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1 ( ) 24 1 ( )

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5 0.5 iii,,, ( 1 ) 1 1,,,,,,, 2 2,,,, ( ),,, 5, E104,,,,,,,,, 0.5, III, C,, 1!! 2,, 2,,,?,,, (),,

6 iv,,!,,?, : 1 1,,,,,,,

7 ,,,,, *1,,,,,,,,,,,,,,,,, 1,,,,,,, *1, 2,,,,,,, 3,,,,,,,,,,,,

8 2 1 GPS (Global Positioning System;, ),,,,,,,,, (LED), 1, ( ),,,,,,,,,,,,,,,,,,,,,,,,,, 2003,,,,,,,,,,,,,,,,,,,,,,,,,,,

9 1.2 3,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 1.2,......,,......,,...,, A B, A B,, A B, B A,,, A B, B A *2,,,,,,,,,,, 3,,,,,, *3,,,,, ( ), *2,, B A,, *3

10 4 1, 1 2? (3), 1,, 1, , (4),,,,,,,,,,,,,, ,,, 15, 15, , 100, 250, (1) (2),, 1, mm,, 1 m 2, 500 g, 1 kg 3000 kg,,? 1000,,, 1, , 1000 /, 1000 *4,,, 3,,,, 1.4,,,,,,,,,, (1) J, 30 30, (2) 5, J 1 *4,?, :,,,,,,,,

11 1.5 5,,,,,,,,,,,,, ( ), *5 ( 1.1),,,,, 1.1,,,,, 60 7,,,,, 1.1 ( ) 6? 7?,,, *5,, : : () 2 () 3 ( : *6 2,,,, 3 (1) 30 30, *6,, 2,

12 J ) (2) 1000/5 200 (3) / (4) 200/50, , 2, 3,, 4 (1) 1/10 1/ (2) ,, 5, 1 m 2 1, = 1500 kg 1000 kg/m 3, 1.5 m 3 1m 2, 1.5 m, 1500 mm,, 1500 mm, *7?, 50 6, 7,, 1.6,,,?,,,,,,, *7,,

13 ,, 4 *1,,, 4,,,,, 14 C 14 N 8, 4?,,,,,,,,, *2 2 A, B, A B, A B, 2.3, M m r,, ( 2.1) 2.2,,,,,, 2.1 F 2 *2, *1,,

14 8 2, F = GMm r 2 (2.1), G, G = m 3 s 2 kg 1 (2.2), (2.1), F 12 = G m 1m 2 (r 2 r 1 ) r 2 r 1 3 (2.3), 2, 1 2, F , m 1, m 2 G, (2.1) (2.1), *3 1 2, r 1, r 2 1 2, (2.1) 2 1, 1, (2.1) *4, M m,, m M,,,,,,, r 2 r 1 r 2 r 1 *6 2.2 (2.4),, m , M, r (2.1), m m, M r, 9 r = 6400 km, M = kg m = 1 kg,, 9.8 N *5 *3?,,,, *4?,?,!,,,, *5 SI km (6400 km) F, m, F = mg (2.5), g (2.1), g GM r 2 (2.6) *7, 9.8 m s 2 *6 r 2 r r 2 r 1 = r (r 2 r 1 )/ r 2 r 1 ), 1 2, 2 1 *7,,, g

15 2.4 9 g? [F ]= MLT 2, [m]=m, [g]=[f ]/[m]=lt 2, g,,, m = 1 kg F = 9.8 N, g = F/m = 9.8 N/kg (2.7), N kg m s 2, N/kg m s 2,, g 9.8 m s 2 (2.8) g, g, g, 10? 11 36,000 km,? (2.1) r r? 12 1/81.3, 1/3.68,,,? (2.1) M r 13, 1000 hpa 1 m 2,? (2.5) 1 m 2 m, mg,, 2.4,,,, (charge) *8, 2 electric force magnetic force,,,, 2 *9,,,, * 10, 2,,,,, : 2,, F, F F = k q 1 q 2 r 2 (2.9), q 1, q 2, r k, 4, k = N m 2 C 2 C, SI, 1 C=1 A s (2.30), (Coulomb s law), (Coulomb force), k, (2.30),,, (2.30),, *8,?,,,, *9, 2, 1, *10

16 10 2,,,,, q 1 q 2, q 1 q 2,, (2.30) F,, 2,, q 1 q 2, F 0, (2.30) (2.30) F, 2, * 11, (2.30),,, (2.30),, : (3) F g F e? 2.5, 4, 4,, 1,, F 12 = k q 1q 2 (r 2 r 1 ) r 2 r 1 3 (2.10), F , r 1, r 2 1 2, (2.3)),,,,, q e q e, 4, q e = C 14? 15? 16 1 m 2, F e, F g (1) F e (2) F g, m e, kg *11, X A, B A B X 1,,, A B, B A A A, B B B A, B A, A, A

17 2.5 11,, B X A,,,, :,,,,, T, : ( ), (),,,, 2,, X X,,,,,, , m,,, mg/2, ,, A, B, m,, B A,, mg/2 T, B 2T , F, F,, F , 19, 2.7,,

18 12 2,, F (x) = F (0)x (2.14) F (0),,,, x F, x F, (2.14) F (0), F (0) k k (2.14), 2.7 2, 1/4 F = kx (2.15), F,, x, k (Hooke) 20,? 2.6,,,, x F F (x),,, F (x + dx) = F (x) + F (x)dx (2.11) (dx ), x = 0,, x = 0, F (0) = ), F (0 + dx) = F (0) + F (0)dx (2.12) F (dx) = F (0)dx (2.13) dx, x, dx x 2.8 Hooke,,, (2.15),,, (x F, x,, F, x, F 21 (1)? (2)? (3)?

19 2.6 13,,,,,,,,,, (x,, * 12 (2.15),,,,,,,, 22?? 23,, 24,?,? , k 2,,, M Mg/(2k) 2 1, 2k Mg, 2, , k 2,, M 2Mg/k 2 1, k/2, Mg,, k, m,? x, x 0 x, kx, mg g,,, 2, kx + mg = 0 (2.16), kx = mg,, x = mg k (2.17) *12,, k a b, bk/a n m A, L X,, X,, X,,, X k, A/L

20 14 2, k = E A L (2.18) 2.7, E 28 (1)? (2) * 13 (3) 10 m, 2 mm 10 kg,? (4) (3) (1 mm)? (5) (3)?,, F = E A L x (2.19) F A = E x L (2.20) F/A, x/l, σ, ɛ,, σ = Eɛ (2.21), 1,,,,, * 14, 2,,,,,,,,,,,,,,,,,,,,, N,,,,,,,,, 29, (1)? (2)? * 15, 2.10, θ, m *15,,, *13 ( ),,, *14,,,,

21 2.8 15, mg,,!,,, 2.12, mg sin θ 2.10, , , 3 kg x kg,,,, x T T T, 2.11 N , , 2.8, 2.10, mg, mg sin θ, mg cos θ,,?,,,,,, (mg cos θ), N ( 2.11),,,,,, 2, N, 2, F s, F s µn (2.22), 2, F m, F m = µ N (2.23)

22 16 2 F s, F m, µ, µ,,,, * 16 (2.22),, 2, 2,, µ, φ µ = tan φ (2.24) , µ k m),,,,,,,,, µn,,, µ < µ,,,,, ,,,,,,,,,, (1)? (2)? (3)? 8 9 (2.1), r km, m, r = m 10 m F F = mg, g, * *17, GM/r 2,, 32 φ, m,,, φ *16,?,,,, g,, g g,,???,?

23 r 1, r 2 r 1 = 6400 km, r 2 =(6400 km)+(36000 km) km, m F 1, km F 2, (2.1),, F 1 = G Mm r1 2 F 2 = G Mm r2 2 F 2 F 1 = r2 1 r 2 2 =, 2 12 ( r1 ) 2 ( 6400 km ) r km (2.25) (2.26) r 1, r 2, M 1, M 2, m F 1, F 2, (2.1),, F 1 = G M 1m r 2 1 F 2 = G M 2m r 2 2 F 2 F 1 = M 2 M 1 (2.27) ( r1 r 2 ) 2 (2.28), M 2 /M 1 = 1/81.3, r 2 /r 1 = 1/3.68, F 2 = 1 F = , 1/6 13 (2.29) 1 m 2 m, F F = mg = Pa 1 m 2 =10 5 N, m = F/g = 10 5 / kg 14 q 1 q 2 r,, F F = k q 1q 2 r 2 (2.30) k, 4, k = N m 2 C 2, 15 q e = C 16 (1) (2.30), F e = ( ) = N (2.31) (2) (2.1), (3) 17 F g = ( ) = N (2.32) F g F e = (2.33) :,, F g F e,, 1,, A, F, F,,, F, F,, F,, 18 2, T,, 2T,, mg

24 18 2,,, 2T mg = 0, T = mg/2, T,, mg/2 19 B, T B, B m mg, 2T mg = 0 T = mg/2,, T, mg/2 20,,,,, 21 (1) x, F, (2) F = kx k (3) [k] = [ F/x] = [F ]/[x] = MLT 2 /L = MT 2 22,, 23,,, 24 m, x 0 kx 0 + mg = 0 (2.34), m, x 1, 12, 1/6, kx 1 + mg/6 = 0 (2.35) 2 mg, kx 1 = kx 0 /6 (2.36), x 1 = x 0 /6,, 1/6 25 A, B A, F, x,, F = kx (2.37),, B, M, 2, 2F, Mg, 2F + Mg = 0 (2.38) F,, 2kx + Mg = 0 (2.39) x = Mg 2k (2.40) 1, x = Mg/k (2.17),, k 2 26 C, D C, F 1, x 1,, F 1 = kx 1 (2.41) D, F 2, x 2,, F 2 = kx 2 (2.42), m, D, F 2, Mg, F 2 + Mg = 0 (2.43), D, C F 1, D D, F 1 + Mg = 0 (2.44)

25 2.9 19,, Mg = F 1 = F 2 = kx 1 = kx 2 (2.45) x 1 = x 2 = Mg k 2 x, x = x 1 + x 2 = 2Mg k (2.46) (2.47) 1, x = Mg/k (2.17),, k 1/ , k a,, k/a 25, k/a b,, bk/a 28 (1) E = kl/a, [k] = MT 2, [L/A] = L 1, [E] = ML 1 T 2 (2) Pa (3) k L = 10 m, A = = m 2, k = EA/L = kg s 2, m x, x = mg k 10 kg 9.8 m s 2 = kg s 2 = m, 1.6 mm ( 2 mm) (4),, 1/4, 4, 6.4 mm (5) Pa , /0.65, 2.5 mm (1) [σ] = [F/A] = MLT 2 /L 2 = ML 1 T 2 (2) [ɛ] = [x/l] = L/L = kg,, (3 kg) g sin(π/6) x,, x g sin(π/3) T,, T = (3 kg)g sin(π/6) = xg sin(π/3), 31 x = (3 kg) sin(π/6) sin(π/3) = 3 kg 1.7 kg (2.48) (1), F s F m, F s µn, F m = µ N, N µ, µ,, (2), µ µ (3) [µ] = [µ ] = [F m /N] = 1 32 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, 33 µ = mg sin φ mg cos φ = sin φ = tan φ (2.49) cos φ ( ) x x = 0, x,, : mg sin θ, : kx 2, F s, mg sin θ kx + F s = 0 (2.50), mg cos θ, F s

26 20 2 µmg cos θ,, F s = mg sin θ + kx µmg cos θ (2.51), µmg cos θ mg sin θ + kx µmg cos θ (2.52), 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 2.10, N... kg m s 2,, 0 F 12 = Gm 1 m 2 (r 2 r 1 )/ r 2 r 1 3 F 12? g GM/r 2? mg g,, F = GMm/r 2 F = kq 1 q 2 /r 2? 2?,,,,,??,,,,,,, q 1 q 2 < 0 F, q 1 q 2 > 0 F? q 1 q 2 < 0 q 1 q 2 F,,,?

27 ,,,? F = kx, F = kx? F = kx,, F = kx,, µ < µ,,,,,,, µ < µ ABS (,,,,,,,,,,?,,,,,,,,,,,,,,,,,,,,,,,,,,,,?,,?,,,

28 22 2,,,,,,,,,,,,,,,?,,,,,, G etc?...,,,, 1,,?,?? 2? 2,,

29 23 3 :, 6.1, 6.2, 6.3, 6.7, ,,, 2,,,,,,,,,,,,,,, 3.1 ( 19),,, T, mg,,, ( 3.1), B x,,, x,, 2 x *1 *1 B x, 1 2 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), (), 0,,

30 24 3 x,,,, (work),,,,,,,,,,,,,,, 2, x = 3 kg () 34 (1)? (2)? (3) SI J (4) , S, l 1, l 2, m 1, m 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) m 1 m 1 gl 1 sin θ, m 2 m 2 gl 2 sin θ (3), m 1 l 1 = m 2 l 2 (3.7) 3.2 2, , A x ( 3.2), B, x, A, 3g{sin(π/6)} x, B, xg{sin(π/3)}( x) x, B, B, (3g sin π 6 ) x (xg sin π ) x = 0 (3.4) 3 3.3,,,,,,,,,,, 0

31 3.2 25,,,,,,,,, , 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 = m, F? 3.2,,,,,, 3.1 T x,, m mg x 2 (3.12),, 0,,,,,,, A B 1000,, 2,, 2, 2 (2 0 ), A 1000, B C, A, B, C,...,,,,,,,,,, SI J ,?,,?

32 26 3,,,,,,,,,,,,,,,,,, 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),, 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, P. 88 (6.11), : a b, W = b a F (x) dx (3.20) x 0 a, x n b, F (x) x (3.20), (3.13),, 3.3 m, mg, h 0 h 1,, (2.5), F = mg (3.21), x 2 = x 2 x 1, x 3, x 4,, x n (n ), F 3, F 4,, F n

33 *2, (3.20), P 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,,, () 3.4 m, M, M R 0 R 1 M m,, (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 () 38 (3.24) k, x 0 x 1, W, W = 1 2 k (x2 1 x 2 0) (3.25) : (2.15) F = kx, (3.20) 39, A, x, x = 0 x, x = h ( 3.5), 3.5 (1) V, V = Ah (2) F 1, F 1 = P A (3.26) (3) *4 F 2, F 2 = P A (3.27) (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) *2 (2.5), *3 (2.1), *4,,,

34 28 3 (6) (3.28), (3.29) : dw = P dv (3.30) (7), V 1 V 2 W : W = V2 V 1 P dv (3.31) (8),, :, P, P = W t (3.36),, t, P = dw dt (3.37),,, P V = nrt (3.32) n, R, T : W = V2 V 1 nrt V dv (3.33) (9),, T, : W = nrt ln V 1 V 2 (3.34) (10) 1 0 ( ) (3.30),,, ( ), dw, dw = dw, dw = P dv (3.35), (3.30) *5 3.3,, t, W 3.5 m, t h W mg h P, P = W t = mg h t t 0, P = mg dh dt (3.38) (3.39) dh/dt v, P = mgv (3.40) (),, ML 2 T 3, SI, kg m 2 s 3,, J s 1, W kg,, 3.0 m/s (3.36),,, *6,,, W, h, W h, 1 W 1, 1 W h *5,, dw, ( dw ) dw, *6,

35 W h, J 3.4, (3.20), W, a b, a, W b, b x, W x W (x), x W (x) *7, (1) U(x) = W (x) (3.41) U(x), W (x), x, ,, h 0 = 0, h 1 h, W (h) = mgh,, (3.41), U(h) = mgh (3.42),,,,,,,,,, 42, (3.41) W (x), 3.6,,,,,, W (x) *8,, W (x) = W (x), (2) U(x) = W (x) (3.43) U(x), W (x), x, 3.6, h, mg, h, W (h) = mgh, (3.43), U(h) = mgh, (3.42) (3.41) (3.43),, 2,,,,, 3, (3) U(x) = W (x) (3.44) U(x), W (x), x, 3.6, h (), mg, h, W (h) = mgh, (3.44), U(h) = mgh *7, *8 W (x) W (x)

36 30 3, (3.42), (3.41), (3.43), (3.44),, (, W (x) = W (x) = W (x) ),,,, 43 k, x, U(x), U(x) U(x) = 1 2 kx2 (3.45), ( ),,,,, SI J C 1 V, 47 (1)? (2) SI (3) 2 V 0.3 C (4) 1 V 1 44 m, M R, U(R) (R = ), U(R), U(R) 45 U(R) = GMm R, (3.46), (1) 10 m, 2 kg, (2) 10 m, 2 mm 1 mm, (3), 46 q, Q R, U(R) : U(R) = k Q q R (3.47), k (2.30), 1 V 1 () J, 1 ev ev, 48 θ, m, L,, (conservative force),,,,,, U(x), x W (x), U(x) = W (x), W (x), W (x) x, U(x), 49

37 (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), 51 θ, µ, m, L, (1),, (2) ML 2 T 2 (3) 1 J = 1 Nm=1 kg m 2 s 2 (4), 0 35 (1), p (1) (2) m 1, m 1 g m 1, *9 h 1, l 1 sin θ, m 1 m 1 gh 1 = m 1 gl 1 sin θ (3.48) m 2, m 2 g m 2 h 2, l 2 sin θ, m 2 m 2 gh 2 = m 2 gl 2 sin θ (3.49) *9 (displacement) (3), 2 0, m 1 gl 1 sin θ m 2 gl 2 sin θ = 0 (3.50), m 1 l 1 = m 2 l 2 (3.51) 36 (1) 2πr,, F, 1, 2πrF (2) 1, y,, mg,, mg y (3) 1,, 0 (4) F = (5), 23 N 2 kg,, 2, 1000 kg 37, 38 (3.20) F (x) = kx, W = 39 x1 x 0 ( kx) dx = 1 2 k(x2 1 x 2 0) (3.52) (1), A, h V = Ah (2) P,,, P A,,, F 1 = P A (3),

38 32 3 F 1 F 2 = F 1 = P A (4), dw, F 2 dh, dw = F 2 dh = P A dh (3.53) (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 (3.54) : 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 , W = 1570 J 3 40 (3.39), m = 2.0 kg, g = 9.8 m s 2, v = 3.0 m s 1, 2 P = 59 W 41 1 W h = 1 J s s=3600 J 42 x W (x), U(x) = W (x) U(x) 43 (3.25), x 0 = 0, x 1 = x, U(x) = W (x) = 1 2 kx2 (3.55) (3.24), R 0 =, R 1 = R, U(R) = W (R) = GMm R (3.56) (1) (3.42), m = 2 kg, g = 9.8 m s 2, U(R) 0 U(x) 0 x R h = 10 m, U = 196 J 200 J (2) 28(3), k, k = kg s 2, x = m, (3.45), U = J (3) M, m, R, G = M = m = R = kg kg m m 3 kg 1 s 2 (3.46), U = J 46 Q, x q x q (0 < x ), F, (2.30), F = k Q q x 2 (3.57)

39 R q,, W = R F dx = R (3.41), U(R) = W = k Q q R ( ) 47 k Q q [ x 2 dx = k Q q x ] R (1) (2) J C 1,, V (3) 0.6 J = k Q q R (3.58) (4) C, 1 V, J 48,, mg sin θ (3.59) (θ L, mgl sin θ (3.60),, W 10 = F m X, W 01 + W 10 = 2F m X, 51 48,,,, 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 ,, F = mg, W = mg(h 1 h 0 ), h,, h 0 < h 1, W, h, 49,,, 50 (1) x 0 x 1 x 0,, x 0,,, 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 /, 4,,,? /?

40 34 3,,,? / /,,... 1,,,, 10,,,,,

41 35 4 :, 3.1, 3.3, 3.5, 8.1, 8.2, 8.3, 8.5, 8.7, 8.8, 12.1 : v x (t) = d x(t) (4.6) dt v y (t) = d y(t) (4.7) dt 4.1,,, ( ),,, 3 (x, y, z 3 ),, ( ), ( ) : t, r(t) = (x(t), y(t), z(t)) (4.1), v(t) = (v x (t), v y (t), v z (t)) (4.2) v z (t) = d z(t) (4.8) dt, (4.5), : a x (t) = d dt v x(t) = d2 x(t) (4.9) dt2 a y (t) = d dt v y(t) = d2 y(t) (4.10) dt2 a z (t) = d dt v z(t) = d2 z(t) (4.11) dt2, (t) 4.2 ( ) *1 ( ) 52, t, v = (v x, v y, v z ), r(t) = (x(t), y(t), z(t)) : x(t) = x(0) + v x t (4.12) y(t) = y(0) + v y t (4.13) z(t) = z(0) + v z t (4.14), a(t) = (a x (t), a y (t), a z (t)) (4.3),, v(t) = d r(t) (4.4) dt a(t) = d dt d2 v(t) = r(t) (4.5) dt2 (4.4), (4.12), (4.13), (4.14), r(t) = r(0) + vt (4.15), r, r(0), v *1,,,,

42 36 4,,, 0 = (0, 0, 0),, 4.3,, ( ),,,??,,, 0, m F, F = ma (4.16), a 2 A, B, A B, A B, 2,,,,,,,,, 1, F a, F = (F x, F y, F z ) (4.17) a = (a x, a y, a z ) (4.18), (4.16), F x = ma x (4.19) F y = ma y (4.20) F z = ma z (4.21) 3 (4.16), : ma = F (4.22) (4.16) *2 (4.16), (4.5),, ( : ),,,, F = m dv (4.23) dt, F = m d2 r, dt 2 (4.24),, MLT 2, (4.16), (M) (LT 2 ), MLT 2, (4.23)(4.24),, *2,, (4.16) (4.22),,,,,, (4.16) (4.22),

43 4.4 37,,,,, F,, v(t) r(t), t, t,,, *3, (4.16) F = 0, 0 = ma m 0,, a = 0 a, 0, ( ),,,,,, 1,?,,,, *4,,,,,, m, v, p = mv (4.25), m, (4.16), F = dp dt (4.26) dp dt = F (4.27), *5, 54,,,, ( 3 ) , m g, t x t,,, x(t), v(t), a(t) x(0) = x 0, v(0) = v 0 (4.28) 53 (1),,, : mg = ma (4.29) (2) (4.29),, g g *3, *4, p *5, m, F = ma F = dp/dt, F = ma F = dp/dt,

44 38 4 (3) (4.29) mg = m dv dt (4.30) (4) (4.30) t = 0 t, v(t) = v 0 gt (4.31) (5) (4.31) v(t) = dx/dt, t = 0 t, x(t) = x 0 + v 0 t g 2 t2 (4.32), m,, (6), 0, 0, v(t) = gt (4.33) x(t) = 1 2 gt2 (4.34),, 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 56, 0,, (1) 10,, (2), 1, 340 m s 1, m, v 0 x,, v ( αv) α (x = 0) t, t = 0 ( 4.1) 57 v 0 = 40 m s 1 (4.35) 140 km h 1, t, x(t)? x 0 = 0 m, v 0 = 40 m s 1, (4.32), x(t) t 58, 4.1 (1) t < 0, 0 = m dv dt (2) 0 < t, αv = m dv dt (4.36) (4.37) (3), v(t)

45 4.5 39, v(t) = v 0 exp ( α ) m t (4.38) (4) v(t), *6,,, *7, 2 * ?? 59,, βv 2 β (1), β v 2 = m dv dt (4.39) (2), β dt = mdv v 2 (4.40) (3), (C ) β t = m v + C (4.41) (4) C = m v 0 (4.42) (5) v(t) = mv 0 v 0 β t + m (6) v(t) (4.43),,,,, m, t v(t) v(0) = 0 αv α v, αv v, (1) mg αv = m dv dt (2) (4.44) dv = dt (4.45) g + αv/m (3) C m v α ln gm + = t + C (4.46) α (4) v, v(0) = 0, v(t) = mg α (5) v(t) { exp ( α ) } m t 1 (4.47) (6), v = mg α (4.48), (, ),,? 62 *6 *7, cm cm s 1 *8,, 63 ( 4.2),,,,, m

46 40 4, v(t) ( β ) v = v tanh v m t (4.54), tanh x,, tanh x = ex e x e x + e x (4.55) tanh , t v(t) βv 2 β,, (1) mg + βv 2 = m dv dt (4.49) (2),,, dv/dt = 0, v, mg v = β (4.50),, (3) v, m dv dt = β(v2 v 2 ) (4.51) (4) p. 94, ( 1 1 ) dv = β dt (4.52) v v v + v 2v m (5) C 1 ln v v β = 2v v + v m t + C (4.53) (6) v(0) 0, (4.6), v x = d dt x(t), v x t + C = x(t) (4.56) (C ) t = 0, C = x(0) (4.57), v x t + x(0) = x(t) (4.58), x(t) = x(0) + v x t (4.59) y z, y(t) = y(0) + v y t z(t) = z(0) + v z t (p), (m) (v) p = mv 55 (1) (2.5), mg, F F = mg (4.60)

47 *9, mg = ma (4.61) (2), a = g, a, g (3) a = dv/dt, (4.30) (4) (5) dv dt, t 0 = g (4.62) dv dt dt = t 0 ( g)dt (4.63) (6.105), v(t) v(0), gt,,, t v(t) v(0) = gt (4.64) v(t) = v 0 gt (4.65) v(t) = dx dt = v 0 gt (4.66) 0 dx dt dt = t 0 (v 0 gt)dt (4.67) x(t) x(0) v 0 t gt 2 /2,, x(t) x(0) = v 0 t 1 2 gt2 (4.68) x(t) = x(0) + v 0 t 1 2 gt2 (4.69) (6) x 0 = 0, v 0 = 0 56 (1) (4.33) t = 10 s, v = 98 m s 1 (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 *9 (2.5) (4.34) t = 35 s, x = 6000 m, 6000 m (4.32), 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 g 2g (4.70) (4.71), t = v 0 /g, v 2 0/2g v 0 = 40 m s 1,, (40 m s 1 ) 2 = 82 m (4.72) m s 2, 80 m 58 (1), F m,, F m x, F, F = F m,, F m = m d2 x dt 2 (4.73) (2), d 2 x/dt 2 = F m /m,, dx dt = F m m t + C 1 (4.74) C 1, t = 0 dx/dt v 0, C 1 = v 0, dx dt = F m m t + v 0 (4.75),, x = F m 2m t2 + v 0 t + C 2 (4.76) C 2 t = 0 x 0, C 2 = 0, x = F m 2m t2 + v 0 t (4.77) 4.3 (3) t, x, dx/dt 0, (4.75), 0 = F m m t + v 0 (4.78)

48 42 4 x(t),, C, αt = m ln v + C (4.82) t C C, ( v = ± exp α m t + C ) (4.83) t = 0 v = v 0, ± exp C = v 0 (4.84), v = v 0 exp ( α ) m t (4.85) (4) 4.4, t = mv 0 /F m (4.77),, x = mv2 0 2F m (4.79) F m = mv2 0 2x (4.80), x = 20 m, m = 20 kg, v 0 = 1.5 m s 1, F m = 1.1 N (4) ( (2.23)), µ, F m = µ N N, N = mg, F m = µ mg,, µ = F m /mg = (5) (4.79), x, 59 v 0 2 x 20 m 25 m, x 1.25, v = 1.12, v m s 1 =1.68 m s 1, 1.7 m s 1 (1) t < 0,, 0,, (4.36) (2) 0 < t F = αv,, (4.37) (3) (4.37), αdt = m dv v (4.81) v(t) v(0) t ,,, 61 (1) 0 < t F = βv 2,, (4.39) (2) (3) (4) (4.41), t = 0, 0 = m/v(0) + C C = m/v(0) (5) (4.42) (4.41) v, (4.43), (6) (1), mg αv,, (4.44)

49 v(t) v(0) ( ± ) (4.92), v = gm ( α exp α ) m t gm α (4.95) 0 t (4.47) (5) (2) (4.44) m, g αv m = dv dt (4.86) dt, g+αv/m, dt = dv g + αv/m (4.87) v(t) 0 -mg/a v=-gt 4.6 t, (4.45) (3) (4.45) : dv g + αv/m = dt (4.88), dv (α/m)(v + gm/α) = m dv α v + gm/α = m v α ln gm + +C 1 α (C 1 ), t + C 2 ((C 2 ), m v α ln gm + + C 1 = t + C 2 (4.89) α C 1, C 2 C 1 = C, (4.46) (4) (4.46), ln v + gm = α (t + C) (4.90) α m, v + gm α = ± exp ( α m (t + C) ), v = ± exp ( α ) m (t + C) gm α t = 0, v = 0, 0 = ± exp ( α ) m C gm α, ± exp ( α ) m C = gm α (4.91) (4.92) (4.93) (4.94) (6) (4.47) t, v mg/α 63,, (4.44) 0, v = mg/α (1), mgβv 2, F = mg + βv 2 (4.96), 1 F = mdv/dt, (2) v v, dv/dt = 0, (4.49), mg + βv 2 = 0,, v = mg/β = v (3) v (4.49) g (4) (5) (6) (4.53), v v ( ( β = ± exp 2v v + v m t + C)) (4.97), t = 0, v = 0 =,,, C = 0, v v ( β ) = exp 2v v + v m t (4.98) v,

50 ( exp v = v ( exp 2v β 2v β m t ) 1 m t ) + 1 ( β ) exp v m t, ( exp v = v ( exp v β v β, ( β ) v tanh v m t ) ( m t exp ) ( m t + exp (mg/b) 1/2 4.7 v(t), 4.7 v=-gt v β v β m t ) m t ) (4.99) (4.100) (4.101) 59, 0,,,,, 0 t,,,,,,,,, /!! / / /, /,,,, 10 t,, 500 kg, 1

51 45 5 :, 5.7, 5.8, 5.9, 5.10,, x, y, z, 5.1,,, 5.1 *2, :,, 0 5.1, x x, x,, kx 5.1 k, x kx = m d2 x dt 2 (5.1) 5.1,, m k,,, x x,,,, *1,, (4.16), m, *1,,,,,, d 2 x dt 2 = k m x (5.2) (5.2), x(t), t, *3,,,,,,, *2,, : (), (), (), 0,,, 0 0,, 0 = 0 *3

52 46 5, x(t) = A cos ωt (5.3) A ω *4 (5.3), 5.2 A 0 2π/ω 4π/ω t -A x 5.2 (5.3)!, A ( ),, 2π/ω ω *5, T, : ω T,,, T = 2π ω (5.4),, ω = 2π (5.5),, (5.12) B = 0 (5.3) T, (5.12) A = 0 B A,, s 1,, Hz (), (5.3) (5.2) (5.9), (5.3) (5.9), (5.12), (5.10), φ = φ π/2, cos *7,, (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 *6 A = 0, (5.3) 0, ω 2 = k/m, k ω = 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), x(t) = A cos(ωt + φ π 2 ) = A sin(ωt + φ ), φ φ (5.11), (5.11), φ = φ + π/2, sin *8, x(t) = A sin(ωt + φ + π 2 ) = A cos(ωt + φ ), φ φ (5.10), (5.10) (5.10), *4 ω P. 149 *5, *6, cos ωt = 0 (5.2), t, t = π/(2ω) t = 3π/(2ω), (5.2) *7 θ, cos(θ π/2) = sin θ *8 θ, sin(θ + π/2) = cos θ

53 5.2 47, (5.12) ( 5.9 ), (5.11), (5.11) (5.10) (5.12), (5.10), (5.11), (5.12) 3,, A φ,,,, (5.12),, t = 0 x(0), (5.12), 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), (5.12), x(t) = x(0) cos ωt + x (0) ω sin ωt (5.16), 5.1,, x(0) x (0) ( ω (5.8), (5.2), (5.8), : d 2 x dt 2 = ω2 x (5.17),,,,,,, , 2, , t = 0 x = X 0, 0,, : (5.16) , t = 0 x = 0, V 0,,, (5.17),, t,,,, 2, 1, 1, 2, 2,,, 2 (5.17), (5.12) A, B, (5.16),,,, F = ma, x, 2, *9 5.2,,,,, x, y, z *9,,,

54 48 5, X = v2 0 sin 2θ g (5.23) 68,, m ( 5.3), x, y t, t = 0 t = 0 x θ, v 0 (7) v 0, θ, X 69, 7.26 kg, 86.7 m,, t r(t), v(t) g r(t) = (x(t), y(t)), xy m,,, F ( 5.4),, (1) r(0) = (0, 0), v(0) = (v 0 cos θ, v 0 sin θ) (2),, 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) 5.4 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 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)

55 * 10, t = 0 x (r, 0), r(0) = (x(0), y(0)) = (r, 0) (5.27) 70 (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 = ( F cos ωt, F sin ωt) (5.32) (3), F cos ωt = ma x (t) (5.33) F sin ωt = ma y (t) (5.34) (4) (5.30) a x (t), a y (t),, F = mrω 2 (5.35) (5) v v v = rω (5.36) (6) (5.35), F = mv2 r (5.37) (7), v 2, (8), r 1/2, 71,,,,,, ( x(t), y(t) ) = (r cos ωt, r sin ωt) (5.38), x x(t) = r cos ωt (5.39), (5.3), d 2 x dt 2 = ω2 x (5.40), y y(t) = r sin ωt (5.41), (5.9), d 2 y dt 2 = ω2 y (5.42),, 2 (x y ),,,,,, 72 ( 69 ), kg 1.5 m 2 (5.37) m v 69 73, r ( ),, M, m G (1), r, ω, m (2), ω GM ω = r 3 (5.43) (3) T, T = 2π r 3 GM (5.44) (4) r, G, M ω,,, *10,, r

56 , 6400 km, (1) 400 km (2) km,,, ω = k/m, m 2, ω 1/ , T, T = 2π/ω, T (5.16), x(0) = X 0, x (0) = 0, x(t) = X 0 cos ωt 5.5 X 0 -X 0 x 0 2π/ω 4π/ω t (5.16), x(0) = 0, x (0) = V 0, x(t) = V 0 ω sin ωt (5.45) 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) x V 0 /ω -V 0 /ω 0 2π/ω 4π/ω t 5.6 (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, y(t) = (v 0 sin θ)t gt2 2 (4) (5) 5.7 v 0 2 sin 2 θ/(2g) 5.7 y (5.50) 0 2 v 0 sin2θ/(2g) x (6) (5.22) y = 0, x = 0 (5.23),, 2 cos θ sin θ = sin 2θ

57 (7) (5.23), θ = π/4, 45 69, θ = π/4, (5.23) X = v0/g 2 v 0 = gx X = 86.7 m, g = 9.8 m s 12, v 0 = 29.1 m s 1 70 (1) P (2) r = (r cos ωt, r sin ωt) = r(cos ωt, sin ωt) (5.51), (cos ωt, sin ωt),,,, ( cos ωt, sin ωt) F,,, F(t) = F ( cos ωt, sin ωt) = ( F cos ωt, F sin ωt) (5.52) (3) x, y (4) (5.30) a x (t), a y (t) (5.33), (5.34), F cos ωt = mrω 2 cos ωt (5.53) F sin ωt = mrω 2 sin ωt (5.54) cos ωt, sin ωt, F = mrω 2, (5.35), (5.35), (5) (5.29) (v x, v y ) (6) (5.36) (5.35) ω (7) (5.37), 4 (8) (5.37), 2 71,,,,,,,,,,,,,,,,,,,,,,, 72, v, (5.37), F = mv 2 /r 69, 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 73 (1) (5.35), mrω 2 (2) GMm/r 2, mrω 2 = GMm r 2 (5.55) ω (3) T = 2π/ω (4) G = N m 2 kg 2, M = kg, r = m, ω = Hz, 2π/ω = s= , 74 (5.44) M (1) r = 6800 km, T = 5580 s 1.6 (2) r = km, T = s 24

58

59 53 6 (1) 6.1, m F (x ),,, F = ma (6.1) a ( 1, F a ),, : F = m dv dt (6.2) t, v (6.1) (6.2),,, (4.16) 1, (6.2), (x ), F (t), m, t 0 t 1, (6.2), (6.2),,,,,,,,, (t = t 0 ) (t = t 1 )?, (6.2),, t, t + dt,, ( ), x(t + dt) = x(t) + x (t) dt = x(t) + v(t) dt (6.3) x(t), : x(t + dt) x(t) = v(t) dt (6.4), dt, dx, x(t + dt) x(t) = dx, dx = v dt (6.5), (6.2) dx F dx = m dv dx (6.6) dt dx, (6.5), F dx = m dv v dt (6.7) dt, t 0 t 1,, (6.7), t 0 t 1, (6.7) *1, x1 t1 F dx = m dv v dt (6.8) x 0 t 0 dt x 0 = x(t 0 ), x 1 = x(t 1 ) t v *2, x1 x 0 F dx = v1 v 0 mv dv (6.9) v 0 = v(t 0 ), v 1 = v(t 1 ), x 0 x 1 W 01 *1 P88 (6.11) *2 dt

60 54 6 (1) ( (3.20) ),, [ ] v1 1 = 2 mv2 = 1 v 0 2 mv mv2 0 (6.10), *4, (6.11),, W 01 = 1 2 mv mv2 0 (6.11),, T (v) 1 m v, T (v) = 1 2 mv2 (6.12) (kinetic energy), (6.11), W 01 = T (v 1 ) T (v 0 ) (6.13),, x(t), v(t) (1) t 0 t 1, F W m, F, a (x ) t : (2) : W 01 = F {x(t 1 ) x(t 0 )} (6.15) W 01 = ma{x(t 1 ) x(t 0 )} (6.16) (3) (6.11), *5 v(t 1 ) 2 v(t 0 ) 2 = 2a{x(t 1 ) x(t 0 )} (6.17) T (v 1 ) = T (v 0 ) + W 01 (6.14),, T,, T, T,,,,,,, (6.12) T (v), v 2, T (v), v ( ), v, *3, ( ) 75, (6.11), *3 () 77 ( 58) v 0 x, (1) F m x (2) mv2 0 (6.18) 0 (3), F m = mv2 0 2x (4.80) (6.19) (4), *4 (6.11), *5 v 2 v 2 0 = 2ax,,, 3, (6.17) 3,,

61 m,,, v 1 M, R, G, (1),, R),, W, W = GMm R (6.20) (2), mv (6.21) (3), v 1 : 2GM v 1 = R (6.22) (4),, v (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),, W 01 = U(x 1 ) + U(x 0 ) (6.26) (6.11), U(x 1 ) + U(x 0 ) = 1 2 mv 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,,,,,,, 79 (1) (2)? (3)?

62 56 6 (1) 80, 56 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) 81, 5.1 t = 0 v = 0, x = x 0, 80, 81,,,, : 82, 58, 77,? 83,,, 50 m, L 20 m,, 60 kg,,, 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),, 1 m,, 0.2 m (1) 1 m k 0? (2) 20 m, 20 m k? 0, T 0 U 0,, E 0 0, x, v, mgx

63 (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,, E, c,,,,,,,,,,,,,,, 6.3,,,,,,,,,,,, +,,, E = mc 2 (6.43),,, *6?,,, (6.12) T (!) 76 (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 ) *6,,,

64 58 6 (1) (3) (6.11) W 01 : 1 2 mv(t 1) mv(t 0) 2 = ma{x(t 1 ) x(t 0 )} 2 m, 77,,,,,,,, (1), F m F 76(1), F m x (2) mv 2 0/2, 0 (3) (6.11), (1) (2) (4) (3), 78 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 = N m 2 kg 2, 79 M = kg, r = m, (6.46) v 1 = m s km/h (1) (2),, (3), 80 (1) (6.12) (6.30), (6.32), (3.42), h x, U(x) = mgx (6.31), (6.33) (2) (6.32) (6.33) 81 (1) ( (6.35) t ) (2) (6.12) (6.36), (6.37), (3.45) (6.35), (6.38) (3) (6.37) (6.38), T + U = 1 2 mx2 0ω 2 sin 2 ωt kx2 0 cos 2 ωt, ω = k/m,, T + U = 1 2 kx2 0 sin 2 ωt kx2 0 cos 2 ωt, T + U = 1 2 kx2 0(sin 2 ωt + cos 2 ωt) = 1 2 kx2 0, 82, mv0/2 2 0,,,,,, mv0/2 2,,,, 83 m, g (1) 0

65 6.5 59, k 0 δ + mg = 0 (6.47) δ, 0.2 m, k 0 = mg δ = 60 kg 9.8 m s 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, 0 = mgx +, ( x 2 2 L + mg k k(x L)2 2 (6.48) ) x + L 2 = 0 (6.49), SI, x 2 48x = 0 (6.50), x =37.3 m, 10.7 m, L =20 m,, x = 37.3 m 50 m,, 6.5,,,

66

67 61 7 (2), 1,, 1 3, *1 1,, 3, v 2 v 2, v 2 = vx 2 + vy 2 + vz 2 (7.4) ( ), 3, (7.1), T (v) = 1 2 mv2 = 1 2 m(v2 x + vy 2 + vz) 2 (7.5) 1, (7.1), (7.1), 1, , m v (1 ), T (v), T (v) = 1 2 mv2 (7.1) ( (6.12)),, 3, (7.1) v 2 v, v 2, v = (v x, v y, v z ) (7.2), v 2 = v v = vx 2 + vy 2 + vz 2 (7.3) 7.2 3, 3,,, 3 F, r F r θ,, r cos θ (7.6), W, W = F r cos θ (7.7),, W = F r (7.8) *2 (7.8), (3.13) 3 (7.8), F, F, *2 (7.8),, *1, 2,

68 62 7 (2), F, (3.16) (3.20), r 0 r n,, r 1, r 2,..., k 1 n, r k 1 r k 2,,, Γ,,, 3, r k = r k r k 1 (7.9) F k W k, (7.8) 7.3 3, 1 (3.41),, (3.43), (3.44) x r, 3, (1 ) 7.1 W k F k r k (7.10), r 0 r n W : n n W W k F k r k (7.11) k=1 k=1 (3.18) 3, Σ, d, r U(r) = W (r) (7.14) U(r), W (r), r, (2 ) U(r) = W (r) (7.15) U(r), W (r), r, W = F dr (7.12) r 0 r n r (3.20) (3 ) 3, r 0 r,, U(r) = W (r) (7.16) Γ, U(r), 3 W (r), r, W = F dr (7.13) Γ,,, 3

69 , 85 F, r 0 r 1, F W 01, W 01 = U(r 0 ) U(r 1 ) (7.17) U 86, 3, 0, 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, (7.21), t 0 t 1, (7.21), t1 F dr = m dv vdt (7.22) dt Γ t 0 (7.13), W 01, v = (v x, v y, v z ) (7.23) 7.4 3, 3 1, (6.6) (6.8), 3, : F = m dv dt (7.18) F, v, m, t,,, r(t) t, dt t + dt, ( ), dr, dr = r(t + dt) r(t) (7.19) dt v, v = dr/dt, dr = vdt (7.20) (7.18) (7.20),, F dr = m dv dt vdt (7.21), (6.7) 3, dv ( dt = dvx dt, dv y dt, dv ) z dt (7.24) (7.22), : t1 t 0 t1 = = = t 0 t1 ( dvx m t 0 vx (t 1 ) v x (t 0 ) dt, dv y dt, dv z dt ( dv x m v x dt + v dv y y dt + v z dv x mv x dt dt + mv x dv x + ) (v x, v y, v z )dt (7.25) t1 dv y mv y dt dt + t 0 vy (t 1 ) v y (t 0 ) ] vy (t 1 ) dv ) z dt (7.26) dt mv y dv y + t1 [ 1 vx (t 1 ) [ 1 [ 1 = 2 x] mv2 + v x (t 0 ) 2 mv2 y + v y (t 0 ) 2 mv2 z = 1 2 m( v 2 x(t 1 ) + v 2 y(t 1 ) + v 2 z(t 1 ) ) dv z mv 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 ) ) (7.27) = 1 2 mv mv2 0 (7.28), (7.22) : W 01 = 1 2 mv mv2 0 (7.29) (7.5), (7.29) W 01 = T (v 1 ) T (v 0 ) (7.30), T (v) (7.30), 1 (6.13), 3,,

70 64 7 (2),, (7.30) (7.17), U(r 0 ) U(r 1 ) = T (v 1 ) T (v 0 ) (7.31), T (v 0 ) + U(r 0 ) = T (v 1 ) + U(r 1 ) (7.32), ( t 0 ) ( t 1 ), 1,, ( ), (7.32), 3 88 P l, m ( 7.3) ( O Q Q, P, O, Q t X(t), OPX θ, θ, , ( m) ( 7.2) A 0, ( B), ( C) C, v h (1) g : 1 2 mv2 = mgh (7.33) (2) h = 5 m, v (3), C? 7.5 (1) t v(t), v(t) = l dθ (7.34) dt t t + dt X, (),, θ(t + dt) θ(t),, t t + dt X dt (2) t T U, T = 1 2 ml2( dθ ) 2 dt (7.35) U = mgl(1 cos θ) (7.36), O U = 0 (3),, ( ), T + U t, T + U t, 0

71 7.6 65, ml 2 dθ d 2 θ dθ + mgl sin θ dt dt2 dt = 0 (7.37) (4), d 2 θ dt 2 = g sin θ (7.38) l (5) θ() 0,, d 2 θ dt 2 = g l θ (7.39) : , (6) ω = g/l (7.39) d 2 θ dt 2 = ω2 θ (7.40), θ(t), (7) θ(t) = θ 0 cos ωt θ 0 (8) τ, g l *3 (9) l = 1.0 m, (10), l? 7.6 (,, 3, ),,,,,,,,,, *4,, (x ), x 0 x 1, F W 01, (6.26), W 01 = U(x 1 ) + U(x 0 ) (7.41) U, x 0 x, x 1 x x + dx (dx 0 ), W 01 = U(x + dx) + U(x) (7.42),, W 01 89, θ 0, l g, m, θ 0, () : W 01 = F dx (7.43) 2, F dx = U(x + dx) + U(x) (7.44) dx, (1), g, l, τ, l τ g (2), U(x + dx) U(x) F = dx dx 0, (7.45) *3 T, T, τ( ) *4 (explicit),,,,

72 66 7 (2) (1 ) F = du dx (7.46),, 3 F r, r + dr dr = (dx, dy, dz) (7.47), W, (7.42), (3 ) ( U F = x, U y, U ) z (7.53) *6, grad, ( U grad U = x, U y, U ) (7.54) z *7, (7.53) : F = grad U (7.55) W = U(r + dr) + U(r) (7.48) U *5, U(r + dr) = U(x + dx, y + dy, z + dz) = U(x, y, z) + U U U dx + dy + x y z dz = U(r) + U x dx + U y (7.48), W = U(r) U x = U U dx x y ( U = x, U y, U z ( U = x, U y, U z U dx y U dy U dy + dz (7.49) z U dy dz + U(r) z z dz ) (dx, dy, dz) ) dr (7.50) Γ, F, F W, W = F dr Γ,, W : W = F dr (7.51), ( U F dr = x, U y, U z ) dr (7.52) dx, dy, dz 0,, 85 r 0 F *6 F = (F x, F y, F z) (7.52) dx 0 dy = dz = 0, F x dx = U U dx,, Fx = x x dy 0 dx = dz = 0, dz 0 dx = dy = 0, F y = U y, F z = U z *5 P.183 (7.53) *7 grad, gradient,, 3

73 W (r 0 ),, U(r 0 ) = W (r 0 ) (7.56), r 1 F W (r 1 ), U(r 1 ) = W (r 1 ) (7.57), r 1, r 0 ( 7.4),, W (r 1 ) = W (r 0 ) + W 01 (7.58) W 01 r 0 r 1 W (r 0 ), W (r 1 ), (7.56), (7.57), U(r 1 ) = U(r 0 ) + W 01 (7.59), U(r 0 ) U(r 1 ) = W 01 (7.60) 86 Γ W r 0, r 0, (7.17), r 1 = r 0, W 01 = W, W = U(r 0 ) U(r 0 ) (7.61) 0,, W = 0 87 (1) C A, 0, mgh mgh C, mv 2 /2, 0 mv 2 /2,, ( ), A C, (2), v = 2gh = m s 2 5 m = 9.9 m s 1 (3) h, h (5.37), C 88, mv 2 /h, N mg,,, N mg, N mg = mv2 h, N = mv2 h (7.62) + mg (7.63) (1) mv 2 = 2mgh, N = 2mg + mg = 3mg (7.64), h, 3,, 3 (1) t t + dt X, l θ(t + dt) θ(t), l θ dt dt (2) T = mv 2 /2 v, T, O X l(1 cos θ), U (3) (4) (5) sin θ θ (6) (7) ( (7.40), ) (8) τ = 2π/ω *8, l τ = 2π g (7.65) (9) l = 1.0 m, g=9.8 m s 2 (7.65), τ = 2.0 s (10) (7.65), τ l, τ, l l 1/4 *8 (5.4)

74 68 7 (2) 89 (1) (7.65) g =, g = 4π2 l τ 2 (7.66) (2) 1/6 (7.65), τ 1/ g, g 1/6 τ 6 = 2.4,,,,,,, 7.8 U 0? U, 0, U 0, U 0, U 0,,,,,,,

75 , m A, m B 2 A, B, t 0, xy, t 1, t 2 t 0 A, B v A, v B, t 2, v, 2,, t 1,,, : 8.1 2,, *1,, (4.25),, t 0, m A v A + m B v B (8.1),,,,, 2,, 2,,,,,,, t 1,,,,,,, A, B, A, B t 2,, (m A + m B )v (8.2), (8.1) (8.2),, m A v A + m B v B = (m A + m B )v (8.3), v = m Av A + m B v B m A + m B (8.4),,! *1,,

76 , m A = m B = 1 kg, t 0 A x 1 m s 1, B y 1 m s 1 (1) v (2) v, m F,,, F = ma = m dv dt = md2 r dt 2 (8.5) t r, v, a,,,,,, m dv dt = F (8.6) dt, m dv = F dt (8.7), t = t 0 t = t 1 t, v(t1 ) v(t 0 ) m dv = t1 t 0 F dt (8.8), [ ] v(t1) mv = mv(t 1 ) mv(t 0 ) (8.9) v(t 0), (8.8), mv(t 1 ) mv(t 0 ) = t1 t 0 F dt (8.10) t 0 t 1,, dt,,, : t m, F(t), t1 t 0 F dt (8.11), t 0 t 1, (8.10), *2,, A, B 2,, 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 (8.12) t1, t 0 F BA dt (8.13) 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 (8.14) F AB A B, F BA B A, F AB = F BA (8.15), F AB + F BA = 0 (8.16), (8.14) 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 ) (8.17), 2 t 1 t 0,, 2 *2,,,,

77 8.2 71,, 3,, 91 3 A, B, C, (8.12), A, B, C,, A F AB + F AC,,,, 2,,,,, ,,,?, (1) (2),, 0,,,,,,, *3,,,,,, 2, 2, *3,,,, :, :, : 94 x, 2 A, B, v A, v B ( 8.2 ), ( 8.2 ), v A, v B *4, x ( 8.2 ) A, B m A, m B 2, 2 ( ) (1), 2 m A v A + m B v B = m A v A + m B v B (8.18) 1 2 m AvA m BvB 2 = 1 2 m Av A m Bv B 2 (2) (8.18), (8.19) : (8.19) m A (v A v A ) = m B (v B v B ) (8.20) m A (v A2 v 2 A ) = m B (v B2 v 2 B ) (8.21) *4,

78 72 8 (3) (8.21) : m A (v A v A )(v A + v A ) = m B (v B v B )(v B + v B ) (8.22) (4) (8.22) (8.20), : v A + v A = v B + v B (8.23) (5) (8.23) : v B v A = (v B v A ) (8.24) (4) m A >> m B, : v A v A (8.31) v B v B + 2v A (8.32) (8.29), (8.30), 2,, ( 8.3),,, (6) (8.24) : v B v A v B v A = 1 (8.25) (8.24),,,, v B v A, v B v A, e, e = v B v A v B v A (8.26), (8.25), e = 1, e 0 1 e = 0, B A ,,,,,, (8.31), (8.32),, ( ), ( ), ( 8.4), (1) (8.23), (8.20) v B : v A = m A m B v A + 2m B v B (8.27) m A + m B m A + m B (2) (8.23), (8.20) v A : v B = m B m A v B + 2m A v A (8.28) m A + m B m A + m B , (3) m A = m B, : v A = v B (8.29) v B = v A (8.30) 96 h 0 0, e

79 (1) v 0 v0 2 = 2gh 0 (8.33) (2) v 1 v 1 = e v 0 (8.34) (3), h 1, v1 2 = 2gh 1 (8.35) (4) h 1 = e 2 h 0 (8.36) (5),,,... n 1, n h n, h n = e 2n h 0 (8.37) (6) e = 0.8, h 0 = 10 m, 0.1 m, 97 2,, h 0, A, m A B, m B B A, m A >> m B A, A B g (1) ( 8.5 ) A v 0 v 0 = 2gh (2) A, B A ( 8.5 ) A v A, B v B : v A = v 0 (8.38) v B = v 0 (8.39) (3), B A ( 8.5 ) B v B,, (8.32), : v B 3v 0 (8.40) (4) B, A, ( h)? 98 (, 1 ) 2, n, h,,, ( 8.6) v n (1) : v n (2 n 1)v 0 (8.41)

80 74 8 (2), ( h)? (3) h = 5 m,, 10 (: 2 ) 8.3,,,,,,,,,,,,,!!?,,,,,,, 0 * (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 ) (8.4), 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 *5 91 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,, 92 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 (1), (2), (3) ( ) (4) (8.27), (8.28) m A, v A = 1 m B/m A v A + 2m B/m A v B 1 + m B /m A 1 + m B /m A 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 v A v B = v A v B v B v B = v B + 2v A, 96 (1), mgh 0, 0 0 mgh 0,, 0, mv 2 0/2 mv 2 0/2, mgh 0 = mv 2 0/2 (2) e

81 (3) (8.33) (4) (8.33), (8.34), (8.35) v 0, v 1 (5), h n = e 2 h n 1 e (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 = n = > 0.01 n = n = < 0.01, n = 11, (1) A,,, 1 2 m Av 2 0 = m A gh (8.42) v 0 (2), (3) ( ) (4) B, (1), 1 2 m Bv 2 0 = m B gh (8.43), H,,, 1 2 m Bv B 2 = mb gh (3), v B = 3v 0, 9 2 m Bv 2 0 = m B gh (8.44) (8.44) (8.43), 9 = H/h (8.45), H = 9h, 9

82

83 77 9 2,,,,, 3 (x, y, z 9.1, (x, ( ),,, y, ), 3 ( 9.1), (),,,, 9.1 A: x,,,, B: y, C: z, ( ),, *1 (9.1),,,,,,, ( ),,, K (9.1) K = 1 2 k BT (9.1), T k B, k B = J K 1 (9.2) (9.1), *2, (9.1),, (9.1), K *1 T T,, T K *2,,

84 78 9 *3 (9.1), 1/2,,, 1,,, (9.5) 9.2 (),,, L 9.2, O, x, y, z (L/2, 0, 0) x A m (v x, v y, v z ),, v x, mvx/2 2, k B T/2, * 4 v x = kb T m (9.3) v y, v z, ( )v, v = vx 2 + vy 2 + vz 2 (9.4), v, v = 3 k BT m, T = 300 K 27 (9.5) (1) H 2, x (2) H 2, (3) N 2,,,, v, m, *3 K,, ( ), (9.1), K, *4, v x (root-mean-square) 9.2, N, : 2,, A, A, A,, ( P ), ( 9.3) :,,, x (x ) v x, x, v x v x,

85 9.2 () 79, mv x (mv x ) = 2mv x (9.9) (9.8), (9.9), A, t 9.3 A (z ) A, A P v x,,,, t, v x t x, t, A, A v x t L 2 v x t (9.6),,,, L 2 v x t L 3 N (9.7) *5, x, x, A, 1 2 L 2 v x t L 3 N = Nv x t 2L (9.8), 1 A, A, ( ), A, v y v z, x v x,, x, mv x, mv x, Nv x t 2L ( 2mv x) = Nmv2 x t L (9.10), A ( =),, A L 2, A () P L 2 (,, x ), P L 2 t (9.11) 2, A : P L 2 t = Nmv2 x t L, (9.12) P L 3 = Nmv 2 x (9.13), L 3 V, (9.13) P V = Nmv 2 x (9.14), (9.3), v 2 x = k BT m, (9.14), : (9.15) P V = Nk B T (9.16),, N n, N = nn A N A, (9.16), P V = nn A k B T (9.17),, : *5 L 3, N, ( ), N/L 3, ( (9.6)), (9.7)

86 80 9 R, (gas constant) : R = N A k B (9.18), J mol 1 K 1, (9.17), : () P V = nrt (9.19)? ( ), *8 (9.18),, (9.20) n, N = nn A, U = F 2 nn Ak B T (9.21) U = F nrt (9.22) 2, 1, U = F RT (9.23) 2, (1 ) U T, 1 1 K,,, C v : 9.3,,,,,,,,,, *6, 1 1, k B T/2, F N U *7, U = F 2 Nk BT (9.20),, *6,,,, *7, U,, U C v = F 2 R (9.24) F, R, F?, (He, Ne ) 3 F = 3, 2 (H 2, O 2, N 2 ), *9 ( ) 2 * 10, 2, F, 3, 2, 5 ( 9.4),, ( ), C v = 3 2 R (9.25) 2, C v = 5 2 R (9.26) *8,,, *9, *10, 3,,,, 1,,

87 (2) (9.5), v, v x 3, v = m s 1 (3) (9.5), v, N 2 (28) H 2 (2) 14, v 1/ 14 = 0.27, v = m s A: x, B: y, C: z, D: x, E: z * 11*12, ( ),,,, (1) k B = J K 1 (2) R = 8.31 J mol 1 K 1 (3) R = N A k B N A 103 (1) k B, 3 (2) R, 3 (3), (1) H 2 2 H 2 1 m, /N A kg N A (9.3), v x = m s 1 *11 (9.26),, (9.26) (9.26) 2, *12, 2,,, ( ),, 2,,

88

89 83 10 :, 12.11, 4. a b b a,, a b = b a (10.5) 10.1, (!!), 3 *1 2 : a = (a 1, a 2, a 3 ) (10.1) b = (b 1, b 2, b 3 ) (10.2), (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ) (10.3), a b (!), a b = (a 1, a 2, a 3 ) (b 1, b 2, b 3 ) := (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ) (10.4), ( ): a, b, 1. a b, a, b *2 2. a b, a b 3. a b, a b *1,, *2 a b sin θ, θ a, b 5.,, a a = 0 (10.6) 4, 2, 3 5, 1 104, a b, a, b (1) a = (1, 2, 0), b = (1, 1, 1) (2) a = (1, 0, 1), b = ( 1, 1, 2) : a(t) = ( a 1 (t), a 2 (t), a 3 (t) ) b(t) = ( b 1 (t), b 2 (t), b 3 (t) ), t ( t ): (a b) = a b + a b (10.7) 10.2,, 3 3,,

90 84 10,, 3, (), *3 p, r,, L = r p (10.8) angular momentum ( 10.1) (2) t p p = mrω( sin ωt, cos ωt, 0) (10.10) (3) L, t ( 10.2 ) L = (0, 0, mr 2 ω) (10.11) r, v, p, L ( ) m ( ), 106 (1)? (2)? (3), 107 3, (x, y, z ; ) xy, r, m ω t = 0 x (1) t r r = r(cos ωt, sin ωt, 0) (10.9), *3,,,,,,, (10.8), 108 xyz, m, t r = (V t, y 0, 0) V y 0 (1) v, (2) p (3) L, t ( 10.3 ) L = (0, 0, mv y 0 ) (10.12),,,? 108, (10.12),, y 0 y 0, x

91 (10.17) 1 v v 0 5, (10.17), mr dv dt, (10.18) m dv dt = F (10.19) (F ), (10.18),,,,,,,,,,,,,,,,,,, * (10.8), m, d dt L = d (r p) (10.13) dt = d (mr v) (10.14) dt = m d (r v) (10.15) dt ( dr = m dt v + r dv ) (10.16) dt = mv v + mr dv (10.17) dt, (10.15) (10.16), (10.7) *4,,, ( (7.53)), r F (10.20), (10.13) (10.17),, d dt L = r F (10.21) ) r F,, (torque), (10.21),,, *5, n k (k = 1, 2,, n) k,,,, m k, r k, v k, L k k F k,, F k = F k1 + F k2 + + F kn + F e k (10.22) F k1, F k2,...,, 1 k, 2 k,... k, F kk,, F kk = 0, F e k, k, 10.4, 3 *5,, 3 ( F = ma),

92 86 10 : 10.4 (F 12 F 21 ) ( ), k, (10.21), (10.22), d dt L k = r k F k (10.23) = r k (F k1 + F k2 + + F kn + F e k) (10.24),, d dt L 1 = r 1 (F 11 + F F 1n + F e 1) d dt L 2 = r 2 (F 21 + F F 2n + F e 2) d dt L n = r n (F n1 + F n2 + + F nn + F e n), n k=1 d dt L k = + n r k (F k1 + F k2 + + F kn ) k=1 n r k F e k (10.25) k=1,, 1 n j, k j k, 1 r j F jk 1 r k F kj, r j F jk + r k F kj (10.26),,,,,, *6, F kj, r j r k F jk, 0 ( 5 ), (10.28) 0, F kk = 0, (10.25), 0, n k=1 d dt L k = n r k F e k (10.29) k=1 t, d dt n L k = k=1 n r k F e k (10.30) k=1,,, (1) ( ),,,,,, 0, 0 = n r k F e k (10.31) k=1,,, 0,, 3, (3.7),, *7, (10.30),, F kj = F jk (10.27), (10.26), (r j r k ) F jk (10.28) r j r k k j *6 *7,,,

93 , (10.30) 0, d dt n L k = 0 (10.32) k=1, t,, (2) ( ) 105 ( ),, 109 (1) (2),, (3) (4) (5) (6) 106 (1) (2) r L, p MLT 1, r p ML 2 T 1 :,, 2, (3) 2, r p r p m 2, 2r,, 2 ω, (1) 1 p p, mrω (2) 1 L L, mr 2 ω (3), (4) 2mr 2 ω (5) T T = mr 2 ω 2 (6), (7), (8), (1), xy, z 0, xy, r x ωt, (10.9) (2) (10.9) t m (3) 108 (1) L = r p = mr 2 ω(cos ωt, sin ωt, 0) ( sin ωt, cos ωt, 0) = (0, 0, mr 2 ω) t, t d dt r = d dt (V t, y 0, 0) = (V, 0, 0), x (2) p = mv = (mv, 0, 0) (3) L = r p = (V t, y 0, 0) (mv, 0, 0) = (0, 0, mv y 0 ) t, t

94 ML 2 T 2 110, 1 r = (r cos ωt, r sin ωt, 0), 2,, 2 (1) p = mr = m( rω sin ωt, rω cos ωt, 0), (2) p = mrω L = r p 3 = (r cos ωt, r sin ωt, 0) m( rω sin ωt, rω cos ωt, 0) = mr 2 ω(cos ωt, sin ωt, 0) ( sin ωt, cos ωt, 0) = mr 2 ω(0, 0, cos 2 ωt + sin 2 ωt) = mr 2 ω(0, 0, 1), L = mr 2 ω (3) z (4) (2), 2,,, 1 2, 2mr 2 ω (5) 1, 1 2 mv2 = 1 2 m(rω)2 = mr2 ω 2 2, 2, T = mr 2 ω 2 (6) Ω, (4), ( r ) 2Ω 2m 2,,, ( r ) 2Ω 2mr 2 ω = 2m 2, Ω = 4ω, 4 (7), (5) ( r ) 2Ω ( r ) 2(4ω) m 2 = m 2 = 4mr 2 ω 2 2 2, 4 (8),,,,, 10.5,!,

95 89 11, (),,, 1,,,,,, 2, ( 11.1) (),, (rigid body),,,,,,, 11.1,,,, 110, 2 ( ) ( 11.2):,,,,,, *1 *1,,,,,,,,,,,,, d, 2,,, ( )v, (5.36) v = rω (11.1),

96 90 11,, (6.12) 1 2 mv2 = 1 2 m(rω)2 = 1 2 mr2 ω 2 (11.2), ( 2 )T, mr 2 ω 2 (11.3), ( ) 2, 2, 3,, 2,,, ω, n, m 1, m 2,..., m n r 1, r 2,..., r n k r k ω, k T k : T k = 1 2 m kr 2 kω 2 (11.4) T,,, (11.5), T = 1 2 Iω2 (11.7) 111 (1) (2)? , 2mr 2, (6.12), v T, T = 1 2 mv2 (11.8), (11.7),,, (11.8) v m, (11.7) ω I,, *2,, *3 T = T 1 + T T n = 1 2 m 1r 2 1ω m 2r 2 2ω m nr 2 nω 2 = 1 2 (m 1r m 2 r m n r 2 n)ω 2 = 1 ( n ) m 2 k rk 2 ω 2 (11.5) k=1, : 113 r, m 3, 11.3, I? m 1, m 2,, m n n, I = n m k rk 2 (11.6) k=1 I, (moment of inertia, r 1, r 2,, r n *2 :,,,,,,... *3,,,

97 r, m n, 11.4, I? 11.4 n 115,, nm M M m n, M, r 117 ρ, b, r ( 11.7), I, I = π ρ b r4 2 (11.11), r r,, M, I = Mr2 2 (11.12),? I = Mr 2 (11.9) ρ, b, r, r, I, V, z, x y V x, y, z, x i, y j, z k (x i, y j, z k ) i, j, k x i y j z k, ρ x i y j z k ρ I = 2π ρ b r 3 r (11.10) ( 11.6) r r r,, 2πrρb r , (z ) r ijk, r 2 ijk = x 2 i + y 2 j (11.13)

98 92 11 z 2 k!) I, (11.6), I = i ρ (x 2 i + yj 2 ) x i y j z k (11.14) j k x i, y j, z k,, : I = V ρ (x 2 + y 2 ) dx dy dz (11.15), V 118 (11.14) (11.15), r 2 ) x 2 + y 2 + z 2 z , ( 11.9) I = Mr2 4 (11.16),,,,,,,,,,,,,, 120 θ, h, r, M, I X X (11.12) (11.16),,,, 3,,, 3,, (1) X U X U = 0 X U?,,, X (2) X 0 X ( 0) E 0?

99 93 (3) X, (X ) v, ω, v = rω (11.17) t (v ω ), X rω t (4) X, X E 1, E 1 = M v2 2 + Iω2 2 (11.18), (5) (E 0 = E 1 ), Mgh = M v2 2 + Iω2 2 (11.19) (6) (3), 2gh = v 2( 1 + I ) M r 2 (11.20) (7), 2g h v = 1 + I/(M r 2 ) (11.21) (8) I 0, v = 2g h M (9) X, M, v = g h (10) X, M, v 121?, 2 A,, B,,, 3 *4, *4,, m 2 2 d 2, ( 11.11) r, d = 2r P.77, T , k B T/2 1, K, K = 1 2 k BT (11.22) *5, I, 112, I = 2mr 2, 122 K = 1 2 Iω2 = mr 2 ω 2 (11.23) 2, (1) : ω = 1 kb T r 2m (2) 16 O? (11.24) (3), d = 0.12 nm (300 K), *6 *5 T, T,, K *6,,,,,,,,,

100 ,, 88,, 123 M,, P ( 11.12) I G P G l G O, G t G(t), OPG θ(t) (4), l = I M l (11.28), l, (7.38) d 2 θ dt 2 = g sin θ (11.29) l (5) θ 0, (11.29) : d 2 θ dt 2 = g l θ (11.30) (6), θ = θ 0 cos ωt, ω, T l T = 2π g (11.31) (7) : ( 2π ) 2l g = (11.32) T 3 (2) I, (11.7),, :, L, (1) t T (t) T (t) = 1 ( dθ ) 2 2 I (11.25) dt (2) t U(t) U(t) = M g l (1 cos θ) (11.26), G O U = 0 (3), I d2 θ = M g l sin θ (11.27) dt2,, L = Iω (11.33) * ,, 1.8 *7, (11.33),, L,, ω,, I, (11.33), L = Iω (11.34)

101 11.1 ( ) 95, 11.1 ( ) (9.1),,?,,, 1, m,, T, h, mgh, P, P, 0 h, h,,, h P (h), ρ(h) H, A h h + dh, AP (h + dh), AP (h), ρa dh g,, AP (h + dh) + AP (h) ρa dh g = 0 (11.35), P (h + dh) P (h) dh = ρg (11.36) dh, dp/dh, dp dh = ρg (11.37), *8,, P = ρk BT m ρ = mp k B T (11.37), (11.38) (11.39) dp dh = mg k B T P (11.40), ( P = P (0) exp mgh ) k B T (11.39), mp (0) ( ρ(h) = k B T exp mgh ) k B T (11.41) (11.42), exp exp mgh,, E *9, E, ( exp E ) k B T (11.43),,, T, E, ( exp E ) k B T (11.44) *8, P V = Nk B T V, N V, P = (N/V )k B T m, P = (mn/mv )k B T mn, mn/v, ρ, P = (ρ/m)k B T, (11.38) *9 U,, U, E

102 (1) (2) (11.6), [I]=[m k r 2 k ]=[m k][r k ] 2 =ML (11.6) n = 2, r 1 = r 2 = r, m 1 = m 2 = m I = m r 2 + m r 2 = 2m r (11.6) n = 3, r k = r, m k = m I = 3m r (11.6), r k = r, m k = m I = nm r nm = M 116 1, b r, 2πr 2πbr r ρ, 2πρbr r M (11.9) I I 117 r n, (11.10), I, I = n 2π ρ b rk r 3 (11.45) k=1 r k k r 0,, : I = R 0 2π ρ b r 3 dr (11.46) R, I = π ρ b R4 2 (11.47) R r, (11.11), M,, (11.11) I = M r2 2, (11.12), (11.9) 118, (11.6), r k,,, r x 2 + y 2 + z 2, (z ) x 2 + y z, x, y b ρ (x, y, z) z x 2 + y 2 I, (11.15) I = r b/2 r 2 z 2 r b/2 r 2 z 2 ρ(x 2 + y 2 ) dx dy dz, x 2 +y 2 x 2,, y, r r 2 z 2 I = b r ρx 2 dx dz r 2 z 2 (11.49) x, I = b = 2bρ 3 r ] [ρ x3 r 2 z 2 r 3 dz r 2 z 2 r r (r 2 z 2 ) 3/2 dz (11.50) z, I = 4bρ 3 r 0 (r 2 z 2 ) 3/2 dz (11.51) z = r sin θ (), dz = r cos θ dθ, 0 θ π/2, (r 2 z 2 ) 3/2 = (r 2 r 2 sin 2 θ) 3/2 = {r 2 (1 sin 2 θ)} 3/2 = (r 2 cos 2 θ) 3/2 = r 3 cos 3 θ M = π ρ b r 2 (11.48)

103 , I = 4bρ 3 = 4br4 ρ 3 π/2 0 π/2,, ( e cos 4 iθ + e iθ θ = 2 0 r 3 cos 3 θ r cos θ dθ cos 4 θ dθ (11.52) ) 4 = e4iθ + 4e 2iθ e 2iθ + e 4iθ 16 = e4iθ + e 4iθ + e2iθ + e 2iθ = I = 4br4 ρ 3 = 4br4 ρ 3 = 4br4 ρ 3 cos 4θ 8 π/2 + cos 2θ ( cos 4θ cos 2θ [ sin 4θ sin 2θ + + 3θ ] π/ π 16 = πbr4 ρ = Mr2 4 4, M = πbr 2 ρ 120 (1) U = Mgh ) dθ (11.53) (11.54) (2) 0, 0 E 0 U (1), E 0 = Mgh (3) rω t/ t = rω (4), Mv 2 /2, Iω 2 /2, (5) (1) (4), (6) (3) v, Mgh = M r2 ω Iω2 2 (11.55) 2 M, (7) (11.20) v = (8) (11.21) I = 0 (9) (11.9) (11.21) (10) (11.12) (11.21), 4g h v = 3 (11.56) 121 (11.21), I M (I/M) I/M, A,, B A, I/M B, B 122 (1) (11.22) (11.23), ω (2) 16, 1 16g, 1, m = kg = kg (3) (11.24) 123 r = d 2 = m 2 = m ω = 23 J K K m kg = s 1 (1),, ω,, θ t, ω = dθ/dt (11.7), (2) ( ),,, ( )P O l P G l, θ, P G, l cos θ, O G l l cos θ, (3), T (t) + U(t), T (t) + U(t) t

104 , d {T (t) + U(t)} dt = d { 1 ( dθ ) 2 } dt 2 I + M g l (1 cos θ) dt ( dθ )( d 2 θ ) = I dt dt 2 + M g l sin θ dθ dt = 0 (4) ( (11.28) (11.27) I (11.29) ) (5) sin θ = θ (6) : ω (1) ω θ = θ 0 cos ωt (11.30), ω 2 θ 0 cos ωt = g l θ 0 cos ωt (11.57) t, ω 2 = g/l T, T = 2π/ω, (7) (11.31) g = 124 L, ω, (11.33), I,, I ,,, 124

105 (x(t), y(t), z(t)), (0, 0, 0), (x, y, z ),, ( ), ( ),,,,,,,,, x, y,!?,,,,,,,,,,,,,, :!,!,,,,,,,,,, 3, 3, 3, 3,, ,,,,,,,,,,,,,,,,,

106 100 12,,,,,,,,,,,,,,,, O, O,,, O, O (0, 0, 0), O (p x + u x t, p y + u y t, p z + u z t) (12.1) (p x, p y, p z ), O, t = 0 O, (u x, u y, u z ), O, O, 3, O O, O (x(t), y(t), z(t)) (12.2), O, (X(t), Y (t), Z(t)) (12.3), x(t) = p x + u x t + X(t) y(t) = p y + u y t + Y (t) (12.4) z(t) = p z + u z t + Z(t), x (t) = X (t) y (t) = Y (t) (12.5) z (t) = Z (t), O O,,, O F x = mx (t) F y = my (t) F z = mz (t),, F x = mx (t) F y = my (t) F z = mz (t) (12.6) (12.7), O, 0, 0, O! 12.3,,, O, O,,, t = 0 O O, 0, O (0, 0, 0), O ( ax t 2 2, a yt 2 2, a zt 2 ) 2 (12.8) (a x, a y, a z ), O, O,, 3, O O, O ( x(t), y(t), z(t) ), O, ( X(t), Y (t), Z(t) ), x(t) = a x t 2 /2 + X(t) y(t) = a y t 2 /2 + Y (t) z(t) = a z t 2 /2 + Z(t), x (t) = a x + X (t) y (t) = a y + Y (t) z (t) = a z + Z (t) (12.9) (12.10) (12.11) (12.12)

107 , O F x = mx (t) F y = my (t) (12.13) F z = mz (t),, F x = mx (t) + ma x F y = my (t) + ma y (12.14) F z = mz (t) + ma z, F x ma x = mx (t) F y ma y = my (t) (12.15) F z ma z = mz (t), O,,, O, O,, ( ma x, ma y, ma z ) (12.16), (F x ma x, F y ma y, F z ma z ) (12.17), (12.15), (12.16),, O,,,,,,,, 125 H2A,, km (1), (2), 12.4, m,, O O,, r(t) = (x(t), y(t)) (12.18) f(t) = (f x (t), f y (t)) (12.19) (t ) O x y, r f, x, y, r(t) r(t), f(t) f(t), r(t) = x(t) + iy(t) (12.20) f(t) = f x (t) + if y (t) (12.21), (2 ),, O, f(t) = mr (t) (12.22), { f x (t) = mx (t) f y (t) = my (t), (12.23) f(t) = mr (t) (12.24) r (t), r(t) 2, r (t) = x (t) + iy (t), O,, ω, x y O t = 0 O O

108 ( 12.1) f(t) = mr (t), f(t) = mr (t)e i ω t + 2imωR (t)e i ω t mω 2 R(t)e i ω t (12.34) (12.30), F (t)e i ω t = mr (t)e i ω t + 2imωR (t)e i ω t mω 2 R(t)e i ω t (12.35) e i ω t e i ω t, F (t) = mr (t) + 2imωR (t) mω 2 R(t) (12.36), 12.1 O, O O O,, R(t) = (X(t), Y (t)) (12.25) F(t) = (F X (t), F Y (t)) (12.26) O, O R(t), F(t) R(t), F (t), R(t) = X(t) + iy (t) (12.27) F (t) = F X (t) + if Y (t) (12.28) O O ω t, O r(t) O R(t) ω t, r(t) = R(t)e i ω t (12.29) f(t) = F (t)e i ω t (12.30), (12.29) t, r (t) = R (t)e i ω t + i ωr(t)e i ω t (12.31) t, r (t) = R (t)e i ω t + 2i ωr (t)e i ω t ω 2 R(t)e i ω t (12.32) F 2imωR (t) + mω 2 R(t) = mr (t) (12.37), O, 2imωR (t) + mω 2 R(t) (12.38) (12.38), R (t), O, i i, y x O O 90,, ω, (12.38) 2, R(t), O, ω 2,, R(t), 126 v, m ω (1) m, ω, v (2)? v =1000 km h 1 (3), 1? m, mr (t) = mr (t)e i ω t + 2imωR (t)e i ω t mω 2 R(t)e i ω t (12.33)

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

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