] M r l I G = 2 5 mr2 O s g Ans. ω = l + r + 2 5 r 2 l+r 17
3.14 [3] [ ] O (1) O I = X i m i r 2 i (2) t θ(t) O (3) θ(t) Ans. ω = r 2g 5l 18
3.15 [4] [ ] (1) O ml 2 Ans. (2) kc 2 > mgl 19
3.16 [2] R M 2 5 MR2 Ans. h = 2 5 R 20
3.17 [2] [ ] M I = 1 3 ML2 k (1) y1(t)+y2(t) 2 y 1 (t), y 2 (t) (2) θ(t) θ(t) Ans. (1) ω = r 2k r 3k M, (2) ω = M ( ) 4.3 10 3 [kg / m] L = 6[m] k = 1.5 10 3 [N / m] 21
3.18 [6] Ans. ω = ( ) M r k r k m, ω = m 2m + M I = 2ml 2 + 1 3 Ml2 22
3.19 [7] θ 1 (t), θ 2 (t) Ans. ω = r g l, ω = r g l + 2k a 2 m l 23
3.20 [8] 24
3.21 [8] (*) (9.2) 25
3.22 [8] l m C G h = 2 π l C I C = ml 2 (1) t θ(t) O G(x(t), y(t)) r, θ(t), h ~ OG = ~ OA + ~ AC + ~ CG (2) I G G E K = 1 2 m {ẋ(t)} 2 + {ẏ(t)} 2 + 1 2 I G{ θ(t)} 2 (3) θ(t) E P (4) d dt (E K + E P ) = 0 θ(t) r g Ans. ω = (π 2)l 26