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I ( ) 2019

i 1 I,, III,, 1,,,, III,,,, (1 ) (,,, ), :...,, : NHK... NHK, (YouTube ),!!, manaba http://pen.envr.tsukuba.ac.jp/lec/physics/,, Richard Feynman Lectures on Physics Addison-Wesley,,,, x χ, ( ),, 2, 1?...,,, 2,...,,,

ii 3,?...,, 4,,?...,,,, 5...,,,,,,,,, 6,?...,,,,,, 7,...,,? 8,,..., 9,...,... 10... III 11,?...,!,, ( ), 12?...,,!,... 13,?...,,, 14,?...,,!,, 15...,, 16,?...!,!!,,, 2019 3 11

iii i 1 1 1.1.............................................. 1 1.2....................................... 3 1.3................................... 3 1.4........................................ 4 1.5.................................... 4 1.6......................................... 4 1.7............................. 5 1.8.................................... 5 1.9................................................... 6 1.10............................................... 6 2 9 2.1?.............................................. 9 2.2 4................................................ 10 2.3........................................... 10 2.4................................................... 10 2.5............................................... 11 2.6............................................... 12 2.7............................................... 13 2.8................................................... 14 2.9............................................... 16 2.10........................................ 16 2.11................................................ 18 2.12.................................... 20 2.13?.............................................. 21 2.14................................................... 22 3 25 3.1............................................. 25 3.2........................................... 27 3.3........................................ 30 3.4................................................. 33 3.5.......................................... 33 3.6.......................................... 34

iv 3.7................................................... 35 4 39 4.1.......................................... 39 4.2.......................................... 40 4.3.............................................. 41 4.4........................................... 41 4.5.............................................. 42 4.6............................................. 44 4.7......................................... 45 4.8.......................................... 47 4.9................................................... 47 5 51 5.1................................................. 51 5.2.............................................. 53 5.3................................................. 54 5.4................................................... 57 6 (1) 59 6.1............................................. 59 6.2........................................ 61 6.3.......................................... 62 6.4......................................... 64 6.5................................................... 65 7 67 7.1.............................................. 67 7.2?................................ 69 7.3.............................................. 72 7.4................................................... 73 7.5 :........................................ 74 8 75 8.1............................................... 75 8.2 ( )............................... 76 8.3................................... 78 8.4.............................................. 79 8.5........................................ 79 8.6............................................. 80 8.7........................................ 80 8.8?.......................................... 81 8.9.............................................. 82 8.10.............................................. 83 8.11................................................ 84 8.12............................................... 85

v 8.13............................................. 85 8.14........................................ 86 8.15................................................... 88 9 (2) 89 9.1 3.................................. 89 9.2 3........................................ 89 9.3 3............................. 90 9.4 3............................. 91 9.5.............................................. 92 9.6.................................. 93 9.7................................................... 94 10 97 10.1................................................... 97 10.2................................................ 97 10.3............................................. 99 10.4................................................... 101 10.5 :.............................................. 102 11 103 11.1........................................... 103 11.2............................................. 103 11.3............................................. 104 11.4..................................... 107 11.5............................................... 108 11.6................................................... 109 12 111 12.1.......................................... 111 12.2.............................................. 112 12.3........................................... 113 12.4........................................ 114 12.5................................................... 116

1 1,,,, *1,,,,,,,,,,,,,,,,,,?,,, 17,...,, ( ), *1,,,,,,,! 18,?...,,,,, ( ),, 1.1,, 1,,,,,,,

2 1, I II,,,,,,, II,,,,,,,,,,, GPS (Global Positioning System;, ) ( II ),, ( ),,,,,, (LED), MRI,,,, ( ) 1, ( ),,,,,,,,,,,,,,,,,,,,,,,,,

1.2 3 19...,,, 1.2,,, 1.1,,,,,,? 1.2,,, ( ),, 1.3,, ( ),,, ), 1.4 2003,,,,,,,,,, 2 1,, 200 400, 1.3,,,,,,,,,,,,,,,,,,,,,,,,,,,

4 1 1.4,,,,,,, (,,, ),,,,,,,,,,,,,,,,,,,,, 1.5,,???,,,,??????, 1.6,, ( ),,,,,,,,,, ( ),,,,,,,,,,,,,,,,

1.7 5... 1.7,,,,,,,,,,,,...,,,,,,,,,,,,, 1.8,...,,...,, A B, A B,, A B, B A,,, A B, B A *2,,,,,,,,, (, x = x 0 + v 0 t + 1 2 at2 ),,,,,,,,,,......,, *2,, B A,,

6 1, *3,,,,, ( ),,, (1) J ; J1, J2, J3, 50 30, 3? (2) 5, J 1, 1 1.9,,,,,,,,,, 2018,,, 1000 15,,, 15, 15, 1000 (3), 1,?, 1, 100 (4),,? 5 2018, 100, 240, (1)? (2)?,, 1 2 6 1500 mm,, 1 m 2, 500 g, 1 kg 3000 kg,,?,,, 1, 1000 1, 1000 /, 1000 *4,,, 20?..., 50 1.10,,,,,,,, 4,,,, *3 *4,?,, :,

1.10 7,,,,,,,,,,,,,,,,,, 1.5,, 100 km/h 45,?,,, ( ), *5 ( 1.1) 1.6,,,,,,,, 7? 8? *5,, 1.1 ( ),,,,,,,, 1.7,,,,, 60 7,,,,,

8 1 1,, 2?,, 1 : : 1 ( ) 2 ( ) 3 ( : *6 3,,,, 6, 1 m 2 1, 0.5 kg 3000 kg/kg=1500 kg 1000 kg/m 3, 1.5 m 3 1 m 2, 1.5 m, 1500 mm,, 1500 mm, *7,,,,,,,,,,? 22,,...,,,!,,,,,, 23,...,, 24,?..., 21,..., 10,,...,,,,,,,,, *6,, 2, *7,,

9 2 2.1?,,,?, ( ) : (1), ( ) (2) ( ) (3) ( )... (8) [ ],, ( ), (8) (1), (2), (3),, (1), (2), (3), (1), (2), (3),,,,,,,,, (8),, (8), 4,,,, 4, 2 kg 3 m s 2,, 2 kg 3 m s 2 =6 kg m s 2 kg m s 2, kg m s 2 N ( ), N=kg m s 2 1 N, 1 kg 1 m s 2, 2 kg 0.5 m s 2 25?...,, 26,?...,,,,, 27,?...,,,,, 28 N...... kg m s 2

10 2 9 2.0 kg, 3.0 N 10,??,?,? 2.2 4,, 4,, :,,, 4,,,,, 137 Cs 137 Ba, 14 C 14 N 11, 4? 2.3,,,,,,,,,,, *1 2 A, B, A B, A B,, 2.4, M m 2 r,, F F = GMm r 2 (2.1), G, : G = 6.6741 10 11 m 3 s 2 kg 1 (2.2) G, (2.1) (2.1),?,,,,,,,,, *1,

2.5 11 1, (2.1)?,?,!,,,,, M m,, m M,,,,,,, (2.1),, (2.3) m, M, r,,, 12 r = 6400 km, M = 6.0 10 24 kg m = 1.0 kg,, 9.8 N :! 2.5, F, m, g, F = mg (2.4), g = F/m (2.5) g, F/m ; g, g?, g, F SI N ; kg m s 2, m SI kg, g F/m SI, kg m s 2 /kg = m s 2, g,, g m, M ( ) r F, (2.1), F = GM r 2 m (2.6), g = GM r 2 (2.7), (2.4) (2.7) G, M, r 12, g = 9.8 m s 2 (2.8) ( ),, g,,,,, (2.3),,,,,, ),, m, (2.5),, g

12 2 1?, (2.8) (2.7)..., g 13,,,? :,,,!, g, g, g, 16 1/81.3, 1/3.68,? (2.1) M r 29,?... 0 30?..., ( ),,, 31?... 14? g, g (2.7), r, g 15 100 m, 10,000 m, 36,000 km, g? (2.7) r r?, 100 m, g, 0.003, 100 m 3 mm 1 70 mm,,, g,,, 2.6,,,,,,,,,,, ( ), 1 1,,, 1 1 1.602 10 19 C (C, SI, 1 C=1 A s ), q e q e, 4, q e = 1.602 10 19 C q e = 1.602 10 19 C, +q e = 1.602 10 19 C,,,

2.7 13, ( ),, 1 1,, 0, 0,, 0 0,,,, 1 2 2,,, F, F F = k q 1 q 2 r 2 (2.9), q 1, q 2 1 2, r k, 4, k = 8.987 10 9 N m 2 C 2 (2.9), (Coulomb s law), (Coulomb force), k, (2.9),,,,, (2.9) (2.1),,,,, 32? 2?...,,,,,,,, q 1 q 2, q 1 q 2,, (2.9) F,, 2,, q 1 q 2, F,, 2, (, ), (2.9) ( ) 17? 18? 19 1.0 m 2, F e, F g (1) F e (2) F g, m e, 9.1 10 31 kg (3) F g F e? 2.7,,,,,,, ( ):, q, 2 E, B, F = qe + qv B (2.10) v E B, ( ), OK,

14 2,,,,,,, 20 1 1.0 m? : B 33,...... (2.10) (2.10),,! 34,?..., 35,?..., 2.8, 4, 4,, 1,, 1 X A, B A B X 1,,, A B, B 2.1 A A A, B B B A, B A, A, A,, B X A,, ( ),, :,,,,, T, : ( ), ( ),,,, 2,

2.8 15, X? X,,,,,,, m,, B A,, mg/2 T, B 2T 21 2.2, F, F,, F 2.4 2.2, 36?... 22 2.3, m,,, mg/2, 23, 2.5,,, 2.3 2.5 2, 1/4 23 2.4,, A, B 24,

16 2? 2.9,, (, ),,,,,,,, SI, N/m 2 N=kg m s 2, kg m 1 s 2 Pa ( ),,,,, 2.1 A =0.50 m 2 m =3.0 kg, g mg,, mg/a = 3.0 kg 9.8 m s 2 /(0.50 m 2 ) 60 kg m 1 s 2 60 Pa ( ) 25,, 1/10 0.050 m 2,? 26, OK 27 1000 hpa hpa h, 100 (1) 1 m 2,? (2) 1 m 2,? 37,,?,,?...,,,,,,,,,,,,, 2.10,,, ( )x F F (x),,, F (x + dx) = F (x) + F (x)dx (2.11) (dx ), x = 0,, x = 0, F (0) = 0 2.6 ), F (0 + dx) = F (0) + F (0)dx, F (dx) = F (0)dx (2.12) dx, x, dx x, F (x) = F (0)x (2.13) F (0),,,, x F, x F

2.10 17, (2.13) F (0), F (0) k k (2.13), F = kx (2.14), F,, x, k (Hooke), cell,,, (x,, *2 (2.14),,,,,,,, 29?? 2.6 30,, 38 F = kx, F = kx?... F = kx,, F = kx,,, (2.14),,, (x F, x,, F, x, F 28,, k, m,? x, x 0 x, kx, mg g,,, 2, kx + mg = 0 (2.15) (1)? (2)? (3) SI?, kx = mg,, x = mg k (2.16),,,,,,, 31, *2,,

18 2?,? 32 2.7, k 2,,, M Mg/(2k) 2 1, 2k Mg, 2, 33 2.7, k 2,, M 2Mg/k 2 1, k/2, Mg (1) SI? (2) *3 (3) 10 m, 2.0 mm 10 kg,? (4) (3) (1.0 mm)? (5) (3)?, F = E A L x, F A = E x L (2.18) F/A,, x/l, σ, ϵ,, σ = Eϵ (2.19) 2.7, 1,,,,, 34 k a, 2 b, bk/a n m A, L X,, X, 36 (1) SI? (2) SI? 39,...,, X,,, X k, A/L, k = E A (2.17) L E 2.11,,, *3 ( ),, 35,

2.11 19,,,,,,,,,,,,,,,,,,, N,,,,,,,,,, *4, 2.8, θ, m, mg, 2.8, mg, mg sin θ, mg cos θ 2.8, 2.9 N 2.10,,,?,,,,,,, (mg cos θ), N ( 2.9),,!,,, 2.10, mg sin θ *4,,,,,,, 37 2.11, 30 60 2, 3 kg m,,,

20 2, m T T T, 2.11 2, 2 40,..., 2.12,,,,, 2, N, 2, F s, F s µn (2.20), 2, F m, F m = µ N (2.21) F s, F m, µ, µ,,,,,?,, (2.20),, 2, 2,,,,,,,,,,, µn,,, µ < µ µ µ,,,,,,,,,,,,,,,... 41 µ < µ...,,,, 42,,, µ < µ?... ABS ( 38 (1)? (2)?

2.13? 21 (3)? (4) 39 ϕ, m,,, ϕ µ, ϕ µ = tan ϕ (2.22) 40 2.12, µ k m) 2.12,,! 41? 43,,,,,?...,,,,?,,,,,,,,,,, 2.13?,, 2.12,,, 0,,, (system),,, 2,,,,,,,,,,,,,,,,,, 44,,,,?...,,,,,,

22 2 3 1 g 1 cm s 2 1 dyn (dyn ) 1.0 dyn N 4 1, 2cm 2, A B A, B ( 10 ) ( )? 5 2.0 N/m, 1.0 N, 1.0 N? 6, a, b, c, a < b < c ( ab ), ( bc ),,? 2.14 9 ( ) 1.5 m s 2 11 12 : (2.1), r km, m, r = 6.4 10 6 m 14 m mg g *5 15 r 1,,, r 2 g 1, *5, GM/r 2,,,,, g,, g g g 2, (2.7), g 1 = G M r1 2, g 2 = G M r2 2,, g 2 g 1 = r2 1 r 2 2 = ( 6400 km r 2 ) 2, r 2 =6,400 km+100 m=6,400.1 km, 0.999968 99.997, r 2 =6,400 km+10,000 m=6,410 km, 0.9969 99.7, r 2 =(6400 km)+(36000 km) 42000 km, 0.023, 2 16 r 1, r 2, M 1, M 2 M 2 /M 1 = 1/81.3, r 2 /r 1 = 1/3.68, m F 1, F 2, (2.1), F 1 = G M 1m, F 2 = G M 2m r 2 1 r 2 2, F 2 = M ( 2 r1 ) 2 1 = F 1 M 1 r 2 81.3 3.682 = 1 6.00, 1/6 17 q 1 q 2 r, F = kq 1 q 2 /r 2 F, k, k = 8.987 10 9 N m 2 C 2 18 q e = 1.602 10 19 C 19 (1) (2.9), F e = 8.987 10 9 N m 2 C 2 (1.602 10 19 C) 2 (1 m) 2 = 2.3 10 28 N (2.23) (2) (2.1), F g = 6.7 10 11 N m 2 kg 2 (9.1 10 31 kg) 2 (1 m) 2 = 5.5 10 71 N (2.24) (3) F g /F e = 2.4 10 43 :,,, 1,

2.14 23 20 (2.10), B = 0 q,, F = F, E = E, E = F/q F (2.9)(q 1 = q, q 2 = q e ), E = F/q = k q e /r 2 q e = 1.602 10 19 C, r = 1.0 m, k = 8.987 10 9 N m 2 C 2, E = 1.4 10 9 N/C 21, A, F, F,,, F, F,, F,, 22 2, T,, 2T,, mg,,, 2T mg = 0, T = mg/2, T,, mg/2 23 B, T B, B m mg, 2T mg = 0 T = mg/2,, T, mg/2 24,,,,, 28 (1) x, F, (2) F = kx k (3) k = F/x, F ( N=kg m s 2 ) x ( m), kg s 2 29,, 30,,, 31 m, x 0, kx 0 + mg = 0, m, x 1, 16, 1/6,, kx 1 + mg/6 = 0 2 mg, kx 1 = kx 0 /6, x 1 = x 0 /6,, 1/6 32 A, B A, F, x,, F = kx (2.25),, B, M, 2, 2F, Mg, 2F + Mg = 0 (2.26) F, 2kx + Mg = 0, x = Mg/(2k) 1, x = Mg/k (2.16),, k 2 33 C, D C, F 1, x 1,, F 1 = kx 1 (2.27) D, F 2, x 2,, F 2 = kx 2 (2.28), M, D, F 2, Mg

24 2, F 2 + Mg = 0 (2.29), D, C F 1, D D, F 1 + Mg = 0 (2.30), Mg = F 1 = F 2 = kx 1 = kx 2, x 1 = x 2 = Mg/k 2 x, x = x 1 + x 2 = 2Mg/k 1, x = Mg/k (2.16),, k 1/2 34 33, k a,, k/a 32, k/a b,, bk/a 35 (1) E = kl/a, k SI kg s 2, L/A SI m/m 2 =m 1, E kg s 2 m 1 kg m 1 s 2 OK! Pa, SI! (2) 2.0 10 11 Pa (3) k L = 10 m, A = 3.14 (0.002 m/2) 2 = 3.1 10 6 m 2, k = EA/L = 6.2 10 4 kg s 2, m x, x = mg k 10 kg 9.8 m s 2 = 6.2 10 4 kg s 2 = 1.6 10 3 m, 1.6 mm ( 2 mm) (4),, 1/4, 4, 6.4 mm (5) 1.3 10 11 Pa 0.65, 0.65 1/0.65, 2.5 mm 36 (1) σ = F/A, σ SI, kg m s 2 /m 2 =kg m 1 s 2 =Pa (2) ϵ = x/l, ϵ, m/m=1,! ( ) 37 3 kg,, (3 kg) g sin(π/6) m,, mg sin(π/3) T,, T = (3 kg)g sin(π/6) = mg sin(π/3), m = (3 kg)g sin(π/6)/{g sin(π/3)} = 3 kg 1.7 kg 38 (1), F s F m, F s µn, F m = µ N, N µ, µ,, (2), µ µ (3) µ = F m /N, F m N, µ µ 39 m, mg sin ϕ F s,,, F s = µn N,, N = mg cos ϕ, F s = µmg cos ϕ, mg sin ϕ = F s = µmg cos ϕ, µ = mg sin ϕ/(mg cos ϕ) = sin ϕ/ cos ϕ = tan ϕ 40 ( ) x x = 0, x,, : mg sin θ, : kx 2, F s, mg sin θ kx + F s = 0 (2.31), mg cos θ, F s µmg cos θ, F s = mg sin θ + kx µmg cos θ (2.32), µmg cos θ mg sin θ + kx µmg cos θ (2.33), µmg cos θ mg sin θ kx µmg cos θ mg sin θ, mg k mg ( µ cos θ sin θ) x (µ cos θ sin θ) k

25 3 :,,?,?, 2 x *1 3.1,,, 2,?,,,,,,?,,?,,,,, 3.1 ( 23),,, T, mg,,, ( 3.1), B x,,, x,, 3.1 x 2 x,,,, T (2 x) + mg( x) (3.1), ( x),,, 0, 2T x mg x = 0 (3.2) T = mg 2 (3.3), ( ) *1 B x, 1 2 x

26 3, 0,, 2 x, x,,,, (work)?,,,,,,,,,,,,,,, 2, (3 kg)g{sin(π/6)} x, B, mg{sin(π/3)} x, B ( ), ( (3 kg)g sin π ) ( x mg sin π ) x = 0 (3.4) 6 3, m = 3 kg ( ) 42 (1)? (2) SI J (3)? 43 3.3, S, l 1, l 2, m 1, m 2 1, 2, 3.3,, θ, (1) h 1, h 2, h 1 = l 1 sin θ (3.5) h 2 = l 2 sin θ (3.6) (2) 1 m 1 gl 1 sin θ, 2 m 2 gl 2 sin θ (3), m 1 l 1 = m 2 l 2 (3.7) 3.2 2, 2 2.11 3.2 37, A x ( 3.2), B, x, A 3.3

3.2 27,,,,,,,,,,, 0,,,,,,,,, 44 3.4, r 1, y (1) F 1, 2πrF (3.8) (2) m 1, mg y (3.9) (3) (4) 2πrF mg y = 0 (3.10) F = mg y 2πr (3.11) (5) m = 1000 kg, r = 0.2 m, y = 0.003 m, F? 45?...,,,? 3.4 46..., 3.2,,,,,, 3.1 2T x,, m mg x (3.12),, 0,,,,,,,? A B 1000,, 2,, 2, 2 (2 0 ), A 1000, B C, A, B, C,...

28 3,,,,,,,,,, SI J 45?,?,,?,,,,,,,,,,,,,,,,,?, F, x x F, W, W = F x (3.13),, x 0, x 1,, x 0 x 1, (,, x 0 x 1 ), F 1 W 1,, W 1 F 1 x 1 (3.14) x 1 = x 1 x 0 x 1, x 1 x 2,, F 2 F 2 F 1 W 2,, W 2 F 2 x 2 (3.15) x 2 = x 2 x 1, x 3, x 4,, x n (n ), F 3, F 4,, F n,, W 3 F 3 x 3 W 4 F 4 x 4 W n F n x n (3.16), n W k k=1 n F k x k (3.17) k=1, x 0 x n, W : W n W k k=1 n F k x k (3.18) k=1 n, x 1, x 2,, x n,, (3.18), W = lim n x k 0 n F k x k (3.19) k=1,, : a b, W = b a F (x) dx (3.20) x 0 a, x n b, F (x) x (3.20), (3.13),,

3.2 29 3.3 m, mg, h 0 h 1,, (2.4), F = mg (3.21), *2, (3.20), W = h1 h 0 ( mg) dx = mg(h 1 h 0 ) = mg(h 0 h 1 ) (3.22) h 0 > h 1, W > 0 h 0 < h 1, W < 0,,, ( ) 47 3.3,?..., F = mg, W = mg(h 1 h 0), h,, h 0 < h 1, W, h, 3.4 m A, M B, B R 0 R 1 B A,, (2.1), F = GMm x 2 (3.23), *3 (3.20), R1 ( W = GMm ) R 0 x 2 dx = ( 1 = GMm 1 ) R 1 R 0 ( ) [ GMm x ] R1 R 0 (3.24) 46 m, h E (1) E, (3.22) (3.24) E 1, E 2 R = 6370 km, M = 5.972 10 24 kg, g = 9.823 m s 2, (2) m=10 kg, h = 100 m ( ), h = 10 km ( ), h = 1000 km ( ), E 1 E 2 3 (3) h E 1 E 2? 47 k, x 0 x 1, W, W = 1 2 k (x2 1 x 2 0) (3.25) : (2.14) F = kx, (3.20) 48, A, x, x = 0 x, x = h ( 3.5), P (1) V, V = Ah (2) F 1, F 1 = P A (3.26) (3) *4 F 2, F 2 = P A (3.27) *2 (2.4), *3 (2.1), *4,,,

30 3 (9),, T, : W = nrt ln V 1 V 2 (3.34) (10) 1 0 ( ) (3.30),,, ( ) 3.5, (4),, x = h + dh dh > 0,, dh < 0 dh, x = h x = h + dh F 1 F 2, dw, dw = F 2 dh = P A dh (3.28) (5) dv, V + dv : dv = A dh (3.29) (6) (3.28), (3.29) : dw = P dv (3.30) (7), V 1 V 2 W : V2 W = P dv (3.31) V 1 (8),, : dw, dw = dw, dw = P dv (3.35), (3.30) *5 3.3, (3.20), W, a b, a, W b, b x, W x W (x), x W (x) *6, (1) U(x) = W (x) (3.36) U(x), W (x), x, 3.5 3.3,, h 0 = 0, h 1 h, W (h) = mgh P V = nrt (3.32) n, R, T : V2 nrt W = dv (3.33) V 1 V *5,, dw, ( dw ) dw, *6,

3.3 31,, (3.36), U(h) = mgh (3.37),,,,,,,,,, 49?, (3.36) W (x), 3.5,,,,,, W (x) *7,, W (x) = W (x), (2) U(x) = W (x) (3.38) U(x), W (x), x, 3.5, h, mg, h, W (h) = mgh, (3.38), U(h) = mgh, (3.37) (3.36) (3.38),, 2,,,,, 3, (3) U(x) = W (x) (3.39) U(x), W (x), x, 3.5, h ( ), mg, h, W (h) = mgh, (3.39), U(h) = mgh, (3.37), (3.36), (3.38), (3.39),, (, W (x) = W (x) = W (x) ),,,, 50 k, x, U(x), U(x) U(x) = 1 2 kx2 (3.40) 51 m, M R, U(R) (R = ), U(R), U(R) *7 W (x) W (x) U(R) = GMm R (3.41)

32 3, (3.41), (3.37), M, R 0, h, h, m, (3.41), U(R 0 + h) = GMm R 0 + h = GMm R 0 R 0 R 0 + h = GMm R 0 1 1 + h/r 0 (3.42), h << R 0,,, 1 1 + h/r 0 1 h R 0 (3.43), U(R 0 + h) GMm R 0 ( 1 h R 0 ) = GMm R 0 + GMmh R 2 0,, GMm R 2 0 (3.44) = mg (3.45) (3.44), U(R 0 + h) GMm R 0 + mgh (3.46), U(R 0 + h) + GMm R 0 (3.47) U(h) ( ), (3.37) 52,, (1) 10 m, 2 kg, (2) 10 m, 2 mm 1 mm, (3), 53 θ, m, L?,?, (conservative force),,,,,,,, U(x), x, W (x), W (x) x, U(x), 54? 55,,,, (1), (2),,, (3),, (4), 56 (1) x 0 x 1 W 01,, x 1 x 0 W 10, 0 = W 01 + W 10 x 0 x 0, x 0 x 1 (2), 57 θ, µ, m, L?,?

3.4 33 3.4,, t, W, P, P = W t (3.48),, t, P = dw dt (3.49),,, 3.6 m, t h W mg h P, 1 W 1, 1 W h 59 1 W h, J 3.5,,,, : 60 Q 1, x, q 2, U(x) : P = W t = mg h t t 0, P = mg dh dt (3.50) (3.51) U(x) = k Q q x (3.53), k (2.9) dh/dt v, P = mgv (3.52) ( ), SI, kg m 2 s 3,, J s 1, W 58 2.0 kg,, 3.0 m/s (3.48),,, *8,,, W, h, W h, *8,, ( ),,,,, SI J C 1 V, 2, SI V 61 60 (1) 1 x (2) 100 ( ),, 0.529 10 10 m V? 2, ( ), 2 (

34 3 ) SI V,,,,,,,,,, 62 (1)? (2)? (3) SI (4) 2.0 V 0.30 C (5) 1.0 V 1 62(5), 1 V 1 1.602 10 19 J, 1 ev ev, 63, α, β, γ α, β, γ ( ) m He, m e (1) m He : He 4 1 mol He 4 g He He, 2, 3.6 P.12,,,, ( ),,, C/s ( ) C/s A ( ) *9 ( ) ( ), V 2 I V I? q V 2 qv (!) t, qv/ t, q/ t,, I, V I V V, J/C, I A, V I J A/C (, V V,? SI,,!), C=A s, J A/C, J/s, W ( )! ( ; ), SI J ( ),, W h ( ) (2) 239 α 64,, α 1 5.5 MeV, SI α (3) 137 β, 1 510 kev m e =9.1 10 31 kg : 65, A h 8000, 36A h *9 A SI, C=A s C, A=C/s A

3.7 35 2 (1) H L (2) h h + dh dv, H, h, dh (h < H, dh ) (3) dm (4) de (5), h = 0 h = H, E L, g, ρ 3.6,,,, A:, C:, F:, J:, N:, V:, W:, Ω: (1) (2),, 12 V (3) LED, 23 W,? 7,, ( 1 a), 50 cm, 5 cm 5 8 60, 100 g g=9.8 m s 2 9 1 L E ρ 3.7 42 (1),, (2) J = N m=kg m 2 s 2 (3), 0 43 (1) : (2) 1, m 1 g 1, * 10 h 1, l 1 sin θ, 1 m 1 gh 1 = m 1 gl 1 sin θ 2, m 2 g 2 h 2, l 2 sin θ, 2 m 2 gh 2 = m 2 gl 2 sin θ (3), 2 0, m 1 gl 1 sin θ m 2 gl 2 sin θ = 0, m 1 l 1 = m 2 l 2 44 (1) 2πr,, F, 1, 2πrF (2) 1, y,, mg,, mg y *10 (displacement)

36 3 (3) 1,, 0 (4) F = (5), 23 N 2 kg,, 2, 1000 kg 45, 46 ( ) (1) E 1 = mgh, E 2 = GMm{1/R 1/(R + h)} (2) h = 100 m, E 1 = 9.82 10 3 J, E 2 = 9.82 10 3 J ( 3 ) h = 10 km, E 1 = 9.82 10 5 J, E 2 = 9.81 10 5 J h = 1000 km, E 1 = 9.82 10 7 J, E 2 = 8.49 10 7 J (3) ( ) 47 (3.20) F (x) = kx, W = x1 x 0 ( kx) dx = 1 2 k(x2 1 x 2 0) (3.54) (5) V = Ah, V + dv = A(h + dh), dv = A dh (6) (7) (3.30) (8), P = nrt/v (9),, W = nrt V2 V 1 dv [ ] V = nrt V2 ln V V 1 = nrt ln V 2 V 1 = nrt ln V 1 V 2 (3.56) : V 1, V 2, (10) n = 1 mol, R = 8.31 J mol 1 K 1, T = 273 K, V 1 /V 2 = 2, ln 2 = log e 2 0.693, W = 1570 J 3 49 x W (x), U(x) = W (x) U(x) 50 (3.25), x 0 = 0, x 1 = x, U(x) = W (x) = 1 2 kx2 (3.57) 3.7 48 (1), A, h V = Ah (2) P,,, P A,,, F 1 = P A (3), F 1 F 2 = F 1 = P A (4), dw, F 2 dh, dw = F 2 dh = P A dh (3.55) 3.7 U(x) 0 x 51 (3.24), R 0 =, R 1 = R, U(R) = W (R) = GMm R 3.8 52 (3.58) (1) (3.37), m = 2 kg, g = 9.8 m s 2,

3.7 37 U(R) R 0 3.8,,, x 0 x 0 W 01 + W 10, 0,, W 01 + W 10 = 0 (2) F m, x 0 x 1 X, W 01 = F m X,, W 10 = F m X, W 01 + W 10 = 2F m X, h = 10 m, U = 196 J 200 J (2) 35(3), k, k = 6.2 10 4 kg s 2, x = 0.001 m, (3.40), U = 0.031 J (3) M, m, x, G = 6.7 10 11 m 3 kg 1 s 2 M = 6.0 10 24 kg m = 7.3 10 22 kg x = 4.0 10 8 m (3.41), U = 7.3 10 28 J 53,, mg sin θ (3.59) (θ L, mgl sin θ (3.60) 54,,, 56 (1) x 0 x 1 x 0,, x 0 57 53,,,, µ mg cos θ,, mg sin θ + µ mg cos θ (3.61) L,, mgl sin θ + µ mgl cos θ (3.62),,,, mgl sin θ 58 (3.52), m = 2.0 kg, g = 9.8 m s 2, v = 3.0 m s 1, 2 P = 59 W 59 1 W h = 1 J s 1 3600 s=3600 J 60 Q, x q x q (0 < x ), F, (2.9), F = k Q q x 2 (3.63) x q,, W = x F dx = x (3.36), U(x) = W = k Q q x ( ) k Q q [ x 2 dx = k Q q x ] x = k Q q x (3.64) (3.65)

38 3 : (3.64), x( dx x), x( x x), x, X s, x,, x 61 (1) (3.53) x q U(x) q = k Q x (3.66) : ev J,,! 65 (1) 36 A h=36 (C/s) 3600 s=130000 C (2) 12 V 36 A h=12 36 3600 V A s=1.6 10 6 J (3) 46 W 12 V, 46 W/(12 V) =3.8 A 36 A h/(3.8 A) =9.4 h 9 LED,, 48 (2) x = 0.529 10 10 m Q = 1.602 10 19 C, (2.9), k = 8.987 10 9 N m 2 C 2 (3.66), 27.2 V 62 (1) (2) 2,, 2 (3) J C 1,, V (4) 0.6 J (5) 1.602 10 19 C, 1 V, 1.602 10 19 J 63,..., 49...,, 50?..., 51,...,, (1) m He = 4 g/(6.0 10 23 ) = = 6.7 10 27 kg (2) α E, v, 1 2 m Hev 2 = E, v = 2E/m He = = 1.6 10 7 m s 1 (3) β E, v,,, v = 2E/m e = = 4.2 10 8 m s 1, c = 299792458 m/s!,, Newton, v = 2.6 10 8 m s 1

39 4 :,,,, 52,,??...,,,,,, 3,,,,,,, r, 3, x, y, z ( ), F, F = (F x, F y, F z ) (4.1) 4.1,,, ( ), ( ) ( ),, F v,,, F v, ( ),,,, ( ), 3 F x, F y, F z,, (4.1) F, F x, F y, F z (4.1), F = (F x, F y, F z ), F = (F x, F y, F z ) 53,,, F = kx,...?..., 1 x, y, z, F

40 4 F = (F x, 0, 0) (4.2) (x ), y z, F x F, F = (F, 0, 0) (4.3), x, F = kx F, ( ),,,, :, 4.2 ( ), ( ) :, t, r(t) = (x(t), y(t), z(t)) (4.4), (t), t,, v(t) = (v x (t), v y (t), v z (t)) (4.5), a(t) = (a x (t), a y (t), a z (t)) (4.6), v(t) = d r(t) (4.7) dt a(t) = d dt 54 d2 v(t) = r(t) (4.8) dt2 d dt,?... t (4.7), : v x (t) = d x(t) (4.9) dt v y (t) = d y(t) (4.10) dt v z (t) = d z(t) (4.11) dt, (4.8), : a x (t) = d dt v x(t) = d2 x(t) (4.12) dt2 a y (t) = d dt v y(t) = d2 y(t) (4.13) dt2 a z (t) = d dt v z(t) = d2 z(t) (4.14) dt2,,, f(t), x a d f(t) dt = f(x) f(a) (4.15) dt, a = 0, x d f(x) = f(0) + f(t) dt (4.16) 0 dt, x t, t t (, 60, t d f(t) = f(0) + f(t) dt (4.17) 0 dt, (4.17) f x ( (4.15) x ), x(t) = x(0) + t 0 v x (t) dt (4.18) (4.9), (4.17) f v x, v x (t) = v x (0) + t 0 a x (t) dt (4.19) (4.12) (4.18), (4.19), r v y z,

4.3 41 : r(t) = r(0) + v(t) = v(0) + t 0 t 0 v(t) dt (4.20) a(t) dt (4.21), 55 x t, f x,,?... 4.3 v = (v x, v y, v z ) ( ) *1 ( ), v x t (4.18), x(t) = x(0) + t 0 v x dt = x(0) + [v x t] t 0 = x(0) + v x t (4.22) y, z, x(t) = x(0) + v x t (4.23) y(t) = y(0) + v y t (4.24) z(t) = z(0) + v z t (4.25), r(t) = r(0) + v t (4.26), r, r(0), v,,, 0 = (0, 0, 0),, 56 0 = (0, 0, 0),? 0 0 m/s?..., 0 0 0, 0 m/s 0 cm/s 0 km/h?,,? 4.4,, ( ), a x t, (4.19), v x (t) = v x (0) + t 0 a x dt = v x (0) + [a x t] t 0 = v x (0) + a x t (4.27) (4.18), x(t) = x(0) + = x(0) + t 0 t 0 v x (t) dt (v x (0) + a x t) dt = x(0) + [v x (0) t + 1 2 a x t 2] t = x(0) + v x (0) t + 1 2 a x t 2 (4.28),, 2, y, z,, v(t) = v(0) + a t (4.29) r(t) = r(0) + v(0) t + 1 2 a t2 (4.30),,,?, 0 *1,,,, 57, (4.27) (4.28)?...,

42 4,, 66, (4.27), (4.28) 4.5,??,,, 0, m F, F = ma (4.31), a 2 A, B, A B, A B,, ( ),,,,,, 2,, ( ),,,,,, 0,,,,, ( 0 ) 67 A, B, ( ) C (1) A C???? (2) B C???? (3) B A???? :,!,,... (*),,, ( ) 2, (*),,,,,,, 0 (, ) 4.1,, (!), ( ),,, (0 ), ( ) 68? :,

4.5 43,, 1, (4.31) F a, F = (F x, F y, F z ) (4.32) a = (a x, a y, a z ) (4.33), (4.31), F x = ma x, F y = ma y, F z = ma z (4.34) 3 (4.31),,, : ma = F (4.35) 58, F = ma ma = F,...,, ( ),, F = ma ma = F, (4.31), (4.8), : F = m dv dt (4.36) F = m d2 r dt 2 (4.37) (4.31), (SI kg) (SI m s 2 ), SI kg m s 2, N (, (4.36)(4.37),,,,,,, F,,, v(t) r(t), t, t,,, *2, (4.31) F = 0, 0 = ma m 0,, a = 0 a, 0, ( ),,,,,, 1,?,,,, *3,,,,,, 69 3 (! F a!) 70,,,, (1) A B, A B, (2), 0 (3) m F, a, F = ma,, : *2, *3, p. 36-37

44 4 m, v, p = mv (4.38), m, (4.31), F = dp dt (4.39) dp dt = F (4.40), *4, 71 5 (p v!),,,,, ( 3 ) 4.6,,,,, P.41 59,,,,?...,,, 60 p, m, v p = mv,?...,, p = mv, p,, p,,, mv!,, ), ( ), ( ),,, ), 72, m g, t x t,,, x(t), v(t), a(t) x(0) = x 0, v(0) = v 0 (4.41) (1), mg = ma (4.42) (2) (4.42),, g,, : v(t) = v 0 gt (4.43) x(t) = x 0 + v 0 t g 2 t2 (4.44) : (4.27), (4.28) (3), 0, 0, v(t) = gt (4.45) *4, m, F = ma F = dp/dt, F = ma F = dp/dt, x(t) = 1 2 gt2 (4.46)

4.7 45, m,, 61, (2.1),,,,?...,, ( 10 km ), ( 10 km ) 6400 km, r = 6410 km r = 6390 km (2.1) 0.6, 2, 73, 0,, (1) 10,,? (2), 1, 340 m s 1?,? (3) 10 km,,? 75,,, m v 0, x t, t = 0 (1) F m, x(t) (2), x(t), 0 (3) v 0 = 1.5 m s 1, 20 m 20 kg F m (4) (5), x = 25 m v 0? (6) 30 kg ( 20 kg ) x = 25 m, v 0?,, (4.28),,, F = ma (4.28),, 74 x, x = 0, v 0 = 40 m s 1 140 km h 1, t, x(t)? x 0 = 0 m, v 0 = 40 m s 1, (4.44), x(t) t, 4.7 76 2.0 kg, 0,,, x 2.0 s, 6.0 N, 4.0 s :, (4.28) (4.28)

46 4 77 m, v 0 x,, ( ) αv α (x = 0) t, t = 0 ( 4.1), β v 2 = m dv dt (4.50) (2), β dt = mdv v 2 (4.51) (3), (C ) β t = m v + C (4.52) 4.1 (4) C = m v 0 (4.53) (1) t < 0, (5) 0 = m dv dt (2) 0 < t, (4.47) v(t) = mv 0 v 0 β t + m (6) v(t) (4.54) αv = m dv dt (4.48) (3), v(t), v(t) = v 0 exp ( α ) m t (4) v(t) (4.49), *5,, 2 *6 78 79?? 77,?, βv 2 β (1) *5 *6,, 62 77, 0??...,,,, 63?..., 0 80,,,,, m, t v(t) v(0) = 0 αv α v, αv v,

4.8 47 (1) mg αv = m dv dt (2) (4.55) dv = dt (4.56) g + αv/m (3) C m v α ln gm + = t + C (4.57) α (4) v, v(0) = 0, v(t) = mg α (5) v(t) { exp ( α ) } m t 1 (4.58) (6), v = mg α (4.59),, ),,,, (, ), 4.8,,,,, 81??? 10 m, x F = Ae kt, A, k, t v t 0 11 2 A, B, A,, 0.50 m s 2 A, B 2.0 kg 3.0 kg, 4.5 N/m : 2 12 ( B747),, 60 m, 5,, v = 10 m/s?, ( ) F, F = ρv 2 SC D /2 ρ, S C D,, 0.5 2, 300 t, 70 m, 8 m 4.9 67 (1) A C, (0 ) (2) B C (3) B A 0 70 (1) A B, (2), a 0,, F 0 (3) (1 ),, F a F a

48 4 72 (1) (2.4), mg, F F = mg *7, mg = ma (2) (4.27), (4.28), a x g, v x v, (3) x 0 = 0, v 0 = 0 73, x t x(t), v(t) (1) (4.45), (4.46) t = 10 s, v = 98 m s 1, x = 490 m 490 m (2) g v(t) = gt t = v(t)/g, v(t) = 340 m s 1, t = 340 m s 1 /(9.8 m s 2 ) =35 s 35 (4.46) t = 35 s, x = 6000 m, 6000 m 74 (4.44), x 0 = 0, x(t) = v 0 t 1 2 gt2 = 1 ( 2 g t 2 2v ) 0 g t = 1 ( 2 g t v ) 2 0 v + 0 2 g 2g (4.60) (4.61), t = v 0 /g, v 2 0/(2g) v 0 = 40 m s 1,, (40 m s 1 ) 2 = 82 m (4.62) 2 9.8 m s 2, 80 m 75 (1), F m,, F m x, F, F = F m a,, F m = ma (4.63) (2), a = F m /m, *7 (2.4) (4.27), (4.28), a x F m /m, v x v, v = v 0 F m m t (4.64) x = v 0 t F m 2m t2 (4.65) x(0) = 0, v(0) = v 0 4.2 4.2 0 x(t) (3) t, x, dx/dt 0, (4.64), 0 = F m m t + v 0 (4.66), t = mv 0 /F m (4.65),, x = mv2 0 2F m (4.67) F m = mv2 0 2x t (4.68), x = 20 m, m = 20 kg, v 0 = 1.5 m s 1, F m = 1.1 N (4) ( (2.21)), µ, F m = µ N (4.69) N, N = mg, F m = µ mg,, µ = F m /(mg) = 5.7 10 3 (5) (4.67), x, v 0 2 x 20 m 25 m, x 1.25,

4.9 49 v 0 1.25 = 1.12, v 0 1.25 1.5 m s 1 =1.68 m s 1, 1.7 m s 1 (6) (4.67) (4.69) F m, F m = µ mg, x = v2 0 2µ g (4.70) x, 30 kg kg, ( ), ( 1.7 m s 1 ) 64 75(6),?...,,,, F = ma F m, ma m, 76 m, t, F (t), F (t) = bt (4.71) (b ) x, x(t), v(t), a(t), F (t) = ma(t) (4.71), x(0) = 0, (4.74), x(t) = t 0 bt 2 bt3 dt = 2m 6m (4.76), (4.71) t =2.0 s, 6.0 N= b 2.0 s, b = 3.0 N/s, m=2.0 kg, t =4.0 s (4.76), 77 x(4.0 s) = (3.0 N/s)(4.0 s)3 6 2.0 kg = 16 m (4.77) (1) t < 0,, 0,, (4.47) (2) 0 < t F = αv,, (4.48) (3) (4.48), αdt = m dv v (4.78),, C, αt = m ln v + C (4.79) C C, ( v = ± exp α m t + C ) = ±C exp ( α ) m t (4.80) t = 0 v = v 0, ± exp C = v 0 (4.81) a(t) = bt m (4.72) (4.19) (a x, v x a, v),, v = v 0 exp ( α ) m t (4) 4.3 (4.82) v(t) = v(0) + t 0 a(t) dt (4.73) v(0) = 0, (4.72), v(t) v(0) v(t) = t 0 bt bt2 dt = m 2m (4.74) 0 t, (4.18), 4.3 x(t) = x(0) + t 0 v(t) dt (4.75) 79 (1) 0 < t F =

50 4 βv 2,, (4.50) (2), (3) (4) (4.52), t = 0, 0 = m/v(0) + C C = m/v(0) (5) (4.53) (4.52) v, (4.54), (6) 4.4, v + gm α = ± exp [ α m (t C) ], v = ± exp [ α ] m (t C) gm α (4.88) (4.89) v(t) v(0) t = 0, v = 0, [ α ] 0 = ± exp m C gm α (4.90) 0 t, [ α ] ± exp m C = gm α (4.91) 80 4.4 ( ± ) (4.89), v = gm α exp [ α m t ] gm α (4.92) (1), mg αv,, (4.55) (2) (4.55) m, g αv m = dv dt (4.83) dt, g+αv/m, dt = dv g + αv/m (4.84) (4.58) (5) 4.5 v(t) 0 -mg/a v=-gt t, (4.56) (3) (4.56) : dv g + αv/m = dt (4.85), dv (α/m)(v + gm/α) = m α = m α ln v + gm α dv v + gm/α +C 1 (C 1 ), t + C 2 (C 2 ), m v α ln gm + + C 1 = t + C 2 (4.86) α C 1, C 2 C 1 = C, (4.57) (4) (4.57), ln v + gm = α (t C) (4.87) α m 4.5 (6) (4.58) t, v mg/α,, (4.55) 0, v = mg/α 81, 65 10 t,,..., 500 kg, 1

51 5 :, 5.1, z,,, *1, :, ( ), 0 5.1 x, x,, kx 5.1 k, x 5.1 kx = m d2 x dt 2 (5.1), 5.1,, m k,,, x x,,,,,,,,,, (4.31), m,,, x, y d 2 x dt 2 = k m x (5.2) (5.2), x(t), t, *2,,,,, *1,, : ( ), ( ), ( ), 0,,, 0 0,, 0 = 0 *2

52 5 x(t) = A cos ωt (5.3) A ω *3 (5.3), 5.2 A 0 2π/ω 4π/ω t -A x 5.2 (5.3)!, A ( ),, 2π/ω ω *4, T, : ω T T = 2π ω ω = 2π T (5.4) (5.5),, s 1,, Hz ( ), (5.3) (5.2), (5.3) (5.2), (5.2) Aω 2 cos ωt (5.6), A k cos ωt (5.7) m (5.2), A = 0, ω 2 = k/m *5 *3 ω *4, *5, cos ωt = 0 (5.2), A = 0, (5.3) 0, ω 2 = k/m, ω = k m (5.8) (A ) (5.8), (5.3) (5.2),, (5.3) ( ) (5.8), (5.3), (5.2), : x(t) = A sin ωt (5.9) x(t) = A cos(ωt + ϕ) (5.10) x(t) = A sin(ωt + ϕ) (5.11) x(t) = A cos ωt + B sin ωt (5.12) (A, B, ϕ, ) (5.2),,,,,,,, (5.12) B = 0 (5.3), (5.12) A = 0 B A (5.9), (5.3) (5.9), (5.12), (5.10), ϕ = ϕ π/2, cos *6, x(t) = A cos(ωt + ϕ π 2 ) = A sin(ωt + ϕ ), ϕ ϕ (5.11), (5.11), ϕ = ϕ + π/2, sin *7, x(t) = A sin(ωt + ϕ + π 2 ) = A cos(ωt + ϕ ), ϕ ϕ (5.10), (5.10) (5.11), t, t = π/(2ω) t = 3π/(2ω), (5.2) *6 θ, cos(θ π/2) = sin θ *7 θ, sin(θ + π/2) = cos θ

5.2 53, (5.12) ( ), (5.11), (5.11) (5.10) (5.12), (5.10), (5.11), (5.12) 3,, A ϕ,,, 82 5.1, 2,??,,,, (5.12),, t = 0 x(0), (5.12), 83 5.1, t = 0 x = X 0, 0,?, : (5.16) x(0) = A (5.13), (5.12), dx dt = Aω sin ωt + Bω cos ωt (5.14) t = 0 x (0), x (0) = Bω (5.15) 84 5.1, t = 0 x = 0, V 0,?,, (5.17),, t, (5.12), x(t) = x(0) cos ωt + x (0) ω sin ωt (5.16),,,, 2, 1,, 5.1,, x(0) x (0) ( ω (5.8), (5.2), (5.8), : d 2 x dt 2 = ω2 x (5.17), 1, 2, 2,,, 2 (5.17), (5.12) A, B, (5.16),,,, F = ma, x, 2, *8,,, 5.2,,,, *8,,,

54 5, x, y, z,, m ( 5.3) r(t), v(t) g r(t) = (x(t), y(t)) (1) r(0) = (0, 0), v(0) = (v 0 cos θ, v 0 sin θ) (2), 5.3 m d2 x dt 2 = 0 (5.18) m d2 y = mg (5.19) dt2 (3) (5.18), (5.19), x = (v 0 cos θ)t (5.20) 1,, 2,,, 85 2, A, B, C (1), A, B, C? (2) A, B, C,,, (, ) (3) A, B, C,,, (, ) (4), A, B, C? 86,, O, x, y t, t = 0 t = 0 x θ, v 0 t y = (v 0 sin θ)t 1 2 gt2 (5.21) (4) t y = (tan θ)x g 2v 2 0 cos2 θ x2 (5.22) (5), x, y (6) x X, X = v2 0 sin 2θ g (5.23) (7) v 0, θ 87, X?, 7.26 kg, 86.7 m,, 5.3, xy m, r

5.3 55 5.4 (3) F = F, v = v F = mrω 2 (5.35) v = rω (5.36) (4) (5.35), F = mv2 r (5.37) (5), v 2,? (6), r 1/2,? 5.4 (5.32), (5.32), F r, t,, r(t) = (x(t), y(t)) (5.24) v(t) = (v x (t), v y (t)) (5.25) a(t) = (a x (t), a y (t)) (5.26) *9, t = 0 x (r, 0), r(0) = (x(0), y(0)) = (r, 0) (5.27) 88 (1) ω ( : ) r(t) = (r cos ωt, r sin ωt) (5.28) v(t) = ( rω sin ωt, rω cos ωt) (5.29) a(t) = ( rω 2 cos ωt, rω 2 sin ωt) (5.30) a(t) = ω 2 r(t) (5.31) (2) t F(t) = (F x (t), F y (t)), F(t) = mω 2 r(t) (5.32) F x (t) = mrω 2 cos ωt (5.33) F y (t) = mrω 2 sin ωt (5.34) *9,, r 89,,? (1) (2), (3) 90,,, P.114,,, (5.31) : a(t) = ω 2 r(t) d 2 r/dt 2,, r = (x, y),, d 2 x dt 2 = ω2 x (5.38) d 2 y dt 2 = ω2 y (5.39), (5.17),

56 5,, 2 (x y ), ( ) x(t), y(t) = (r cos ωt, r sin ωt) (5.40), x x(t) = r cos ωt (5.41), (5.3), y y(t) = r sin ωt (5.42), (5.9),,,,, 91 ( 87 ),? kg? 1.5 m 2 (5.37) m v 87? 6400 km, (1) 400 km (2) 36000 km,,, 66,?,?...,,,,, GPS ( ), 92, r ( ),, M, m G (1), r, ω, m (2), ω ω = GM r 3 (5.43) (3) T, T = 2π r 3 GM (5.44) (4) r, G, M ω,?,, 67...,, 68...,, 69,?,,,?...,,,,, 93,,

5.4 57 70...,,,,, 84 (5.16), x(0) = 0, x (0) = V 0, x(t) = V 0 ω sin ωt (5.45) 5.6 x V 0 /ω 13,,... (1) ( ) (2) ( ),,,,? (1)! (3),,? 5.4 82 ω = k/m, m 2, ω 1/ 2 0.71, T, T = 2π/ω, T 2 1.4 83 (5.16), x(0) = X 0, x (0) = 0, x(t) = X 0 cos ωt 5.5 x X 0 0 2π/ω 4π/ω t -X 0 -V 0 /ω 0 2π/ω 4π/ω t 5.6 85 : ( ) 86 r(t) = (x(t), y(t)), v(t) = (x (t), y (t)) (1) t = 0 r(0) = (0, 0), v(0) = v 0, v(0) x θ, v(0) = (v 0 cos θ, v 0 sin θ) (5.46) (2) x, (5.18) y mg, (5.19) (3) (5.18) 2, x(t) = C 1 t + C 2 (5.47) C 1, C 2 x(0) = 0 C 2 = 0 x (0) = v 0 cos θ, C 1 = v 0 cos θ, x(t) = (v 0 cos θ)t (5.48) (5.19) 2, y(t) = gt2 2 + C 3t + C 4 (5.49) C 3, C 4 y(0) = 0 C 4 = 0 y (0) = v 0 sin θ, C 3 = v 0 sin θ, 5.5 y(t) = (v 0 sin θ)t gt2 2 (5.50)

58 5 (4) (5) 5.7 v 0 2 sin 2 θ/(2g) 5.7 y 0 2 v 0 sin2θ/(2g) x (6) (5.22) y = 0, x = 0 (5.23),, 2 cos θ sin θ = sin 2θ (7) (5.23), θ = π/4, 45 87, θ = π/4, (5.23) X = v 2 0/g v 0 = gx X = 86.7 m, g = 9.8 m s 2, v 0 = 29.1 m s 1 88 (1) (2) F = ma (5.31) (3) (5.32), (5.29) (4) (5.35) (5.36) ω (5) (5.37), 4 (6) (5.37), 2 90,,,,,,,,,,,,,,,,,,,,,, 91, v, (5.37), F = mv 2 /r 87, m = 7.26 kg, v = 29.1 m s 1 r = 1.5 m, F = 4100 N, g = 9.8 m s 2, 420 kg 420 kg 92 (1) (5.35), mrω 2 (2) GMm/r 2, mrω 2 = GMm r 2 (5.51) ω (3) T = 2π/ω (4) G = 6.67408 10 11 N m 2 kg 2, M = 1.9891 10 30 kg, r = 1.4960 10 11 m, ω = 1.99125 10 7 Hz, 2π/ω = 3.15539 10 7 s= 365.207 =365.21 : 365.25, r : 5, 1, 5 93 (5.44) M (1) r = 6800 km, T = 5580 s 1.6 (2) r = 42000 km, T = 87000 s 24

59 6 (1) 3,,,,,,,,,,, ( ), (x,, m F,, F = ma (6.1) a ( F, a, 1, F a ),, :?,,,,,...!,, (t = t 0 ) (t = t 1 )?, (6.2),,, 6.1 1 m v, (kinetic energy) 1 2 mv2 (6.3) F = m dv dt (t, v ) (6.2) 94 (6.3) 5 (6.1) (6.2),,, (4.31) 1, (6.2), (x ), F (t), m, t 0 t 1, (6.2), (6.2),,,,, v ( m,,, m ), T (v) T (v) = 1 2 mv2 (6.4), x v x t

60 6 (1)!, dx dt = v (6.5) dt, dx = v dt (6.6), x dx, v dt, (6.2) dx F dx = m dv dx (6.7) dt dx, (6.6), : F dx = m dv v dt (6.8) dt, t 0 t 1,, (6.8), t 0 t 1, (6.8), x1 x 0 F dx = t1 t 0 m dv v dt (6.9) dt x 0 = x(t 0 ), x 1 = x(t 1 ) t v *1, x1 x 0 F dx = v1 v 0 mv dv (6.10) v 0 = v(t 0 ), v 1 = v(t 1 ), x 0 x 1 W 01 ( (3.20) ),, = [ ] v1 1 2 mv2 = 1 v 0 2 mv2 1 1 2 mv2 0 (6.11), W 01 = 1 2 mv2 1 1 2 mv2 0 (6.12)! (6.12), (6.4), W 01 = T (v 1 ) T (v 0 ) (6.13),, T (v 1 ) = T (v 0 ) + W 01 (6.14), *1 dt,,,,,, (6.3) v 2,, v ( ), v,,,,, ( ) 95 (6.12) : (6.5), (6.12), ( (6.12), ), (6.12),, 96 m, F, a (x ) t x(t), v(t) (1) t 0 t 1, F W 01 : (2) : W 01 = F {x(t 1 ) x(t 0 )} (6.15) W 01 = ma{x(t 1 ) x(t 0 )} (6.16) (3) (6.12), *2 97 v(t 1 ) 2 v(t 0 ) 2 = 2a{x(t 1 ) x(t 0 )} (6.17) ( 75) *2 v 2 v 2 0 = 2ax,,, (6.17),,

6.2 61 v 0 x, (1) F m x (2) 0 1 2 mv2 0 (6.18) 0 (3), F m = mv2 0 2x (4.68) (6.19) (4), 2 6.2 (6.13), (6.14), T,,, : x W (x), U(x), U(x) = W (x), W (x 1 ), x 0 W (x 0 ), x 1 W 01,, W (x 0 ) + W 01 = W (x 1 ) (6.23), W 01 = W (x 1 ) W (x 0 ) (6.24),, W (x 0 ) = U(x 0 ), W (x 1 ) = U(x 1 ) (6.25) 98 m,,, v 1 M, R, G, (1),, R),, W, W = GMm R (6.20) (2), mv 2 1 2 0 (6.21) (3), v 1 : 2GM v 1 = R (6.22) (4),, v 1,, W 01 = U(x 1 ) + U(x 0 ) (6.26) (6.12), U(x 1 ) + U(x 0 ) = 1 2 mv2 1 1 2 mv2 0 (6.27), 1 2 mv2 0 + U(x 0 ) = 1 2 mv2 1 + U(x 1 ) (6.28), T (v), T (v 0 ) + U(x 0 ) = T (v 1 ) + U(x 1 ) (6.29) x 0,, x 1,,,, x 0 x 1,,,,

62 6 (1),,, 99 (1)? (2)? (3)? 100, 73 t = 0 v = 0, x = 0, v(t) = gt (6.30) x(t) = 1 2 gt2 (6.31) (1) t T U, T = 1 2 mg2 t 2 (6.32) U = 1 2 mg2 t 2 (6.33) (2), T + U = 0 (6.34) 101, 5.1 t x(t), v(t) t = 0 v = 0, x = x 0, x(t) = x 0 cos ωt (6.35) ω = k/m (1) t v(t) x(t) t v(t) = x 0 ω sin ωt (6.36) (2) t T U, T = 1 2 mx2 0ω 2 sin 2 ωt (6.37) U = 1 2 kx2 0 cos 2 ωt (6.38), x = 0 U = kx 2 /2 x x(t) (3), T + U = 1 2 kx2 0 (6.39) 100, 101,,,, : 102, 75, 97,? 6.3,,, 2000 km,,? 2, R = 6400 km, g = 9.8 m s 2, m, v, h G, M,, U h, U 0!

6.3 63 0,, U h = U 0 + 1 2 mv2,, U h U 0 = 1 2 mv2 (1),, U h = GMm R + h, U 0 = GMm R, * 3, U h U 0 = GMmh R(R+h) GMm, mg, GM R R 2 = gr, U h U 0 = grmh R+h (1),, v = 2ghR h + R grmh R+h = mv2 2, v2 = 2ghR h+r,, R = 6400 km, g = 9.8 m s 2, h = 2000 km,! v = 2 9.8 m s 2 2000 km 6400 km 2000 km + 6400 km, 2 9.8 2000 6400 m s 2 km 2 = 8400 km, 2 9.8 2000 16 m s 2 km = 21, 2 9.8 2000 16 m s 2 10 3 m = 21 (km k ) SI 2 9.8 2.0 16 m 2 s 2 10 6 = 21 10 2 9.8 2.0 16 = 10 3 m s 1 21 10, = 5.5 10 3 m s 1 :!,,, v =,,!!, ( ), (10 ),, h [m], [m]? h = 2000 km h = 2000000 m,,, h [m],,, ( ), 103,,, 50 m, L 20 m, 60 kg,,,? *3 : 6.1 103

64 6 (1),, 1.0 m,, 0.20 m (1) 1.0 m k 0? (2) 20 m, 20 m k? 0, T 0 U 0,, E 0 0, x, mgx v (3) x, E 1, E 1 = mv2 2 mgx (6.40), (4) x, E 2, E 2 = mv2 2 mgx + k(x L)2 2 (6.41) (5), E 3 E 3 = mgx + k(x L)2 2 (6.42) (6), (E 0 = E 3 ), x,? 6.4,,,,,,,,,,,, +,,, E = mc 2 (6.43), E, m, c,,,,,,,,,,,,,,,,,, *4 *4,,,

6.5 65?,,, 104,,?,,,,, 14 40 N/m 10 cm, 4.0 g,?,, :, 15 4.0 mm, 10 cm, 1.0 kg, 30 cm 10,,,?, ( ),,, ρ = 7.8 g cm 3, 460 J kg 1 K 1 6.5 94 (6.4) T (!) 96 (1) F, W 01 = x(t1 ) x(t 0 ) F dx = (2) F = ma [ ] x(t1 ) F x = F {x(t 1) x(t 0 )} x(t 0 ) (3) (6.12) W 01 : 1 2 mv(t 1) 2 1 2 mv(t 0) 2 = ma{x(t 1 ) x(t 0 )} 2 m, 97,,,,,,,, (1), F m F 96(1), F m x (2) mv 2 0/2, 0 (3) (6.12), (1) (2) (4) (3), 98 x = mv2 0 2F m (6.44), x v 2 0 (1) (3.24), R 0 =, R 1 = R (2) (3) (6.13), (6.20) (6.21),, GMm R = 1 2 mv2 1 (6.45) v 1 = 2GM R (4) G = 6.67 10 11 N m 2 kg 2, 99 M = 5.97 10 24 kg, R = 6.38 10 6 m, (6.46) v 1 = 1.12 10 4 m s 1 40000 km/h (1)

66 6 (1) (2),, (3), 100 (1) (6.4) (6.30), (6.32), (3.37), h x, U(x) = mgx (6.31), (6.33) (2) (6.32) (6.33) 101 (1) ( (6.35) t ) (2) (6.4) (6.36), (6.37), (3.40) (6.35), (6.38) (3) (6.37) (6.38), T + U = 1 2 mx2 0ω 2 sin 2 ωt + 1 2 kx2 0 cos 2 ωt, ω = k/m,, T + U = 1 2 kx2 0 sin 2 ωt + 1 2 kx2 0 cos 2 ωt, T + U = 1 2 kx2 0(sin 2 ωt + cos 2 ωt) = 1 2 kx2 0, 102, mv0/2 2 0,,,,,, mv0/2 2,,,, 103 m, g (1) 0, k 0 δ + mg = 0 δ, 0.2 m, k 0 = mg δ = 60 kg 9.8 m s 2 0.2 m = 2940 N/m (2), 20, k = k 0 /20 = 147 N/m (3) mv 2 /2,, mgx, E 1 = mv 2 /2 mgx (4) x L, k(x L) 2 /2 (5), 0, v = 0 (6) E 0 = E 3, k(x L)2 0 = mgx +, 2 ( x 2 2 L + mg ) x + L 2 = 0 k 2, x = L + mg 2mgL k ± k + m2 g 2 k 2 (6.47), x =37.3 m, 10.7 m, L =20 m,, x = 37.3 m 50 m,, 104 2, 40 km/h 60 km/h 1.5, 1.5 2 = 2.25,,,,,,, 71 U 0?... U, 0, U 0, U 0, U 0

67 7,,,,,,,,,, 7.1 2,,, 7.1 7.1, 2 A, B, A, B m A, m B t 0, A, B v A, v B xy t 1 2 t 2, v v,?,,,,,, t 1,,,,,, 2, t 1,,, :,, *1,, (4.38),,,, t 0, m A v A + m B v B (7.1), A, B, A, *1,,

68 7 B t 2,, (m A + m B )v (7.2), (7.1) (7.2),, m A v A + m B v B = (m A + m B )v (7.3), v = m Av A + m B v B m A + m B (7.4),,, 105 7.1, m A = m B = 1.0 kg, t 0 A x 1.0 m s 1, B y 1.0 m s 1 (1) v (2) v,! m F,, F = ma = m dv dt = md2 r dt 2 (7.5) t r, v, a,,,,, m dv dt = F (7.6) dt : m dv = F dt (7.7) t = t 0 t = t 1 t, v(t1) v(t 0 ) m dv =, t1 t 0 F dt (7.8) [ ] v(t1) mv = mv(t 1 ) mv(t 0 ) (7.9) v(t 0), (7.8), mv(t 1 ) mv(t 0 ) = t1 t 0 F dt (7.10) t 0 t 1,, t, F(t), t1 F dt (7.11) t 0, t 0 t 1 106 5, (7.10),,,, 72...,,,,, A, B 2 t = t 0 t = t 1 A B F AB,, B A F BA, (7.10) : m A v A (t 1 ) m A v A (t 0 ) = m B v B (t 1 ) m B v B (t 0 ) =, t1 t 0 F AB dt (7.12) t1 t 0 F BA dt (7.13) m A v A (t 1 ) m A v A (t 0 ) + m B v B (t 1 ) m B v B (t 0 ) = = t1 t 0 F AB dt + t1 t1 t 0 F BA dt t 0 (F AB + F BA ) dt (7.14), F AB = F BA (7.15)

7.2? 69, F AB + F BA = 0 (7.16), (7.14) 0 (0 0), m A v A (t 1 ) m A v A (t 0 ) + m B v B (t 1 ) m B v B (t 0 ) = 0, m A v A (t 1 ) + m B v B (t 1 ) = m A v A (t 0 ) + m B v B (t 0 ) (7.17), 2 t 1 t 0,, 2,,, 3,,,! 107 3 A, B, C, (7.12), A, B, C,, A F AB + F AC 108 2 A, B, x,, A, B 2.0 kg 3.0 kg, A, B 4.0 m s 1, 5.0 m s 1, 2 1, 7.2?,,,?, 109 105 (1) (2),,?,,,,,!, 0 ( )?,,,,, ( =!),, 0,,?,,,,,,,,,,,,,,, ( ),, ( ),, 2...,,

70 7 3 (6) (7.24) :... (7.17) v B v A v B v A = 1 (7.25) 110? (7.24),,,, v B v A, v B v A, e, e = v B v A v B v A (7.26) 7.2 1 2 :, :, : 111 x, 2 A, B, v A, v B ( 7.2 ), ( 7.2 ), v A, v B,, x ( 7.2 ) A, B m A, m B 2, 2 ( ), (7.25), e = 1, e 0 1 e = 0, B A 0.8 73 e,?...,, e, 0 1 e = 2.718 (1), 112 111 2 m A v A + m B v B = m A v A + m B v B (7.18) 1 2 m AvA 2 + 1 2 m BvB 2 = 1 2 m Av A 2 1 + 2 m Bv B 2 (7.19) (2) (7.18), (7.19) : m A (v A v A ) = m B (v B v B ) (7.20) m A (v A2 v 2 A ) = m B (v B2 v 2 B ) (7.21) (3) (7.21) : m A (v A v A )(v A + v A ) = m B (v B v B )(v B + v B ) (4) (7.22) (7.20) : (7.22) v A + v A = v B + v B (7.23) (5) (7.23) : v B v A = (v B v A ) (7.24) (1) (7.23), (7.20) v B : v A = m A m B v A + 2m B v B (7.27) m A + m B m A + m B (2) (7.23), (7.20) v A : v B = m B m A v B + 2m A v A (7.28) m A + m B m A + m B (3) m A = m B, : v A = v B (7.29) v B = v A (7.30) (4) m A >> m B, : v A v A (7.31) v B v B + 2v A (7.32)

7.2? 71 (7.29), (7.30), 2, ( 7.3),,, h 1, v1 2 = 2gh 1 (7.35) (4) h 1 = e 2 h 0 (7.36) (5),,,... n 1 7.3 1 2,,,,, (7.31), (7.32),, ( ), ( ), ( 7.4),, n h n, h n = e 2n h 0 (7.37) (6) e = 0.8, h 0 = 10 m, 0.1 m,? 114 2,, h 0, A, m A B, m B B A, m A >> m B A, A B g 7.4 1 2, 113 h 0 0, e (1) v 0 v0 2 = 2gh 0 (7.33) (2) v 1 v 1 = e v 0 (7.34) (3), 7.5 2 114 (1) ( 7.5 ) A v 0 v 0 = 2gh (2) A, B A ( 7.5 ) A v A, B v B : v A = v 0 (7.38) v B = v 0 (7.39) (3), B A

72 7 ( 7.5 ) B v B,, (7.32), : v B 3v 0 (7.40) (4) B, A, ( h)? 7.6 16 7.3,,,,,,,,,,,,,!!?,,,,,,, 0 *2 16 2, n, h,,, ( 7.6) v n *2 (1) : v n (2 n 1)v 0 (7.41) (2), ( h)? (3) h = 5 m,, 10 ( : 2 ) 17 200 km/h 200 km/h, 200 km/h, 3.0 ms,,,? 60 g 18 50 kg, 10 kg, 1.0 m s 1, 19 M, u,, M,, 0, M 0 v M :,, M, M + dm (dm < 0) dm ( ), v u

7.4 73 7.4 105 (1) m A = m B = 1 kg, v A = (1 m s 1, 0 m s 1 ), v B = (0 m s 1, 1 m s 1 ) (7.4), v = (0.5 m s 1, 0.5 m s 1 ) (2) v = (0.5 m s 1, 0.5 m s 1 ) = 0.71 m s 1 106 107 m A v A (t 1 ) m A v A (t 0 ) = m B v B (t 1 ) m B v B (t 0 ) = m C v C (t 1 ) m C v C (t 0 ) = t1 t 0 t1 t 0 t1 t 0 (F AB + F AC )dt (F BA + F BC )dt (F CA + F CB )dt F AB + F BA 0,, 109 m A v A (t 1 ) + m B v B (t 1 ) + m C v C (t 1 ) = m A v A (t 0 ) + m B v B (t 0 ) + m C v C (t 0 ) (1) A, B : 0.5 J, 1 J (2) (m A + m B ) v 2 /2 = 0.5 J 110 111 112 (1), (2), (3) ( ) (4) (7.27), (7.28) m A, v A = 1 m B/m A 1 + m B /m A v A + 2m B/m A 1 + m B /m A v B v B = m B/m A 1 2 v B + v A 1 + m B /m A 1 + m B /m A, m A >> m B, m B /m A 0, 2, v A 1 0 1 + 0 v A + 2 0 1 + 0 v B = v A v B 0 1 1 + 0 v B + 2 1 + 0 v A = v B + 2v A, 113 (1), mgh 0, 0 0 mgh 0,, 0, mv 2 0/2 mv 2 0/2, mgh 0 = mv 2 0/2 (2) e (3) (7.33) (4) (7.33), (7.34), (7.35) v 0, v 1 (5), h n = e 2 h n 1 e 2 (6) h 0 = 10 m, h n = e 2n h 0 < 0.1 m, e 2n < 0.01 e = 0.8, 0.8 2n < 0.01 n = 10 0.8 2n = 0.0115 > 0.01 n = 11 0.8 2n = 0.0074 < 0.01, n = 11, 11 114 (1) A,,, 1 2 m Av 2 0 = m A gh (7.42) v 0 (2), (3) ( ) (4) B, (1), : 1 2 m Bv 2 0 = m B gh (7.43), H,,, 1 2 m Bv B 2 = mb gh (3), v B = 3v 0, 9 2 m Bv 2 0 = m B gh (7.44) (7.44) (7.43), 9 = H/h, H = 9h, 9

74 7 7.5 :, ( ),,,,,,, *3,,, II,,,,, 1 2,,,,,, (x, y, z) *4,,,, ( ) E E = h 2π ω (7.45) h, h = 6.626068 10 34 J s (7.46) h/(2π), 2π,, ħ, ħ := h 2π ( ) (7.47), (7.45) : E = ħω (7.48),,,,,, ( ) ω, ω, 2π/ω c λ, 2π ω c = λ (7.49) (7.45) ω, ( ) : E = hc λ (7.50),, (7.50),, ( ),, (7.50), (7.45) (7.48) t ω, *3,,, *4??,

75 8 I II,, 8.1,,, ( 8.1), (= ) 3 (x, y, z, (x y, ), 3,,,,,,, 8.1 A: x, B: y, C: z,,?, *1 ( ), K K = 1 2 k BT (8.1), T k B, k B = 1.38065... 10 23 J K 1 (8.2) (8.1), (8.2), K *2 *1 T T,, T K *2 K,, (8.1),,,,,,,, *3, (8.1) (8.1),, (8.1),, II (8.1), ( ), (8.1), K, *3,,!

76 8 1/2,,, 1,,, z (L/2, 0, 0) x A 115 5 116 m (v x, v y, v z ),, v x, mv 2 x/2, k B T/2, * 4 v x = kb T m (8.3) v y, v z, ( )v, v = v 2 x + v 2 y + v 2 z (8.4), v, v = 3 k BT m, T = 300 K 27 : (1) H 2 x (2) H 2 (3) N 2 (8.5) 8.2, N, : 2 *5,, A,, A,, (P ), ( 8.3),,, v, m,, (8.5) 8.2 ( ),,, L 8.2, O, x, y, *4, v x (root-mean-square) 8.3 A (z ) A, A P *5,,,,, OK

8.2 ( ) 77 :,,, x (x ) v x, x, v x v x, x v x,,,, t, v x t x, t, A,, A ( = ),, A L 2, A ( ) P L 2 (,, x ), P L 2 t (8.11) (8.10) (8.11), A : A v x t L 2 v x t (8.6) P L 2 t = Nmv2 x t L, (8.12),,, L 2 v x t L 3 N (8.7) P L 3 = Nmvx 2 (8.13), L 3 V, (8.13) P V = Nmvx 2 (8.14), (8.3), *6, x, x, A, v 2 x = k BT m (8.15) 1 2 L 2 v x t L 3 N = Nv x t 2L (8.8), (8.14), :, 1 A, A, ( ), A (v y v z ), A v x,, x, mv x, mv x,, 1 : mv x (mv x ) = 2mv x (8.9) 1, (8.8) t A, : P V = Nk B T (8.16),, N n, N = nn A N A, (8.16), P V = nn A k B T (8.17),, : R, (gas constant) : R = N A k B (8.18) Nv x t 2L ( 2mv x) = Nmv2 x t L (8.10), 8.314472... J mol 1 K 1, (8.17), : *6 L 3, N, ( ), N/L 3, ( (8.6)), (8.7) ( ) P V = nrt (8.19)

78 8 117? (8.18), 118 ( ) U = F nrt (8.22) 2, 1, 119 (1) k B, 3 (2) R, 3 (3), U = F RT (8.23) 2, F?, (He, Ne ) 3 F = 3, 2 (H 2, O 2, N 2 ), 8.3,, *9 ( ) 2 * 10, 2, F, 3, 2, 5 ( 8.4),,,,,,,,,, 1 1, k B T/2, F N U *7, U = F 2 Nk BT (8.20),,, *8, (8.20) n, N = nn A, 8.4 2 A: x, B: y, C: z, D: x, E: z,, U = 3 nrt (8.24) 2 2, U = 5 nrt (8.25) 2 * 11,,,, 2,, U = F 2 nn Ak B T (8.21) *7, U,, U *8,,, *9, *10, 3,,,, 1,, *11, 3/2 5/2,,

8.4 79 * 12, ( ),,,, 74 H 2O 3?... V, 3 3 3, F = 6 (8.23), U = (6/2)nRT = 3nRT, CO 2, 3 2, 2 F = 3 + 2 = 5 U = (5/2)nRT 8.5,,,,,, 48,,!,,,, 8.4,, dq dt, dq/dt ( ), dq dt,, X, dq dt, dq/(x dt ) ( ), X, X,, C v v, volume v volume,, n U, (8.22), U = F nrt (8.26) 2 dq dt, U + dq, T + dt, U + dq = F nr(t + dt ) (8.27) 2 (8.27) (8.26), dq = F nr dt (8.28) 2 120,, SI,,, J K 1, J K 1 kg 1, J K 1 mol 1 C v = dq n dt = F 2 R (8.29) 75?...,, *12, 2,,, ( ),, 2,,, C v = 3 2 R (8.30) 2, C v = 5 2 R (8.31), (8.29) (8.22), U = n C v T (8.32),,

80 8,,,,,?,, 8.6,,,,, : U = Q + W (8.33), U,, U, Q, W 121 ( (8.33)) 5!, U,,,,, d (8.33), du = dq + dw (8.34) dq, dw, (3.30), dv,, dw = P dv (8.35), du = dq P dv (8.36), : du + P dv = dq (8.37), P,, U + P V = Q (8.38),,, 8.7,,,,, C p p, pressure p pressure, C p, (8.37), (8.32), dq = nc v dt + P dv (8.39),, P V = nrt P n, R, P dv = nr dt, dq = nc v dt + nr dt = n(c v + R) dt (8.40), dq/(n dt ) = C v + R C p,, C p = C v + R (8.41),, 122 :, C p = 5 2 R (8.42) 2, C p = 7 2 R (8.43)

8.8? 81 8.8?,,,,,??,,,,,,,,,,?,,,,,,?,,,, 2:1,,?,,, (8.32) * 13, ( ) U, (8.33),, (8.33) U = 0 T!,, (8.32) U, * 14 U = 0,,?,,,?,,,, U 0, (8.32)?,,,,,,,,,, (8.32) *13, V 3, C v 3R *14 C v, T (8.32) U

82 8,,,,,,,, (8.32),,,,!,, P V V, P V,,,, ( ),,,,,? 5000,, 10 30,,,,,,,, 8.9, 1 I,,,,, 1?,,,, m, g, h mgh, h,,,, h, mgh!!,, U, P, V, H := U + P V (8.44) ( ) 76 (8.38) Q U, P V... ( ) ( ),!?,,, U U, U,!! 123 (8.44) 5,,, H, ( U, P, V, H) ( U + U, P + P, V + V, H + H),, H = U + P V (8.45)

8.10 83, H + H = (U + U) + (P + P )(V + V ) (8.46), H = U + P V + V P + P V (8.47),, P = 0,, H = U + P V (8.48) (8.38), H = Q (8.49),,, ( ),,,,, 1 S 1, 2 S 2,, S 2 S 1 S, S, : 77,?? S = 2 1 dq rev T (8.50)?...,,,,,,,,,,,,,,, 1 2, Q rev, T, 1 2,, S ds,, T,,,,,,! ds = dq rev T (8.51), U,,,, 8.10,,,,,,!,, (8.50), (8.51),,,, 124 (8.50) 5

84 8 8.11,,,,, ( ),, ( ) 8.1 (= = ),,,,,, 8.2,,,,,,,,,,,,,? 8.1,,!, (8.50),, (8.50) dq rev 0, 0,, 0,,? 8.3,,,,,,,,,,,,, 0,,,,,,,,,,?,, (8.37), : du + P dv = dq (8.52) U, P V, U, du = 0 P dv = dq dq dq rev, (8.50), : S = 2 1 dq rev T = V2 V 1 P dv T (8.53), V 1 V 2, 1, 2 ( ), P = nrt/v (n ), S = V2 V 1 n R dv V = n R ln V 2 V 1 (8.54)

8.12 85 ( ) 125 300 K 2.0 mol, 50 L 100 L,?, 2,,,,, (8.50) (8.54), 8.12, : 8.4 V 2,, V 1, n, T, ( 1 ),,, ( 2 ) 1 2?,,,, ( ), S? 0, (8.50) dq rev = 0, S = 0?,, ( 0 ) dq rev (8.50)? 1 2,,, 2 1 (8.22), U T ), 1 2,,,,,,,, 8.3 S = V2 V 1 n R dv V = nr ln V 2 V 1 (8.55), 8.4,, 8.13 8.3,, (8.54) V 2 > V 1, S > 0, V 2 < V 1, S < 0,,,,,,,,,,,,,,, 8.4?,, V 2 > V 1, V 2 < V 1,,, (8.55) 0,,, 0,,,

86 8, 78?...,,,, 8.4, :,,,, 126 3, 8.3,,, 8.4, 79 8.4,!?...,,,,,, 80 8.4,,?...,,,,,,, 8.14,,,,,,,?,!!,,,,, G := U + P V T S, U, P, V, T, S 127 3,, (8.47), ( U, P, V, T, G) ( U + U, P + P, V + V, T + T, G + G),, G = U + P V T S (8.56), G + G = (U + U) + (P + P )(V + V ) (T + T )(S + S) (8.57), G = U + P V + V P + P V T S S T T S (8.58),,,, P = 0

8.14 87 T = 0,, G = U + P V T S (8.59) 1 2, (8.38),,, ( ) G = Q T S (8.60),,, dg = dq T ds (8.61) dq, dq, (, ), dq/t, ds,, 3, n,,, T, P, V, S R, C p, C v 1, 2, 20 1 2, P 1 V γ 1 = P 2V γ 2 P V γ γ := C p /C v : (8.36) dq = 0 ( ), du = P dv, (8.32), du = n C v dt, P dv = n C v dt P P = nrt/v,, dv/v = (C v /R) dt/t ds dq (8.62) T, 0, ds dq T 0 (8.63) T T 0, : T ds dq 0 (8.64), dq T ds 0 (8.65), (8.61), dg 0 (8.66) (8.63),,,,,,,,,,,, dg = 0, G,, : 1 2 C v /R, V 1 T C v/r 1 = V 2 T C v/r 2 T 1, T 2, (8.41), P 1 V γ 1 = P 2V γ 2 21 1 2, P 1 = P 2, 1 2,, 1 3, 3 2 2 (1) P 1 V 1 = P 3 V 3 (2) P 3 V γ 3 = P 2V γ 2 (3) : V 3 = ( V1 ) 1 1 γ V γ 2 (4) : (8.67) S 3 S 1 = n(c v + R) ln V 2 V 1 (8.68) (5) S 2 S 3 = 0

88 8 (6) ( ): S 2 S 1 = n C p ln V 2 V 1 (8.69) (7) 1 2,, Q, : dq = n C p dt (8.70) (8) 1 2,,, : 2 1 dq T (9) (8.71) (8.69) 22 (8.71) 1 2, V 1 = V 2, 1 2,, 1 3, 3 2 2 (1) P 1 V 1 = P 3 V 3 (2) P 3 V γ 3 = P 2V γ 2 (3) :, : 2 1 dq T (9) (8.76) (8.74) (8.76) 21, 22,,,, 8.15 116 (1) H 2 2 H 2 1 m, (2 10 3 /N A ) kg N A (8.3), v x = 1.1 10 3 m s 1 (2) (8.5), v, v x 3, v = 1.9 10 3 m s 1 (3) (8.5), v, N 2 (28) H 2 (2) 14, N 2 v, H 2 v 1/ 14 = 0.27, v = 5.2 10 2 m s 1 119 (1) k B = 1.38 10 23 J K 1 (2) R = 8.31 J mol 1 K 1 (3) R = N A k B N A ( P2 ) 1 γ 1 V 3 = V 1 P 1 (8.72) (4) : S 3 S 1 = nc v ln P 2 P 1 (8.73) (5) S 2 S 3 = 0 (6) ( ): S 2 S 1 = nc v ln P 2 P 1 (8.74) (7) 1 2,, Q, : dq = n C v dt (8.75) (8) 1 2,,

89 9 (2) 6, (1 ),,, 3,,, 1, 3 9.1 3 6, m v (1 ), T (v), T (v) = 1 2 mv2 (9.1) ( (6.4)),, 3, : F, r F r θ,, r cos θ (9.4), W, W = F r cos θ (9.5),, W = F r (9.6) *1 (9.6), (3.13) 3 81,... 3, (9.1) v 2 v, v 2, v = (v x, v y, v z ), v 2 = v v = vx 2 + vy 2 + vz 2, v 2 v 2, v 2 = vx 2 + vy 2 + vz 2 (9.2) ( ), 3, (9.1), T (v) = 1 2 mv2 = 1 2 m(v2 x + vy 2 + vz) 2 (9.3) 1, (9.1), (9.1) 9.2 3, 3,, (9.6), F, F,, F, (3.16) (3.20), r 0 r n,, r 1, r 2,..., k 1 n, r k 1 r k 2 r k = r k r k 1 (9.7) 2 F k *1 (9.6),,,

90 9 (2) 9.1 2 W k, (9.6), : W = F dr (9.11) Γ 9.1 W k F k r k, r 0 r n W : W n W k k=1 n F k r k (9.8) k=1 (3.18) 3, W = lim n r k 0 n F k r k (9.9) k=1 Σ, d!), W = r r 0 F dr (9.10) r n r 82!?, dx dt, d x t, dr, d!?...,, (, ) (, ) (3.20) 3, r 0 r,, Γ, : Γ F, r,,,, Γ,,, 3, 128 3 9.3 3, 3,, 1 (3.36),, (3.38), (3.39) x r, 3, U(r), 1 : U(r) := W (r) (9.12) W (r), r, 2 : U(r) := W (r) (9.13) W (r), r, 3 : U(r) := W (r) (9.14) W (r), r,, 3, 129 F, r 0 r 1, F W 01, W 01 = U(r 0 ) U(r 1 ) (9.15)

9.4 3 91 U 130, 3, 0 9.4 3, 3 1, (6.7) (6.9), 3, : F = m dv dt (9.16) F, v, m, t,,, r(t) t, dt t + dt, ( ), dr, dr = r(t + dt) r(t) (9.17) dt v, v = dr/dt, : dr = vdt (9.18) (9.16) (9.18), : F dr = m dv dt vdt (9.19), (6.8) 3, t 0 t 1 r 0 = r(t 0 ), r 1 = r(t 1 ), v 0 = v(t 0 ), v 1 = v(t 1 ), r 0 r 1 Γ Γ, (9.19), t 0 t 1, (9.19), Γ F dr = t1 t 0 m dv dt vdt (9.20) (9.11), W 01, v = (v x, v y, v z ), dv ( dt = dvx dt, dv y dt, dv ) z dt (9.21) (9.20), : t1 t 0 t1 = = = t 0 t1 ( dvx m dt, dv y dt, dv ) z (v x, v y, v z )dt (9.22) dt t 0 vx(t 1) v x (t 0 ) ( dv x m v x dt + v dv y y dt + v z dv t1 x mv x dt dt + dv y mv y t 0 mv x dv x + vy(t 1) [ 1 vx(t 1) [ 1 = 2 x] mv2 + 2 mv2 y v x(t 0) v y (t 0 ) ] vy(t 1) dv ) z dt (9.23) dt dt dt + mv y dv y + v y(t 0) = 1 2 m( v 2 x(t 1 ) + v 2 y(t 1 ) + v 2 z(t 1 ) ) t1 + [ 1 2 mv2 z mv z dv z dt dt t 0 vz(t 1) v z (t 0 ) ] vz(t 1) v z(t 0) mv z dv z 1 2 m( v 2 x(t 0 ) + v 2 y(t 0 ) + v 2 z(t 0 ) ) (9.24) = 1 2 mv2 1 1 2 mv2 0 (9.25), (9.20) : W 01 = 1 2 mv2 1 1 2 mv2 0 (9.26) (9.3), (9.26) W 01 = T (v 1 ) T (v 0 ) (9.27), T (v) (9.27), 1 (6.13), 3,,,, (9.27) (9.15), U(r 0 ) U(r 1 ) = T (v 1 ) T (v 0 ) (9.28), T (v 0 ) + U(r 0 ) = T (v 1 ) + U(r 1 ) (9.29), ( t 0 ) ( t 1 ), 1,, ( ), (9.29), 3

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