/Volumes/NO NAME/gakujututosho/chap1.tex i

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/Volumes/NO NAME/gakujututosho/chap1.tex i

iii 1 7 1.1............................... 7 2 11 2.1........................................... 11 2.2................................... 18 2.3................................... 19 2.4....................... 25 3 31 3.1................................... 31 3.2.................................... 37 3.3............................ 40 4 45 4.1............................... 45 4.2...................................... 46 4.3...................................... 49 4.4................................... 52 5 59 5.1.................................... 59 5.2.............................. 65 6 73 6.1................................... 73 6.2...................................... 88 7 95 7.1...................................... 95 7.2...................................... 100 7.3.................................... 106 8 115

iv 8.1........................................ 115 8.2........................................ 120 8.3................................... 123 9 131 9.1.................................... 131 9.2............................ 134 9.3.................................... 137 10 143 10.1.................................... 143 10.2......................... 147 10.3................................... 153 11 161 11.1 2............................... 161 11.2............................ 163 11.3................................... 166 11.4.................................... 171 12 177 12.1............................ 177 12.2......................................... 179 12.3............................... 180 13 185 13.1................................... 185 13.2.............................. 187 13.3......................................... 189 13.4...................................... 194 13.5................................. 195 14 199 14.1.......................... 199 14.2................................... 200 15 205 15.1......................................... 205 15.2.......................... 208 15.3.......................... 210 15.4........................................ 215

v 16 217 16.1....................... 217 16.2............................... 219 16.3........................................ 222 17 229 17.1 1................................... 229 17.2 2................................... 229 17.3 3................................... 230 17.4 4................................... 231 17.5 5................................... 233 17.6 6................................... 234 17.7 7................................... 235 17.8 8................................... 238 17.9 9................................... 238 17.10 10.................................. 239 17.11 11.................................. 240 17.12 12.................................. 241 17.13 13.................................. 243 17.14 14.................................. 243 17.15 15.................................. 244 17.16 16.................................. 245

1 1.1 L, T, M ML 2 /T 2 m cm 100

8 1 [ ] v [v] m/s v E v L / T 10 m/s m, cm, SI m (), kg ( ), s ( ), A () MKSA cm m h 2gh g [g] L/T 2 h [h] L 2gh (L/T 2 L) 1/2 = L/T 2gh 2gh/m m g l A T g, l g, l T = [g x l y ] = (L/T 2 ) x L y = L x+y T 2x x + y = 0, 2x = 1 x = 1/2, y = 1/2 T l/g 2π l/g W L ρ v F /W = kρ x v y L z (1.1) k (ML/T 2 ) L 1 = (ML 3 ) x (L/T ) y L z

1.1 9 x = 1, y = 2, z = 1 F = kρv 2 LW (1.2) k 0.5 k 1 A 1. m k (a) L, M, T (b) m x k y x, y 2. k 1, k 2, k 3 (a) h, g ρ v = k 1 g x h y ρ z x, y, z 2 (b) v p, ρ v = k 2 p x ρ y x, y (0o C), 1 v = 340 m/s k 2 (c) v S ρ v = k 3 g x S y ρ z x, y, z 2

2 2.1 12, 15 2.1 4 m = ρv (2.1)

12 2 6 1 kg (SI ) 1kg 9.8N m kg F = mg(n) (g = 9.80655 m/s 2 ) (2.2) g (1m 2 ) Hooke s law l F x F x F = k x (2.3) k k m kg mg = kx x = mg k 2.2 (2.4) 1) 2) A 2.3 1) A 2)

2.1 13 2.2 2.3 A B A + B 2.4 A B A B C = A B (2.5) B C + B = A B + B (2.6) = A A B B A 2.5 sin cos tan (

14 2 2.4 2.5 B A C A B 2.6 θ sin θ = b c cos θ = a c tan θ = b a = sin θ cos θ sin 0 = cos 90 = tan 0 = 0 sin 90 = cos 0 = 1, tan 45 = 1 2.1 θ = 30, 45, 60 sin cos 1/2, 2/2, 3/2 3 ( 1, 2, 3 ) sin cos 3/2, 2/2, 1/2 tan sin cos θ 90 sin, cos, tan 2.7

2.1 15 2.6 2.1 θ sin θ cos θ tan θ 0 0 1 0 30 1 3 1 2 2 3 45 2 2 60 3 2 2 2 1 2 1 3 90 1 0 ± 90 θ 2 θ 2 = θ 1 + 90 (2.7) sin θ 2 = sin(θ 1 + 90 ) = cos θ 1 (2.8) cos θ 2 = cos(θ 1 + 90 ) = sin θ 1 (2.9) cos θ 3 = θ 1 + 180 (2.10) sin θ 3 = sin(θ 1 + 180 ) = sin θ 1 (2.11) cos θ 3 = cos(θ 1 + 180 ) = cos θ 1 (2.12) sin, cos

16 2 2.7 θ 90 sin, cos, tan A, B (x y ) A, B x, y A x, A y, B x, B y A = (A x, A y ) B = (B x, B y ) (2.13) 2.8 A + B x x A + B x = (A + B) x = A x + B x A + B y = (A + B) y = A y + B y A + B = (A x + B x, A y + B y ) (2.14) x, y, z A + B = (A x + B x, A y + B y, A z + B z ) (2.15) 1 x y e x, e y A = A x e x + A y e y B = B x e x + B y e y (2.16) A + B = (A x + B x )e x + (A y + B y )e y (2.17)

2.1 17 2.8 2.1 x 30 A A A x y (A x, A y ) x y 30 x y A (A x, A y) A x = A cos 30 3 = 2 A, Ay = A sin 30 = A 2 A x = A cos 60 = A 2, A y = A sin 60 3 = 2 A

18 2 2.2 0 F = F 1 + F 2 + = 0 (2.18) F 1, F 2, 2.9 ( M) F = F 1 + F 2 + F 1, F 2, 1 2 x y F = F x e x + F y e y (2.19) (2.18) F 1x + F 2x + = 0 F 1y + F 2y + = 0 (2.20) 0 2.10 A B

2.3 19 A B F 12 B A F 21 F 12 F 21 (2.21) 3 2.3 m F = mg (2.22) N N mg = 0 (2.23) mg R, M mg = G mm R 2 (2.24) G 2 g = G M R 2 (2.25) 2.10

20 2 2.11 2.13 θ ( ) θ c mg sin θ c f 0 mg sin θ c = f 0 (2.26) θ c f 0 f 0 N f 0 = µn (2.27) µ N N = mg cos θ c (2.28) 2.12

2.3 21 2.13 (2.26) (2.27) (2.28) µ = tan θ c (2.29) N f = µ N (2.30) µ µ < µ 2.2 2.2 I II I II µ µ 0.78 0.42 0.94 0.40 0.68 0.48 µ > µ. F F = cv (2.31) c ()

22 2 S N S N f 0 S f 0 N Si Al 1 4 =1m 2 Pa 1 P a = 1 N 1 m 2 (2.32) 2.14 p h h 2.15 h ρ h p 0 p p = p 0 + ρgh (2.33) 2.16 a S p = p 0 + ρgh

2.3 23 2.14 2.15 h 2.16 p = p 0 + ρg(h + a)

24 2 p p = ρga (2.34) F = (p p)s = ρg(as) (2.35) as ρas 2.17 V F F = ρgv (2.36) 2.17 4 1. 2.

2.4 25 3. 4. 10 15 m β 2008 2.4 1. 2. 2 2 2.18 (a) (b) (c) (d) (e) (a), (b), (c)

26 2 r N = F r (2.37) N 2.18 (c) F N = F r (2.38) N 2.18 2.19

2.4 27 0 F 1 + F 2 + F 3 + = 0 (2.39) N 1 + N 2 + N 3 + = 0 (2.40) F 1 + F 2 = 0 N 1 + N 2 0 r i, m i r G r G = m 1r 1 + m 2 r 2 + m 1 + m 2 + (2.41) 2.22 N G = N 1 + N 2 + N 3 + = 0 (2.42) 2.20

28 2 2.21 2.22 AB CD 2.22

2.4 29 2 A 1. 60 kg 10 [kgw] (a) (b) N (c) (d) 2. l µ 0.5

3 3.1 600 km 3.1 2 2 r = (x, y) (3.1) r θ 2 ( 3.2 ) r = (r, θ) (3.2) r θ x = r cos θ, y = r sin θ (3.3) 3 3.3 3 (r, θ, ϕ) ϕ r x-y

32 3 3.2 x ϕ θ x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ (3.4) 3.3 3 t r t (t > t) r R = r r (3.5)

3.1 33 t t r r t = t t r = r r (3.6) δ 3.4 v = r t α = v t (3.7) (3.8) x v = x t, α = v (3.9) t x t x(t) x = x(t) 3.5 t x t, r t, v t t x(t) 1. x(t) = t 2 (3.10)

34 3 3.5 v = x t tan θ x(t + t) = (t + t) 2 = t 2 + 2t t + t 2 t 2 + 2t t v = x t = x(t + t) x(t) t 2t t t = 2t (3.11) 2. x(t) = t n (n = 1, 2, 3, ) x(t + t) = (t + t) n = t n + nt n 1 t + t n + 2t n 1 t v = x x(t + t) x(t) = ntn 1 t t t t = nt n 1 (3.12) x(t + t) x(t) + x t t (3.13) x(t + t) = f(t + t)g(t + t) x f(t + t)g(t + t) f(t)g(t) = t t x(t) = f(t)g(t) (3.14) ( f(t) + f ) ( t t g(t) + g ) t t f g g(t) + f(t) t t (3.15) (3.16) x(t) = f(at) (a ) (3.17) x(t + t) = f(a(t + t)) = f(at + a t) f(at) + f a t a t T = at x t = a f(t ) (3.18) T

3.1 35 x(t) = f(g(t)) T = g(t) x t = g t f(t ) T (3.19) t, x, f x t dx d dt v v V = v + v (3.20) 3.6 v v u = v v (3.21) v = v u = 0 3.1 l v 0 v v 0 v V θ tan θ = V v 0 = p v2 V 2 0 v 0

36 3 3.7 v 0 v v x, v y 3.8 r r v v θ p θ = tan 1 v2 V0 2 v 0! tan 1 x tan x 3.8 r 1 v 1 1 2πr, 2πv α T = 2πr = 2πv v α α = v2 r (3.22)

3.2 37 3.9 3.2 2 3 1 : 2 : m F α mα = F (3.23) 3 : 1 2 F 12 2 1 F 21 F 12 + F 21 = 0 (3.24) d(mv) = F dt v 40 3.10

38 3 1 3.10 2 2 m dv dt = F (3.25) p = mv (3.26) 10 dp dt = F (3.27) (3.25) (3.27) (3.27) p = F t (3.28)

3.2 39 3.2 v 0 1 σ 3.11 m = m 0 + σt d(mv) dt = 0 mv = m 0 v 0 v = m 0v 0 m 0 + σt (3.29) 1 2 3.12 F 12 + F 21 = 0 F 12 + F 21 = F 0 1 2 1 2 F F

40 3 F 12 + F 21 = 0 3.12 N 2008?? 3.12 1 2 3.3 v v v = (3.30) v v { v = v (3.31) 3.13(b) v t = 0 v 0, v 0 t 0 x x = v 0 t 0 (3.32) x-t tan θ = v 0 v-t v-t t v = v 0 = t = t 0 v 0 t 0 v-t

3.3 41 3.13 (a) (b) 3.14 (a) x-t (b) v-t α α = (3.33) g α = g = 9.81m/s 2 (3.34) g (2.2) 3[m/s 2 ]

42 3 2[m/s 2 ] 0.4[m/s 2 ] g 3 3.15 t v αt v 0 v = v 0 + αt (3.35) 3.16 v 0 x = v 0 t + 1 2 αt2 (3.36) d(v 0t) = v 0 dt d(αt 2 ) = 2αt dt dx dt = v = v 0 + αt (3.37) x v x(t) t = v v 0 α x(t) = v 0 t + 1 2 αt2 = v 0 v v 0 α + 1 2 α ( v v0 α ) 2 2αx = v 2 v 2 0 (3.38)

3.3 43 v 0 = 0 α = g h 2gh = v 2 v = 2gh (3.39) 3.16 v-t

44 3 3 A 1. x(t) = a(sin t) 2 2. 1 3. 2 3.17 4. v 0 h 5. F m t 1 (m = m 0 + σt) t = 0 (a) v(t) (b) v(t) t t = 0 t

4 4.1 x 4.1 mg x θ mg sin θ (3.25) m dv = mg sin θ dt α = dv (4.1) dt = g sin θ g g sin θ θ 0 0 g g sin θ cos θ sin θ θ 0 x 0 sin θ 4.1 µ 2 m dv dt = mg sin θ µ (mg cos θ) (4.2)

46 4 mg mg cos θ 4.1 N µn µ mg cos θ α = g(sin θ µ cos θ) (4.3) g g(sin θ µ cos θ) t x x = 1 2 g(sin θ µ cos θ)t 2 (4.4) m 1 m 2 4.2 B, C B T 1 T 2 mα = T 1 + T 2 0 T 1 = T 2 T T A, D T m 1 α = m 1 g T (4.5) m 2 α = m 2 g T (4.6) 2 (m 1 + m 2 )α = (m 1 m 2 )g (4.7) α = m 1 m 2 m 1 + m 2 g (4.8) g m 1 m 2 m 1 + m 2 g 4.2 2 x-y y m dv dt = mg (4.9)

4.2 47 4.2 (g = (0, g) ) x, y m dv x dt m dv y dt = 0 = mg (4.10) 1 2 2 x, y v x = v 0x (4.11) v y = v 0y gt (4.12) (v 0x, v 0y ) v 0 t = 0 4.3 θ v 0 t = 0 v 0x = v 0 cos θ (4.13) v 0y = v 0 sin θ (4.14) x = v 0x t (4.15) y = v 0y t 1 2 gt2 (4.16)

48 4 4.3 (cos θ) 2 cos 2 θ cos θ 2 cos(θ 2 ) cos θ θ sin θ 2 θ = 45 2 2 (4.15) (4.16) x = 0 y = v 0y t = x v 0x x v 0x 1 2 g x2 v 2 0x y = x tan θ 1 2 g x 2 v 2 0 cos2 θ x = 2v2 0 tan θ cos 2 θ = 2v2 0 sin θ cos θ = 2 v 0x v 0y g g g 0 4.3 y = 0 (4.17) (4.18) x = 2v2 0 sin θ cos θ g sin θ cos θ = 1 sin 2θ 2 θ = 45 v 0 2 /g β 4.4 y = x tan β θ = 45

(4.17) x tan β = x tan 45 1 2 g x 2 v 2 0 cos2 45 x = 2(tan 45 tan β)v 2 0 cos 2 45 /g = (1 tan β) v2 0 g x tan β (4.19) 4.3 49 β = 10 tan 10 0.18 18% 100 90 m 82 1/ cos β 100 8 4.4 4.3 v F v F v = Cv (4.20) C 3.3 m dv = mg Cv (4.21) dt 2 0 Cv ( ) = dv dt = g (4.22) v 2

50 4 340 m/s = 1200 km/h 1 1225 km/h v = gt m h 3.3 v = 2gh (4.23) 1 v = 2gh 440 m/s 1600km/h (4.24) 1.3 5 2 Cv mg = Cv (4.25) mg/c v v = mg C (4.26) - t t v v = gt t v = v 4.5 v t 4.5 4.5 v-t

4.3 51 v = v Ae Bt (4.27) t v = v t e Bt 1 Bt v v A(1 Bt) = v A + ABt (4.28) gt A = v, B = g/v (4.29) v = mg/c B = C/m v = v (1 e (C/m)t ) (4.30) v (4.21) dv dt = 0 v d dt e (C/m)t (4.31) ( = v C ) e (C/m)t (4.32) m (4.21) mv ( C m ) e (C/m)t = mge (C/m)t v = mg/c ( (4.26)) (4.21) (4.30) mg Cv (1 e (C/m)t ) = mg Cv +Cv e (C/m)t = 0+mge (C/m)t (4.30) 1 (4.30) 4.1 v 0 m 1. 2. v F v = Cv v mg + F v = mg Cv = 0 v = mg/c

52 4 (4.30) v(t) = A Be C m t (4.33) m dv dt = mg Cv BCe C m t = mg C(A Be C m t ) A = mg/c v(t) = mg C Be C m t = v Be C m t (4.34) t = 0 v(t) = v 0 v 0 = mg/c B B = mg/c + v 0 = v + v 0 v(t) = v (v + v 0 )e C m t = (v + v 0 )(1 e C m t ) v 0 (4.35) v = 0 e C m t = 0 = v (v + v 0 )e C m t v v + v 0 C m t = log v v + v 0 t = m C log v + v 0 v (4.36) (4.37) x 1 log(1 + x) x x 2 /2 v v 0 log v + v 0 = log(1 + v 0 /v ) v 0 /v 1 v 2 (v 0/v ) 2 t m C v 0 v «1 v0 = v0 2v g «1 v0 2v (4.38) t = v 0 /g 4.4 4.7,

4.4 53 1 (360 ) 2π, 90 π/2 = π (4.39) 180 θ 1 4.6 sin θ θ, θ 1 (4.40) (cos θ) 2 = 1 (sin θ) 2 cos θ 1 θ 2 1 θ2 2, θ 1 (4.41) 2 x 1 (1 + x) n = 1 + nx + n(n 1)x 2 + 1 + nx 2 n (1+x) k 1+kx, ( x 1 kx 1) 4.6 θ mg mg sin θ = mg sin θ (4.42) x l x x = lθ (4.43) m d2 x = mg sin θ (4.44) dt2

54 4 θ l d2 θ = g sin θ (4.45) dt2 4.7 4.8 ( θ tω = dθ ) t dt ω ω ωl dω dθ = g sin θ dt dt ω 2 /2 ( ) d 1 dt 2 ω2 = ω dω dt (4.46) l 2 d dt ω2 = g sin θ dθ dt (4.46) (4.47) (4.48)

4.4 55 ω 2 = X l dx + g sin θdθ = 0 (4.49) 2 l 2 dx + g sin θdθ = 0 (4.50) l 2 X g cos θ = l 2 ω2 g cos θ = K( ) (4.51) t (4.46) ω = ± K + 2g l cos θ (4.52) K = 2K/l ω = dθ/dt dθ dt = ± K + 2g l cos θ dθ = dt ± K + 2gl cos θ dθ = dt + L( ) ± K + 2gl cos θ dθ t + L = ± t = ± K + 2g l cos θ dθ L K + 2gl cos θ (4.53) (4.54) θ t I θ sin θ θ (4.45) l d2 θ = gθ (4.55) dt2 5 Gradshteyn 4.9 l m

56 4 4.9 θ 1 θ 2 2 4.10(a), (b) 4.10

4.4 57 4 A 1. 2 θ 1, θ 2 m 1, m 2 0 2. v 0 (4.18) (a) (4.18) θ (b) x θ (c). 3. α v 0 θ θ > α (a) (b) θ θ 4. 2 π 4 ρ a 2 v 2 ρ a (a) (b) (c) a = 1 cm 5. v M v v

58 4 M dv(t) = av, a > 0 dt (a) (b) (c) t = 0 v 0 v(t) v 0 > 0 v(t) t t = 0 v 0 2 M dv(t) = av bv 2, a > 0, b > 0 dt d (d) dt = dx d dt dx = v d dx M dv = (a + bv) dx v x (e) t = 0 v 0

5 5.1 ( 5.1) ( ) x F = k x (5.1) k 5.1 5.2 1 m x 0 ( ) x 0 0 m

60 5 (x < 0) 0 5.2 5.2 x() cos sin sin cos x = A sin(ωt + θ) (5.2) A ω ν ω = 2πν (5.3) T t = T ωt = 2π (5.4) 5.3 T = 2π ω = 1 ν (5.5) x = A sin(ωt + θ) A x 5.4 v v = 2πA T (5.6)

5.1 61 5.3 ω = 2π T (5.7) m d2 x dt2 = kx (5.8) x = A sin(ωt + θ) mω 2 A sin(ωt + θ) = ka sin(ωt + θ) (5.9) ω 2 = k m, ω = k m (5.10) 5.4

62 5 5.1 θ (4.55) sin θ θ l d2 θ dt 2 = gθ l (5.8) m l k g x θ (5.10) ω = r s g l, T = 2π l g (5.11) 3 2 5.5 x 1 x 2 1 3 2 x 1 x 2 x 2 x 1 5.5 1 2 1 : kx 1 + k(x 2 x 1 ) 2 : kx 2 k(x 2 x 1 ) m d2 x 1 dt 2 = kx 1 + k(x 2 x 1 ) (5.12)

5.1 63 m d2 x 2 dt 2 = kx 2 k(x 2 x 1 ) (5.13) m d2 dt 2 (x 1 + x 2 ) = k(x 1 + x 2 ) (5.14) x G = x 1 + x 2 2 (5.15) m d2 dt 2 x G = kx G (5.16) k ω G = (5.17) m 1 x 1 = x 2 2 2m 2k 5.6 5.6 x 1 = x 2 (5.12) (5.13) m d2 dt 2 (x 2 x 1 ) = k(x 2 x 1 ) 2k(x 2 x 1 ) = 3k(x 2 x 1 ) 2 X = x 2 x 1 (5.18) m d2 X = 3kX (5.19) dt2 3k ω R = (5.20) m 1 2

64 5 x G = x 2 + x 1 2 = A 1 sin(ω G t + θ 1 ) X = x 2 x 1 = A 2 sin(ω R t + θ 2 ) x 1 = A 1 sin(ω G t + θ 1 ) A 2 2 sin(ω Rt + θ 2 ) (5.21) x 2 = A 1 sin(ω G t + θ 1 ) + A 2 2 sin(ω Rt + θ 2 ) (5.22) 3 2 k m 5.7 x 0 y 0 ky 0 mg ky 0 + mg = 0 y 0 = mg k (5.23) 5.7 x 0 + y 0 x k(y 0 + x) k(y 0 + x) + mg m d2 x dt 2 = k(y 0 + x) + mg = kx (5.24)

y 0 = mg k 5.23 5.2 65 5.2 4.3 v F v = Cv (5.25) A 0 5.8 A(t) = A 0 e κt (5.26) κ 5.8 x = A 0 e κt sin ω t (5.27) ω ω = k/m m d2 x dx dt 2 = kx Cv = kx C dt (5.28) sin ω t ω t + θ

66 5 (5.27) dx dt = A 0( κe κt sin ω t + ω e κt cos ω t) d 2 x dt 2 = A 0(κ 2 e κt sin ω t 2κω e κt cos ω t ω 2 e κt sin ω t) 5.28 ma 0 ((κ 2 ω 2 ) sin ω t 2κω cos ω t)e κt = ka 0 e κt sin ω t CA 0 e κt ( κe κt sin ω t + ω e κt cos ω t) sin ω t cos ω t sin ω t : m(κ 2 ω 2 ) = k + Cκ (5.29) cos ω t : 2mκω = Cω (5.30) (5.30) κ = C 2m (5.29) ω 2 ω = = κ 2 + k m Cκ m = C2 4m 2 + k m C2 2m 2 = k m C2 4m 2 k m C2 4m 2 (5.31) (5.32) C = 0 ω = ω = k/m 5.9 k m ω = ω = k m (5.33)

5.2 67 5.9 l m g ω = ω = l (5.34) ω 5.10 k ω m d2 x = kx + a sin ωt (5.35) dt2 a m d 2 x dt 2 = ω2 x + A sin ωt (5.36)

68 5 A = a/m a sin ωt ω x = B sin ωt (5.37) ω (5.36) Bω 2 sin ωt = ω 2 B sin ωt + A sin ωt (5.38) B = A ω 2 ω2 (5.39) A x = sin ωt (5.40) ω2 ω 2 ω ω 1940 5.11 5.11

5.2 69 5.12 f n (n = 0, 1, 2, ) f i f j f = f i f j f i > f j f 5.13

70 5? 5.14

5.2 71 5 A 1. α k m (a) (b) 2. (5.27) x = Ae κt sin ω t, κ = C k 2m, ω = m C2 4m 2 1 2 2 3 3. (5.2) x(t) = A sin(ωt + θ) (a) t =0 x = a, v = dx = 0 A, θ dt (b) t =0 x = 0, v = dx dt = v 0 A, θ

6 6.1 6.1 F x W = F x (6.1)

74 6 mg h W = mgh (6.2) α x 6.2 W = mgh = mgx sin α = mgx cos θ (6.3) θ θ = π/2 α 6.2 α = 0 F r θ F W = F r cos θ (6.4) F, r F, r 6.3 r cos θ r W = F r cos θ A B θ A B = AB cos θ (6.5)

6.1 75 6.3 W = F r cos θ A, B A, B B = A θ = 0, cos θ = 1 A A = A 2 (6.6) x y 1, e x, e y 6.4 e x e x = 1, e y e y = 1, e x e y = e y e x = 0 (6.7) 6.4 e x, e y A x y A x, A y 6.4 x OC = A x e x (6.8) y OD = A y e y (6.9)

76 6 2.1 A = OC + OD = Ax e x + A y e y (6.10) A B = (A x e x + A y e y ) (B x e x + B y e y ) A = A x e x + A y e y (6.11) B = B x e x + B y e y (6.12) = A x B x e x e x + A x B y e x e y + A y B x e y e x + A y B y e y e y (6.7) A B = A x B x + A y B y (6.13) F r W = F r cos θ = xf x + yf y (6.14) F x, F y, x, y F, r x, y 3 W = F r cos θ = xf x + yf y + zf z (6.15) W = = (6.16) W = F z (z ) F = mg z ( mg) x mgµ µ x mgµ x x mgµ

6.1 77 x, y W W = x F x + y F y (6.17) x, y r W = F r (6.18) E E h m A h mgh A B A B h 6.5 B mgh A B A h 20 E = mc 2 E = mgh (6.19) h m E = mgh k m x F = 0 + kx = kx 2 2 (6.20) F +kx

78 6 6.5 A B B h x x E = F x = kx2 (6.21) 2 x W W = F x = kx x (6.22) 6.6 x W = 1 2 kx2 (6.23) 6.6 sum

6.1 79 W = W = F x = kx x = 1 2 kx2 (6.24) x dx W = dw = F dx = kxdx = 1 2 kx2 (6.25) dx 3.1 W = F dx W F kx 2 /2 x kx 1. F = x n W = x n dx = 1 n + 1 xn+1 (6.26) dw dx = 1 n + 1 (n + 1)xn = x n (6.27) 2. sin x, cos x W 1 = W 2 = sin xdx = cos x cos xdx = sin x (6.28) dw 1 /dx = d( cos x)/dx = sin x dw 2 /dx = d(sin x)/dx = cos x 3. e x 4. W = e x dx = e x (6.29) (a) I (b) (c) (d) Mathematica 6.7 Gradshteyn v v

80 6 6.7 (?) Cv + Dv 3 + E(v) E( v) v E(v) v E(v) = Av 2 + Bv 4 + (6.30) A v m 6.8 JR F m dv = F (6.31) dt 6.8 v F F

6.1 81 dv W = F dx = m dx = m dv dx (6.32) dt dt dx/dt = v dv dx dx = dv dt dt 0 W = m v dv = 1 x, v, t v 2 mv2 (6.33) x v = v x 1 2 mv2 E = 1 2 mv2 (6.34) 6.30 A = m/2, B = 0 B = 3 8c 2 m c Bv 4 Av 2 3 4 c2 E = mc 2 / p 1 v 2 /c 2 v = 0 K E = mc 2 v 2 K = 1 2 mv2 (6.35) 2400 V K h v 6.9K + V 0 E t E s K + V = E t + E s (6.36) K E mc 2 2400

82 6 6.9 K + V E t E s V K E = K + V K + V = E (6.37) h V = mgh (6.38) h v K = 1 2 mv2 (6.39) K + V 0 + mgh }{{} h K + V = 1 2 mv2 + 0 }{{} K + V = (6.40) 6.10 mgh = 1 2 mv2 v = 2gh (6.41) 3.2 h v (3.39)

6.1 83 6.10 6.11 A V = 1 2 ka2 K + V v 0 0 + 1 2 ka2 }{{} A K + V = 1 2 mv2 0 + 0 }{{} K + V (6.42) 6.10 A v 0 1 2 mv2 0 + 0 }{{} K + V = 0 + 1 2 ka2 }{{} B K + V (6.43) 1 2 ka2 = 1 2 mv2 0 (6.44) k v 0 = m A (6.45) k x = A sin ωt = A sin m t (6.46) v v = dx k k dt = A m cos m t (6.47) E = K + V = 1 2 mv2 + 1 2 kx2 = m 2 A2 ( k m ) 2 ( cos ) 2 ( k m t + k ) 2 k 2 A2 sin m t

84 6 6.11 ( = k ) 2 ( ) 2 k k 2 A2 cos m t + sin m t (6.48) (cos θ) 2 + (sin θ) 2 = 1 (6.49) E = k 2 A2 = (6.50) mgh mg 1 2 mv2 kx 6.12 V = mgh h m A B mgh A C B A C C B mg sin θ h mgh sin θ A B C

6.1 85 6.12 A B A C B F V dv (x) F = dx kx 1 2 kx2 dv (x) dx (6.51) = 1 k (2x) = kx (6.52) 2 V 3 x, y, z V (x, y, z) F = (F x, F y, F z ) F x = V x, F y = V y, F z = V z (6.53) x x V = 1 2 k(x2 + y 2 + z 2 ) (6.54) x y, z x ( ) 1 x 2 k(x2 + y 2 + z 2 ) = kx + 0 + 0 = kx 6.53 F x = kx, F y = ky, F z = kz (6.55) F = k(x, y, z) = kr (6.56)

86 6 = ( x, y, z ) F = V ( V F = x, V y, V ) z (6.57) V V F = gradv (grad = ( x, y, z )) (6.58) V x, y, z V (x, y, z) = F dx (6.59) F F : dx F F dx = F x dx + F y dy + F z dz (6.60) V (x, y, z) = (x, y, z) z V (x, y, z) = mgz (6.61) ( V F = x, V y, V z = (0, 0, mg) ) (6.62) V (x, y, z) = z = F 6.13 F s s 6.13

6.1 87 6.13 1 t E(t) E(t) = mv(t)2 + V (x(t)) (6.63) 2 v(t), x(t) t de(t) dt ( mv(t) 2 = d dt 2 = mv(t) dv(t) ( dt = v(t) m dv(t) dt ) + V (x(t)) dv dx) F (x(t)) = 0. + dx(t) dt (6.64) dv (x) F (x) = E(t) dx F x 0 v 0 F x 1 v 1 1 2 mv2 1 1 2 mv2 0 = = O x 1 O x 1 x 0 = + = x 0 x 0 O O x 1 x 0 F dx (6.65) + x 1 O

88 6 1 x 1 2 mv2 1 F dx = 1 x 0 2 mv2 0 F dx 1 2 mv2 1+V (x 1 ) = 1 2 mv2 0+V (x 0 ) O O (6.66) 6.2 N i 1 i 1,i + 1 n (3.27) dp i dt = F i1 + F i2 + + F in = j i F ij (6.67) P dp dt = P = p 1 + p 2 + + p n = N p k (6.68) i=1 n F ij = F 12 + F 13 + + F 1N i=1 j i + F 21 + F 23 + + F 2N + F 31 + F 32 + + F 3N + F ij = F ji (6.69) 0 ( ) dp = 0 (6.70) dt 2 2

6.2 89 2 2 2 2 i f m 1 v 1i = m 1 v 1f + m 2 v 2f (6.71) m 1 v 2 1i = m 1v 2 1f + m 2v 2 2f (6.72) 2 2 2 3 2 2 V G V G = m 1v 1i, v ij = v ij + V G (i = 1, 2 j = i, f) (6.73) m 1 + m 2 v 1i = v 1i V G = m 2v 1i (6.74) m 1 + m 2 v 2i = V G = m 1v 1i m 1 + m 2 (6.75) m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f = 0 (6.76) m 1 v 2 1i 2 (6.76) + m 2v 2 2i 2 = m 1v 2 1f 2 + m 2v 2 2f 2 (6.77) v 2i = m 1 v 1i, v 2f = m 1 v 1f (6.78) m 2 m 2 (6.77) v 1i = v 1f, v 2i = v 2f (6.79) ( ) 1. (6.78) 2. (6.79)

90 6 6.1 N p i (i = 1,, N) m i p i +q i (i = 1,, N) 1. E 2. V E 1. E = = NX i NX i (p i + q i ) 2 2m i p i q i m i + NX i NX i q 2 i 2m i p 2 i 2m i 2. V V m i V E = = NX (p i + q i m iv ) 2 NX (p i m iv ) 2 2m i 2m i i NX i (p i m i V ) q i m i + i NX i i q 2 i 2m i NX NX = E V q i = E V NX NX (p i + q i ) = i i p i i q i NX q i = 0 V E 2 1 d 2 r 1 m 1 dt 2 = F (6.80) d 2 r 2 m 2 dt 2 = F d 2 dt 2 (m 1r 1 + m 2 r 2 ) = 0 (6.81) i dr i dt = r i

6.2 91 p = m 1 ṙ 1 + m 2 ṙ 2 (6.82) R ( 2 (2.41) ) R = m 1r 1 + m 2 r 2 (6.83) m 1 + m 2 d 2 R dt 2 = 0 (6.84) V G = Ṙ (6.80) d 2 ( 1 dt 2 (r 1 r 2 ) = + 1 ) F m 1 m 2 µ def = ( 1 r def = r 1 r 2 (6.85) m 1 + 1 m 2 ) 1 = m 1m 2 m 1 + m 2 (6.86) µ d2 r dt 2 = F (6.87) µ F (r) 6.2 M E M S 33 (6.86) M E /M S = 1/330000 µ = M E µ = M E M S M E + M S 1 = M 1 + M E 330000 E 330001 = 0.999997 M E M S µ M E

92 6 6.3 2 V, v v 1, v 2 2 V v m 1 v 2 1 2 + m 2v 2 2 2 V v (6.83) (6.85) v 1 = V + m 1v 2 1 2 V = m 1v 1 + m 2 v 2 m 1 + m 2, v = v 1 v 2 m 2 m 1 + m 2 v, v 2 = V m 1 m 1 + m 2 v + m 2v 2 2 2 m 1 + m 2 V 2 + µ 2 2 v2 2 µ 2 2

6.2 93 6 A 1. M θ 1 m µ 2. k m A (a) K 0 V 0 (b) (c) (d) K V v A 1/2 3. l m v 0 (a) (b) v 4. V = C/r r = x 2 + y 2 + z 2 V V = C r F

94 6

7 7.1 16 1609 2 7.1 1 : 1 2 : 7.1 10 3 3 : T 2 a

96 7 3 T 2 a 3 (7.1) 3 10 1 4 7.2 M m F 7.3 T r 3 T 2 = kr 3 (k : ) (7.2)

7.1 97 7.2 r ω t ω t r ω t v t v = rω (7.3) 3 (3.22) v = r ω = = rω 2 (7.4) ω T ω = 2π T rω 2 (7.5) mrω 2 = F (7.6) (7.2) T ( ) 2 2π mr = F (7.7) T F = mr 4π2 kr 3 = 4π2 m k r 2 (7.8) m r 2 F M G F = G mm r 2 (7.9)

98 7 7.3 universal ( ) G G = 6.67 10 11 N m 2 /kg 2 (7.10) 2 F = G mm r 2 r r (7.11) m F = mg (g = 9.8 m/s 2 ) (7.12) R M G mm R 2 = mg (7.13) g = GM R 2 (7.14) 18 2 5.5

7.1 99 7.4 mg G 7.1 7.13 7.13 M = gr2 G = 9.8 (6.4 106 ) 2 6.67 10 11 kg = 6.0 10 24 kg (7.15) V F V F = gradv (7.17) V = G mm r (7.18) (7.17) r = x 2 + y 2 + z 2 ( ) 1 r = ( ) 1 r r x r x = 1 r 2 2x 2 x 2 + y 2 + z 2 = 1 r 2 x r ( ) 1 r = 1 ( y y r 2 r, 1 ) r z (7.11) = 1 r 2 z r (7.19) (7.20) 6.4

100 7 ρ r ω r M(r) M(r) = 4πr3 ρ 3 M(r) G M(r) r 2 = G 4πrρ 3 (7.4) rω 2 G 4πrρ = rω 2 3 4πGρ ω = 3 (7.16) 33 (7.18) (7.21) V M m r V (6.4) cos θ = 1 r r [ 1 V = F dr = GmM dr = GmM 1 ] r = GmM 1 r2 r r (7.21) V 7.2

7.2 101 F = f(r)r, f(r) = GmM r 3 (7.22) r r L = rp (7.23) p 7.5 p r mv mv L = r p (7.24) 7.5 2 1 A B = AB cos θ = A x B x + A y B y + A z B z (7.25)

102 7 7.6 : AB sin θ A B = : A, B (7.26) : A B A B A//B 2 0 A B = 0 (7.27) 0 A A = 0 (7.28) 7.6 A B A B A B = B A (7.29) x y z 1 e x e y e z e x e y = e z (= e y e x ) e y e z = e x (= e z e y ) e z e x = e y (= e x e z ) (7.30) A B A = A x e x + A y e y + A z e z B = B x e x + B y e y + B z e z (7.31) 7.30 A B = (A y B z A z B y, A z B x A x B z, A x B y A y B x ) (7.32)

7.2 103 N N = r F (7.33) 0 2 7.7 S = 1 2 rv (7.34) v v L = rp = mrv (7.35) S L (7.36) 2 7.7 m d2 r dt 2 = F (7.37) r mr d2 r dt 2 = r F (7.38)

104 7 1 3 2 2 F F //r r F 0 mr d2 r dt 2 = 0 (7.39) L = r p dl = dr dt dt p + r dp dt = r dp (7.40) dt dr/dt = v, v//p (7.39) dl/dt dl dt = 0 (7.41) L = (7.42) 46 L 7.8 L 50

7.2 105 7.8 1 7.9 7.9 7.10 r A, r B A m r A > r B v A, ω A, v B, ω B v A = r A ω A, v B = r B ω B (7.43) mr A v A, mr B v B

106 7 7.10 mr A v A = mr B v B mr 2 Aω A = mr 2 Bω B r 2 Aω A = r 2 Bω B ω B ω A = ra 2 rb 2 (7.44) 7.3 K V K + V = ( 1 2 mv2 + G mm r ( 1 2 mv2 = E + G mm r ) = E = (7.45) ) 0 (7.46)

7.3 107 ( E + G mm r ) ( 0) (7.47) r V (r) 7.11 E > 0 E < 0 E < 0 E V > 0 y E x V (r) ( ) E > 0 r 7.11 E < 0 r r m E 0 v 7.11 E V 0 x, y θ r 7.12 l r = (ɛ 0) (7.48) 1 + ɛ cos θ ɛ < 1 L E l, ɛ l = L 2 GMm 2 (7.49)

108 7 ɛ = 1 + 2L2 E G 2 M 2 m 3 (7.50) ɛ 7.12 1 cos θ 1 ɛ 1 ɛ < 1 r r = r max = l (7.51) 1 ɛ r = r min = l (7.52) 1 + ɛ r min, r max ɛ = 1, ɛ > 1 7.13 ɛ ɛ = 0 (7.51) r max = 1 + ɛ (7.53) r min 1 ɛ ɛ ɛ 1 2 v 1 =7.9 km/s 7.14 1 v 2 =11.2 km/s 2 1

7.3 109 7.13 R = 6400 km m v2 1 R = mg v 1 = gr (7.54) 2 E E = m ( 2 v2 2 + G mm R ) (7.55) 0 0 0 E = 0 0 = m ( 2 v2 2 + G mm ) R (7.56) (7.13) v 2 = 2GM R (7.57) v 2 = 2gR = 2v 1 (7.58) g=9.8 m/s 2, R = 6.4 10 6 m v 1 = 7.9 10 3 m/s, v 2 = 1.1 10 4 m/s 7.15 h R (24 ) T v s v s = 2π(R + h) T (7.59) 1 1

110 7 7.14 1 2 7.15 R = 6.4 10 6 m, T = 24( ) 60( ) 60( ) = 86400 s 10 5 s 10 1 3 10 7 s 3 v 2 s M = G R + h (R + h) 2 (R + h)vs 2 = MG = gr 2 (7.60) ( gr 2 T 2 ) 1/3 R + h = 4π 2 R + h 42,000 km h 36,000 km 70,000 km 140,000 km 1 300,000 km

7.3 111 0.5 7.2 R 6R 1 1/ 6 v 2 6R = G M r(6r) 2 r GM gr v = 6R = 6 (7.61) V v + V E = m 2 (v + V )2 + G mm «6R E 0 (7.62) v + V V = r gr 3 (7.63) 2 1 6 p gr (7.64) 1 2 1 7.16

112 7 7.16 7 A 1. 1 r T 2. v 0 v R b v 0 3. M R G (a) g M, R, G

7.3 113 (b) m v i. r ii. T iii. r T (c) g=9.8 m/s 2 R=6400 km (d) 1 (e) (f) 1/4 1/6 (g) 4. (a) M R G g M, R, G t = 0 v 0 m 3.(a) (b) (c) (d) (e) (f) (g) z ( R + z) m, g, R, z z ( R + z) v 0, g, R, z v 0 = V V V g, R V z dz 2gR dt = 2 z + R ( R 3 3 ) ( 2/3 z = + 2gR2 t R = R 1 + 3 ) 2/3 2g 2 2 R t R z t 2 t = 0 1 2 3

114 7 5. (a) (b) (1 + x) 2/3 = 1 + 2 3 x 1 9 x2 + O(x 3 ) R ω 2 R 3.9 10 5 km ω = 2π/T T = 28 GMm/R 2 (M m ) GM R 2 = GM ( ) 2 ( R R R 2 = g R R ) 2 R 6400 km (a)

8 8.1 1.6 km/h/s 0.44 m/s 2 20 1? 8.1 100 α t l = 1 2 αt2 (8.1) A 160 km 8.1 m A

116 8 x x 8.2 x O, O t O O l = 1 2 αt2 8.2 x = x + 1 2 αt2 (8.2) 3.2 m d2 x dt 2 = F (8.3) x 8.2 x x x (8.2) m d2 x dt 2 = m d2 ( x + 1 2 αt2) dt 2 = F (8.4) m d2 x dt 2 = F mα (8.5) F m

8.1 117 F I F I = mα (8.6) 8.3 A α 8.3 m N mα = N mg(= F ) (8.7) B B 0 F mα 0 = N mg mα (8.8) N = mg + mα (8.9) α N α = g N 0 30

118 8 8.1 60kg 0.98m/s 2 kg N N mα (α = 0.98m/s 2 ) mg 0 = mg N mα N = mg mα = 60 (9.8 0.98) = 530[N] = 54kg 8.4 8.4 1 A x x = A sin(2πνt) α α = A(2πν) 2 sin(2πνt) (8.10) ν N = mg α = mg + ma(2πν) 2 sin(2πνt) (8.11)

8.1 119 N 0 A(2πν) 2 sin(2πνt) g (8.12) 0 sin(2πνt) = 1 A(2πν) 2 A(2πν) 2 sin(2πνt) g (8.13) (2πν) 2 g/a (8.14) ν g/4π 2 A A g/(2πν) 2 A 8.5 8.5 m

120 8 F = mg = G Mm r 2 (8.15) m m G m I m G = m I (8.16) 8.2 8.6 A 8.7 A A m B B m K r v v 2 /r F I = mα = m v2 (8.17) r

8.2 121 8.6 8.7 A 8.7 v ω v/r t ω t h K h (r + h) cos(ω t) = r (8.18) t cos(ω t) 1 (ω t) 2 /2 (4.41) h 1 2 r(ω t)2 (8.19) α α t 2 /2 1 2 r(ω t)2 = 1 2 α t2 (8.20) α = rω 2 h r ω t 1 h(ω t) 2

122 8 F I = mα = mrω 2 (8.21) F I = mω 2 r (8.22) x-y F I = mω 2 (x, y) (8.23) 1 1 10 (=20π ) ω = 20π s 1 (8.24) 50 cm=0.5 m α = rω 2 (8.25) = 0.5 (20π) 2 (8.26) = 1974 m/s 2 (8.27) = 201g (8.28) 200 200 200 ( 8.8)

8.3 123 8.8 8.2 1/300 x, nx 1 (1+x) n 1+nx (1 1/300) 2 1 1/150 1/150 = 0.67% g 0.67/100 = 0.066 m/s 2 v 2 /R = Rω 2 ω = 2π/24/60/60 = 0.000073 / s 0.034m/s 2 8.3 8.9 8.9 a, b, c, d, ω r 8.10 O v A

124 8 8.9 A A r/v A A 8.10 A A 8.10 A A β βt 2 /2 O

8.3 125 A r/v h = 1 2 βt2 = 1 ( r ) 2 2 β ( O ) (8.29) v 8.10 A A t h = AA = rωt = ωr2 v 1 ( r ) 2 2 β = ωr2 v v β = 2ωv (8.30) (8.31) F c = mβ = 2mωv (8.32) v ω ω ω 8.11 ω z ω z 8.11 F c v y ω z x F c v, ω F c 7.2 F c = 2m(v ω) (8.33)

126 8 8.3 v m z x y α ω = ω(0, cos α, sin α) v = (v, 0, 0) a = 2v ω = 2vω(0, sin α, cos α) (8.34) z y mv 2 r r = = 2mvω sin α v 2ω sin α 2ω sin α z x y α ω = ω(0, cos α, sin α) ż v = (ẋ, ẏ, 0) m d2 x dt 2 = mg x l + 2mωẏ sin α (8.35) m d2 y dt 2 = mg y 2mωẋ sin α (8.36) l 1 y 2 x xÿ ẍy = 2ω(xẋ + yẏ) sin α (8.37) xẏ ẋy = ω(x 2 + y 2 ) sin α + (8.38) (x, y) = (0, 0) 0 0, (x, y) = (r cos φ, r sin φ) (8.38) ẋ = d dt (r cos φ) = ṙ cos φ r φ sin φ (8.39) ẏ = d dt (r sin φ) = ṙ sin φ + r φ cos φ (8.40) r 2 φ = ωr 2 sin α (8.41) φ = ω sin α (8.42)

8.3 127 ω sin α 30 2 1 ( 7 ) H) L) H L 8.12 8.12 1 1 8.4 12 11 1 (8.33)

128 8 8 A 1. l m ω θ 2. l m a 8.13 (a) (b) θ (c) 3. ω y z z = f(y) m

8.3 129 8.14 ( m)g ( m)yω 2 (a) (b) z y = yω2 g z = f(y)

9 9.1 = ω 9.1 m ω 9.1 ω n N N = n (9.1) ω r l 7.2 dl dt = n (9.2)

132 9 ( m)r 2 dω dt = n (9.3) ω dω/dt I dω dt = N (9.4) I ( m)r 2 N I dω dt 1 (9.5) I dω dt I m dv dt = F (9.6) F dv dt = F (9.7) m m m m I v ω F N N F (9.2) Iω p m I v ω p Iω F N (9.8) K K = 1 2 mv2 (9.9) (K) = 1 2 Iω2 (9.10)

9.1 133 I ω ω I ω I I = mr 2 (9.11) r 9.2 a I 9.2 (b) 9.1 9.2

134 9 m dv/dt = dp/dt I dω dt = N (9.12) L dl dt = N, (L = Iω) (9.13) L = l = ( m)r 2 ω (9.14) N = 0 L = 7.2 I I I ω L = Iω = (9.15) N N = r F (9.16) L ω L = Iω (9.17) dl dt = N (9.18) I dω dt = N (9.19) m dv dt = dp dt = F (9.20) (9.19) dω ω ω dt 2 O 2, N 2 9.2 I mr 2 l m

9.2 135 I = 2 m ( ) 2 l = ml2 2 2 (9.21) M l 9.3 x x M/l ( M m = l I = ( m)r 2 ( ) M = x 2 x l ) x (9.22) x 2 x x 2 x = I = M l l/2 l/2 x 2 dx = l3 12 (9.23) (9.24) l 3 12 = 1 12 Ml2 (9.25) 9.3 9.1 9.1

136 9 9.1 (9.27) (9.28) I 1 l 12 Ml2 1 ( a, b) b 12 Ma2 1 ( a) 6 Ma2 1 a 2 Ma2 a Ma 2 a a 2 5 Ma2 2 3 Ma2 9.4 l h Y { l 2 h I = M l = M l 1 3 0 l x 2 dx + 0 2 +h x 2 dx { ( ) 3 ( l l 2 + h + 2 h } ) 3 } = Ml2 12 + M l (lh2 ) = I G + Mh 2 (9.26) 9.4 l h Y I G Y h Y

9.3 137 9.5 I = I G + Mh 2 (9.27) 9.5 x, y z 9.6 I x, I y, I z I z = I x + I y (9.28) 9.6 I x, I y, I z 9.3 a M I G = 1 2 Ma2 m 9.7

138 9 I dω dt = N (9.29) T N = a T (9.30) 9.7 m dv = mg T (9.31) dt N, T I dω ( dt = a mg m dv ) (9.32) dt aω mga = 1 dv dv Ma + ma 2 dt dt dv dt = m m + M g (9.33) 2 9.8 a M T N = at ( ) 1 dω 2 Ma2 = at (9.34) dt

9.3 139 9.8 dv dt M dv = Mg T (9.35) dt aω = v ( 1 dv Ma 2 dt = a Mg M dv ) dt 3 dv 2 dt = g dv dt = 2 3 g (9.36) 3 2 T (9.34) T = M 2g 2 3 = Mg 3 (9.37) θ 9.9 M a F N = af (9.38) I dω = af (9.39) dt M dv = Mg sin θ F (9.40) dt

140 9 9.9 F aω = v M dv dt = Mg sin θ I dω a dt = Mg sin θ I dv a 2 dt ( M + I a 2 ) dv dt dv dt = = Mg sin θ Mg sin θ M + I/a 2 (9.41) Ma 2 /2 dv dt = 2 g sin θ (9.42) 3 (9.36) g g sin θ 9 A 1. M a 2. 9.1

9.3 141 3. a M (a) (b) (9.41) θ

10, 6.022 10 23 NO 3 16 2 17 10.1 A B A B A B A B B T A A T B B A B A B A B B A

144 10 10 m 100 Pa (N/m 2 ) 1 3.2 L x, L y, L z L x x L y y L z z 10.1 m v = (v x, v y, v z ) x mv x ( mv x ) = 2mv x v x > 0 1 v x /L x v x /(2L x ) 1 F F = 2mv x v x = mv 2 x (10.1) 2L x L x L y L z V P = N F L y L z = mv 2 x V P = N mv x 2 V v 2 x v 2 x N (10.2) (10.3) 10.1

10.1 145 v 2 = v x2 + v y2 + v z2 = 3v x 2 (10.4) k B T = mv x2 = mv2 3 (10.5) 0 (10.5) 0 T = + 273.15 (10.6) [K] k B k B = 1.3806503 10 23 J/K (10.7) P V = Nk B T (10.8) (N A ) (n) P V = nn A k B T = nrt (10.9) R = N A k B = 8.314 J/K (10.10) (10.9) U N 1 U = 2 mv i 2 (10.11) v i i 2 v 2 = i=1 1 N N i=1 v i 2 (10.12) 1 1.013 10 5 Pa (=N/m 2 ) P V = RT 0 1 V = RT/P 2.24 10 2 m 3 =22.4 (10.5) U U = N 1 2 mv2 (10.13) U = 3N 2 k BT = 3n RT (10.14) 2

146 10 P V = 2 3 U (10.15) 3 6 3 5 2 2 z 2 x y 2 3 2 5 3 U = 3N 2 k BT 2 U = 5N 2 k BT = 5n 2 RT (10.16) k B T/2 f U = fnk BT 2 (10.17) 1 2 2 (10.9) V V bn (10.18) 2 N/V 2 ( ) 2 N P P + a (10.19) V ( P + a ( N V ) 2 ) (V bn) = nrt (10.20)

10.2 147 10.2 10.2 P V 10.2 (I) (II) 10.2 Q = U + W = U + P V (10.21) Q U W = P V 1 1 ( (10.21)) W = 0 (10.22) W Q = U W V 1, V 2 W = P (V 2 V 1 ) (10.23)

148 10 10.3 P V P V 10.3 W = P V = nrt V (10.24) V W = dx x V2 V 1 nrt V dv = nrt V2 V 1 dv V (10.25) d log x dx log x = y x = e y d log x dx = dy dx = 1 dx/dy = 1 e y = 1 x (10.26) d log x dx = 1 x (10.27) 1 dx = log x + x (10.28) V2 ( ) dv V = [log V ]V 2 V2 V 1 = log V 1 (10.29) V 1 = nrt log ( V2 V 1 ) (10.30)

10.2 149 U = 0 (10.31) 10.21 Q W (10.30) ( ) V2 Q = nrt log V 1 (10.32) V 1 V 2 Q = 0 10.21 U + P V = 0 (10.33) (10.15) (P V ) = (P + ( ) 3P V P )(V + V ) P V = U = 3 P V + P V + P V 2 2 P V + 3 P V (10.34) 2 2 0 = 5 2 P V + 3 2 P V P V = 5 3 P V P P = 5 3 V V (10.35) r( 1) 5r/3 10.4 P V = P P = 5 V? 3 V

150 10 (10.27) x log x x 1 (10.36) x (10.35) log P = 5 3 log V (log P + 53 ) log V = 0 ( ) log(p V 5/3 ) = 0 (10.37) P V 5/3 = (10.38) 5/3 10.4 (I) (II) P V γ = (10.39) γ 2 3 γ = C P /C V (10.40) 10.1 1.1 2

V 0 P 0 V 1 P 1 P 0V 5/3 0 = P 1V 5/3 1 P 1 = P 0 «5/3 V0 1.1 1.1 5/3 0.853 2 2 5/3 0.315 V 1 10.2 151 10.2 2 (10.37) 3 P V = U( 10.14) 2 2 U = 5 nrt (10.16) 2 5 2 P V = U (10.41) P V P V 7/5 = 2 (10.42) (10.41) (10.34) «5P V U = 5 2 2 P V + 5 P V = U 2 U = P V 7 2 P V + 5 2 P V = 0 log P + 7 5 log V = 0 log P + 75 «log V = 0 log (P V 7/5 = 0 P V 7/5 = C Q = C T C = Q (10.43) T Q Q = U + P V = U (10.44)

152 10 C V = U (10.45) T U = 3nRT/2 C V = 3nR/2 2 U = 5nR/2 c Q = U + P V (10.46) P P V = (P V ) P V = (P V ) = (nrt ) = nr T (10.47) Q = U + nr T = (C V + nr) T (10.48) C P = Q T = C V + nr (10.49) C P C P C V 1 c P = c V + R (10.50) 10.3 2 P V 1.39 = (10.39) (10.40) 2 3 γ 5/3( 1.667), 7/5(= 1.4), 8/6 1.333 2 10.4 2 m, 20 m 2 1 J 40m 3 1 22 10 3 m 3 1800 2 5R/2 5R 2 1800 = 3.7 104 J

10.3 153 10.1 25 J/K mol c V () c P () c V ( ) c P ( ) He 3R/2 12.5 5R/2 20.8 12.5 20.1 H 2 5R/2 20.8 7R/2 29.1 21.1 29.4 1 kw 37 10 6 1 k W 10.3 A 0 A 1 A 2, A i A 0 T V P 2 P V P V P V ( 10.5) 10.5 P V 10.6 P 0, V 0 ( A 0 ) V 1

154 10 A 1 V 0 A 2 A 2 A 0 10.6 A 0 A 1 U 0 W (10.30) W = nrt log(v 1 /V 0 ) P 0 V 0 = nrt Q A0 A 1 = W = P 0 V 0 log(v 1 /V 0 ) (10.51) Q A1 A 2 = 5nR(T T )/2 T = P 0 V 0 /nr, T = T (V 0 /V 1 ) A 1 A 2 A 1 P 1 P 0 V 0 = P 1 V 1 P 0 V 0 /V 1 W A1 A 2 W A1 A 2 = P 1 (V 0 V 1 ) = P 0V 0 (V 0 V 1 ) (10.52) V 1 V 0 < V 1 A 1 T = P 0 V 0 /nr A 2 T T = T (V 0 /V 1 ) U U = 3nR ( ) V0 V 1 T = 3P ( ) 0V 0 V0 V 1 (10.53) 2 2 V 1 Q A1 A 2 = U + W A1 A 2 = 5P ( ) 0V 0 V0 V 1 (10.54) 2 A 2 A 0 0 T 2 T U U = 3nR 2 = 3P 0V 0 2 (T T ) ( ) V1 V 0 V 1 V 1 V 1 (10.55)

10.3 155 10.2 10.6 U f W Q P 0 V 0 U W Q A 0 A 1 0 log(v 1 /V 0 ) log(v 1 /V 0 ) 3(V 0 V 1 ) (V 0 V 1 ) 5(V 0 V 1 ) A 1 A 2 2V 1 V 1 2V 1 3(V 1 V 0 ) 3(V 1 V 0 ) A 2 A 0 0 2V 1 2V 1 Q A2 A 0 = U + 0 = 3nR 2 (T T ) = 3P 0V 0 2 ( ) V1 V 0 V 1 (10.56) 10.2 2 3 2 0 1 = (10.57) η η = (10.58) 10.6 ( P 0 V 0 log(v 1 /V 0 ) + 3(V ) 1 V 0 ) 2V 1 ( P 0 V 0 log(v 1 /V 0 ) (V ) 1 V 0 ) V 1 (10.59) (10.60) η = log(v 1/V 0 ) (V 1 V 0 ) V 1 log(v 1 /V 0 ) + 3(V1 V0) 2V 1 (10.61)

156 10 10.5 2 2 5 + 2 2 2 = 5 + /2 /2 5 V 1 V 0 log(v 1 /V 0 ) (V 1 V 0 )/V 1 + (V 1 V 0 ) 2 (10.61) = (V 1 V 0 )/V 1 5 2 + 2 2 η /5 η = 1 5 1 T «T 2V 2 1 2 2 (10.62) 10.6 4 10.7 I) P 0, V 0 T 0 P 1, V 1 V 1 > V 0 II) P 2, V 2 V 2 > V 1 T III) T 2 P 3, V 3 IV) P 0, V 0 10.7 P V I) III) II)

10.3 157 10.3 U W f Q U W Q I) 0 nrt log(v 1 /V 0 ) nrt log(v 1 /V 0 ) II) C V (T T ) C V (T T ) 0 III) 0 nrt log(v 2 /V 3 ) nrt log(v 2 /V 3 ) IV) C V (T T ) C V (T T ) 0 += 0 (10.63) C V (T T ) P 0 V 0 = nrt, P 2 V 2 = nrt 10.3 T I) Q I = nrt log(v 1 /V 0 ) (10.64) W = nrt log(v 1 /V 0 ) nrt log(v 2 /V 3 ) (10.65) P V γ = (10.37) P = nrt/v II) IV) 2 T V γ 1 = (10.66) T V γ 1 1 = T V γ 1 2 (10.67) T V γ 1 0 = T V γ 1 3 (10.68) (V 1 /V 0 ) γ 1 = (V 2 /V 3 ) γ 1 V 1 V 0 = V 2 V 3 (10.69) W = nr(t T ) log(v 1 /V 0 ) (10.70)

158 10 1 19 20 20 CPU T Q Q = Q W = nrt log(v 1 /V 0 ) (10.71) Q η = W Q = T T T = 1 T T (10.72) T, T Q Q

10.3 159 T T W Q (10.72) W = η Q W = η Q = η (Q + W ) Q = 1 η W = T η T T W (10.73) T T 10.6 27 37 1 J Q = 1[J] W f = ηq /(1 η) = Q (T T )/T = 1 10/300 = 1/30[J] W Q Q = 1 η W T = T T W (> W ) (10.74) W = Q 1/η = T/(T T ) 2 5 5 1 10.7 7 27 1 J ) 20/300=1/15 J

160 10 10 A 1. A(T A, V A, P A ) B(T B, V B, P B ) (a) W AB (b) U (c) 300K V 0 8V 0. 2. (a) 15 C 20% 80% (b) 1/6 45 C 2

11 11.1 2 P, V, T, U 11.1 V A B A B A B md 2 x/dt 2 = F t t 11.1 A B

162 11 2 2 2 2 1. : 2. : 2 11.2 T, T T > T T Q T Q W Q Q Q Q W 11.2

11.2 163 2 2 1 2 1 1 2 2 11.2 C 1 T 1 Q 1 2 T 2 Q 2 i T i Q i Q i < 0 11.3 11.2 N = 2 N i Q i T i 0 (11.1)

164 11 11.3 C T 0 T 0 T 1 i C i 11.4 T 0 > T i T i T 0 > T 1, T 2,, T N C i i C Q i ( ) = ( ) C i Q i T 0 Q i 10.72 Q i = T 0 T i Q i (11.2) W i = Q i Q i = T 0 T i Q i Q i (11.3) C W = i Q i (11.4) T 0 Q i = T 0 Q i T i i i W + W i i W + W i = T 0 Q i 0 T i i i Q i 0 T i i (11.5)

11.2 165 11.4 C, C i 11.4 W + W i i T 0 Q i W + W i = Q i (11.6) i i Q i T 0 i W + W i (11.6) i Q i i T 0 Q i = T 0 Q i 0 (11.7) T i i i C Q i 0 T i i i i Q i T i 0 (11.8) i i Q i T i 0 (11.9) Q i T i = 0 (11.10)

166 11 W /Q η = Q Q = 1 Q (11.11) Q Q Q T = Q T (11.12) η = 1 T (11.13) T Q T + Q T < 0 η = W Q = Q Q Q T T Q Q < 0 T T < Q Q (11.14) = 1 Q Q < 1 T T = η (11.15) = 11.3 Q i < 0 (11.16) T i i i Q i T i = 0 (11.17) Q (11.18) T dq T (11.19)

11.3 167 T S A B = S(B) S(A) = B A dq T S A B = S(B) S(A) = Q A B T S = Q T (11.20) (11.21) (11.22) n V 1 T T V 2 V V + V Q Q = U + W nrt V = P V = (11.23) V S = nr V (11.24) V V 1 V 2 V2 ( ) nrdv V2 S 2 S 1 = = nr log (11.25) V V 1 V 1 C T T dq = CdT S = T T dq T = T T C dt T = C log(t /T ) (11.26)

168 11 11.1 n T 1 T 2 n C V = 3n 2 R (11.26) S 2 S 1 = 3nR 2 «T2 log T 1 (11.27) 10.3 I) nrt log(v 1 /V 0 ) T S I = nr log(v 1 /V 0 ) (11.28) II) Q = 0 0 III) nrt log(v 3 /V 2 ) T S III = nr log(v 2 /V 3 ) (11.29) IV) S = nr(log(v 1 /V 0 ) + 0 log(v 2 /V 3 ) + 0) (11.30) (10.69) V 1 /V 0 = V 2 /V 3 S = 0 (11.31) dq/t 10.6 10.2

11.3 169 A 0 A 1 T S(A 1 ) S(A 0 ) = Q A 0 A 1 T = P 0V 0 T log(v 1 /V 0 ) = nr log(v 1 /V 0 ) = nr log(t/t ) (11.32) T = T (V 0 /V 1 ) A 1 A 2 dq = 5nR dt 2 S(A 2 ) S(A 1 ) = 5nR 2 A 2 A 0 T T dt T dq = 3nR 2 S(A 2 ) S(A 1 ) = 3nR 2 T T dt T = 5nR 2 log(t /T ) (11.33) dt = 3nR 2 log(t/t ) (11.34) nr log(t/t ) + 5nR log(t /T ) + 3nR log(t/t ) = 0 (11.35) 2 2 1 0 Q/T A 0 A 1 A 1 A 2 T Q A0 A 1 T = nr log(v 1 /V 0 ) = nr log(t/t ) (11.36) T ( ) 5P 0 V 0 V0 V 1 2 V 1 T = 5nR 2 Q A1 A 2 T = A 2 A 0 Q A2 A 0 T V 0 V 1 V 0 = 5nR 2 T = P 3(V 1 V 0) 0V 0 2V 1 T = 3nR 2 V 1 V 0 V 1 = 3nR 2 T T T (11.37) T T T (11.38)

170 11 Q A0 A 1 T = nr log(t/t ) + 5nR 2 + Q A 1 A 2 T + Q A 2 A 0 T T T T + 3nR 2 T T nr x = T/T T (11.39) f(x) = log x + 5 3(x 1) (1 x) + (11.40) 2 2x f(1) = 0 x > 1 df(x)/dx = 1 x 5 2 + 3 ( ) 1 2x 2 = x 1 + 3 ( ) 1 2 x 2 1 < 0 f(x) < 0(x > 1) P = F P (V, T ) (11.41) U = F U (V, T ) (11.42) V V = F V (P, T ) (11.43) U = F U (V (P, T ), T ) = G U (P, T ) (11.44) U V 2 1. A B

11.4 171 2. B A B A B dq T + A = dq T 0 (11.45) 2 S(A) S(B) B A dq T + S(A) S(B) 0 (11.46) S(B) S(A) B A dq T (11.47) 0 S(B) S(A) (11.48) 11.4 A B 20 1 1 L 20 1 2 L A B 11.2 A B (11.25) n

172 11 U = Q P V (11.49) (11.22)Q = T S U = T S P V (11.50) U F = U T S (11.51) (T S) = (T + T )(S + S) T S = T S + T S + T S T S + T S F = (U T S) = U (T S) T S P V T S S T = P V S T (11.52) H = U + P V (11.53) H H = (U + P V ) = U + (P V ) T S P V + P V + V P = T S + V P (11.54) F P V G G = F + P V = U T S + P V (11.55) G = (F + P V ) P V S T + P V + V P = S T + V P (11.56) 4

11.4 173 U T S P V (S, V ) F U T S S T P V (T, V ) G U T S + P V S T + V P (T, P ) H U + P V T S + V P (S, P ) F, G T A B (11.47) T (S(B) S(A)) Q (11.57) U(B) U(A) Q = U(B) U(A) + W T (S(B) S(A)) U(B) U(A) + W (11.58) F (A) F (B) W (11.59) A B F W = 0 F (A) F (B) (11.60) B A T F (A) F (B) W = P (V (B) V (A)) (11.61) F (A) + P V (A) F (B) + P V (B) G(A) G(B) (11.62) A A V A, T A

174 11 11.4 U U = T S P V (11.63) V S ( ) U = T (11.64) S (S, V ) V ( 2 ) ( ) U T = (11.65) V S V S U V ( ) U = P (11.66) V S P (S, V ) S ( 2 ) ( ) U P = (11.67) S V S ( ) ( ) T P = V S S V F, G, H ( ) ( ) S P = V T T V ) V ( ) S = P T ) ( V S P = ( V T S P ( ) T P S V (11.68) (11.69) (11.70) (11.71) ( U/ V ) T ( ) ( ) U S = T P (11.72) V T V T ( ) S (11.69) V T ) ) ( U V T = T ( P T V P (11.73)

11.4 175 11 A 1. C T T (> T ) 2 (T + T )/2 2. T -S (a) P -V S T -S (b) 3. (11.69), (11.70), (11.71)

12 12.1 PC 1 e = 1.60217646 10 19 C (12.1) e C +e e 2e/3 u, e/3 d uud udd

178 12 Q Q F = k Q 1Q 2 r 2 (12.2) 14 Q 1, Q 2 2 r 2 k = 8.987742438 10 9 Nm 2 /C 2 (12.3) ɛ 0 k = 1 4πɛ 0 (12.4) ɛ 0 SI ɛ 0 = 8.854187817 10 12 C 2 /(Nm 2 ) (12.5) c (7.9) 10 7 4πc 2 C2 /(N m 2 ) 10 7 /4π 12.1 F Q1 = 1 Q 1 Q 2 4πɛ 0 r12 3 r 12 = 1 Q 1 Q 2 4πɛ 0 r12 2 ˆr 12 (12.6) F Q2 = 1 Q 1 Q 2 4πɛ 0 r12 3 r 12 = 1 Q 1 Q 2 4πɛ 0 r12 2 ˆr 12 F Q1 Q 1 F Q2 Q 2 r 12 Q 2 Q 1 r 12 = r 1 r 2, ˆr 12 = r 12 r 12 (12.7) ˆr 12 Q 2 Q 1

12.2 179 F q q r R 1 O r 1 q 1 F q 1 12.1 12.2 Q 2 E = 1 Q 2 4πɛ 0 r12 2 ˆr 12 (12.8) Q 1 F Q1 = Q 1 E (12.9) 2 1 r i Q i (i = 1, 2,, N)

180 12 r E = 1 Q 1 4πɛ 0 R1 2 N 1 = 4πɛ 0 i=1 ˆR 1 + 1 4πɛ 0 Q 2 R 2 2 Q i Ri 2 ˆR i ˆR 2 + + 1 4πɛ 0 Q N R 2 N ˆR N (12.10) R i = r r i r Q QE 12.3 (12.10) = (12.11) ( ) N Φ(r) = i=1 1 4πɛ 0 Q i R i (12.12) E = grad Φ(r) (12.13) e (12.10) 10 10 m ( 10 10 m )

12.2 12.3 181 12.2 ρ(r) r V Q Q = ρ(r) V (12.14) r i V i Q i ρ i = Q i / V i Φ(r) = 1 Q i 4πɛ 0 r r i i = 1 4πɛ 0 i ρ i V i r r i (12.15) Φ(r) = 1 dq(r ) 4πɛ 0 r r = 1 ρ(r )dv 4πɛ 0 r r (12.16) dq(r ) = σ(r )ds (12.17) dq(r ) = λ(r )dr (12.18) Φ(r) = 1 dq(r ) 4πɛ 0 r r = 1 4πɛ 0 σ(r )ds r r (12.19) Φ(r) = 1 4πɛ 0 dq(r ) r r = 1 4πɛ 0 λ(r )dr r r (12.20)

182 12 grad 6 gradv (V ) V ( (12.13) ) r r = R 1 R = ˆR R 2 (12.21) (12.13) E(r) = 1 4πɛ 0 ˆR R 2 ρ(r )dv (12.22) E(r) = 1 ˆR 4πɛ 0 R 2 σ(r )ds, (12.23) E(r) = 1 ˆR 4πɛ 0 R 2 λ(r )dr, (12.24) (6.59) (12.13) Φ = E dr (12.25) 0 O Φ = r E(r ) dr (12.26) 12.1 z x, y z (z > 0 z E z σ z r r + r 2πr r (12.19) Φ(r) = σ 4πɛ 0 Z ds r r = σ 4πɛ 0 Z 2πr dr r 2 + z 2 (12.27) r 0 0 Λ Z Λ 0 r dr r 2 + z 2 Φ(r) = = p r 2 + z 2 Λ = p Λ 2 + z 2 z (12.28) 0 σ 2ɛ 0 ( p Λ 2 + z 2 z ) (12.29) Φ x, y 0

12.3 183 Λ E z = lim Λ σ 2ɛ 0 E x = Φ x = 0, Ey = Φ y = 0 8 p Λ2 + z z 2 z «>< = z >: σ 2ɛ 0 z > 0 σ 2ɛ 0 z < 0 (12.30) 12.2 λ z (12.20) Φ(r) = λ 4πɛ 0 Z dz r r = λ 2πɛ 0 Z 0 dz r2 + z 2 (12.31) z Λ Z dz 1 r2 + z = 2 ln(z + p z 2 + r 2 ) (12.32) Φ = λ 2πɛ 0 (ln(λ + p Λ 2 + r 2 ) ln r) (12.33) ( ) E r E r Λ Φ E r = lim Λ r = lim Λ = λ 2πɛ 0 r λ 2πɛ 0 1 Λ + Λ 2 + r 2 r Λ2 + r 2 1 r «(12.34) ln(z + p z 2 + r 2 ) z r 12 A 1. 0.5Å = 5 10 11 m M p = 1.7 10 27 kg, m e = 9.1 10 31 kg

184 12 2. (12.13) (12.12) 3. A B C Q (> 0) ABCD O 1 2a A, B, C, D Q R E E = k Q R 2 0 Q Φ Φ = k Q R (a) O (b) D (c) (d) D (e) m q (> 0) O A (-a, a) D (a, a) O (-a,-a) B (a,-a) C 4. σ ( (12.30)) λ (12.34) (12.23), (12.24)

13 q (12.10) 13.1 Q r Q/4πɛ 0 r 2 Q/ɛ 0 13.1 4.4 cos θds dω = r 2 (13.1) ds θ cos θ r 2 2π 4π 4π E n 160000 400

186 13 13.1 n 13.2 n E n = E n (13.2) E n Q cos θ E n ds = E nds = 4πɛ 0 r 2 ds (13.3) cos θ/r 2 ds dω E n ds = dω = 4π Q 4πɛ 0 dω (13.4) E n ds = q ɛ 0 (13.5) 1 N Q 1,, Q N E n ds = Q ɛ 0, Q = N Q i (13.6) ρ E n ds = i=1 ρ ɛ 0 dv (13.7)

13.2 187 13.2 13.2 1 λ r h 2πrhE r = hλ (13.8) ɛ 0 E r = λ (13.9) 2πɛ 0 r 12 12.2 (12.34) 13.3 r h

188 13 2 2 ρ R r r > R, r < R 1. r > R 2. r < R 4πr 2 E = Q ɛ 0 E = = 4πR3 ρ 3ɛ 0 (13.10) Q 4πɛ 0 r 2 (13.11) 4πɛ 0 r 2 E = 4πr3 ρ 3 E = (13.12) Q 4πɛ 0 R 3 r (13.13) Φ (12.26) Φ(r) = r > R r < R Φ = R Φ(r) = E(r ) dr r R r Q 4πɛ 0 r 2 dr = E(r ) dr = Q 4πɛ 0 R Q 4πɛ 0 r r ( ) 3 2 r2 2R 2 E(r ) dr (13.14) (13.15) 13.4

13.3 189 (Earnshaw) : ( ) (test charge) ( ) 0 13.5 (a, 0, 0), ( a, 0, 0), (0, a, 0), (0, a, 0) x, y z 13.3

190 13 (13.7) 0 1. 2. 3. 4. σ S 13.6 ds E n = + + = E S + 0 + 0 = σs ɛ 0 E = σ ɛ 0 (13.16) 0 0 12 12.1 (12.30) 2 0 1. 2. 0

13.3 191 13.6 S 0 0 0 Q 0 d 0 Q Q Q 2d ±Q F = Q 2 4πɛ 0 (2d) 2 = Q 2 16πɛ 0 d 2 2 r ± = (z d) 2 + r 2, Φ(r) = Q ( 1 1 ) (13.17) 4πɛ 0 r + r

192 13 13.7 r = x 2 + y 2 ( E = 0, 0, Qd ) 2πɛ 0 R 3, R = r 2 + d 2 (13.18) (13.16) σ = ɛ 0 E z = Qd 2πR 3 (13.19) 13.1 σ Q (13.18) Z Q = dsσ = = Z 0 Z 0 σ2πrdr Qd dr2πr 2π r 2 + d 23» 1 = Qd r2 + d 2 0 = Q (13.20) a q r q 13.8 0 (z ), (0, 0, r I ) q I

13.3 193 0 q q 0 = 4πɛ 0 Φ(r) = + I a 2 + rq 2 2ar q cos θ a2 + ri 2 2ar I cos θ q I r I (13.21) = q a, r q a = a r I (13.22) ( ) a q I = q (13.23) r q r I = a2 r q (13.24) 0 1 a O q I r I q r q 13.8 13.2 Q q Φ = Q q I 4πɛ 0 a r I q I 0 q I Q Q q I Q q I 0 a

194 13 1 13.4 2 V Q ( +Q Q ) Q V [F] 1 Q = CV (13.25) C R V = Q 4πɛ 0 R C = 4πɛ 0 R (13.26) 13.9 2 2 R d A 0 B V B Q B Q B = CV (13.27) B 0 A V

13.5 195 A Q A Q A = C V (13.28) C C = C 2 C 13.3 C = ɛ 0S (13.29) d S d S = 1cm 2 d = 1mm C σ (12.30) (13.16) E = σ 2ɛ 0 (13.30) 2 σ = Q/S V = Ed V d = Q ɛ 0S E = σ ɛ 0 (13.31) Q = ɛ0s d V (13.32) S = 1 cm 2 d = 1mm C 0.9 10 12 F =0.9 pf 13.5 Q 1 Q 2 2 R W W = R d rf (r) = R dr 1 4πɛ 0 Q 1 Q 2 r 2 = Q 1Q 2 4πɛ 0 R (13.33) F (r) Q 2 U U = Q 1Q 2 4πɛ 0 R (13.34) U U = 1 N N Q i Q j (13.35) 2 4πɛ 0 R ij j=1 i=1,i j 1/2 2 i = 1, j = 2 i = 2, j = 1 Q 1Q 2/4πɛ 0R 12 Q 2 Q 1 /4πɛ 0 R 21 2

196 13 Φ i i Q i (12.12) U = 1 N Q i Φ i 2 i (13.36) Q 1, Q 2 R 2 Q 1 (Q 2 ) Q 2 Q 2 K K = mv2 2 = Q 1Q 2 4πɛ 0 R (13.37) (13.29) Q Q + Q Q V U = Q 0 dq V = Q 0 dq Q C = Q2 2C (13.38) E = σ = Q ɛ 0 ɛ 0 S C = ɛ 0S d C, Q U = S d ɛ 0E 2 2 (13.39) u = ɛ 0E 2 2 (13.40) 13 A 1. 1 N +e N e a N U = e2 4πɛ 0 a 2 ln 2

13.5 197 2. 3. 1 µm 0.1µ m e J? 4. R Q 0 (a) (b) (c)

14 14.1 2 d q (0, 0, d/2) q (0, 0, d/2) (12.12) Φ(r) = q ( 1 1 ) 4πɛ 0 r + r (14.1) r ± = x 2 + y 2 + (z d/2) 2 r = x 2 + y 2 + z 2 d r ± = x 2 + y 2 + (z d/2) 2 = r 1 + zd + d2 /4 r 2 r zd 2r (14.2) r ± 1 r ± zd 2r2 Φ(r) = (1 + δ) n 1 + nδ 4πɛ δ 1, nδ 1 0 r 3 (14.3) n = 1/2 p = q(0, 0, d) p 2 2 z d d 2 /4

200 14 14.1 (14.3) E(r) = Φ(r) = 1 4πɛ 0 [ 3(r p)r r 2 ] p r 5 (14.4) 14.1 E 2 2 d θ d(1 cos θ)/2 d(1 cos θ)/2 14.2 2 d (1 cos θ) qe = qd(1 cos θ)e (14.5) 2 qd(1 cos θ)e qde U = qde cos θ = p E (14.6) 14.2

14.2 201 14.2 κ κ Q 0 κq 0 Q 0 (κ 1)Q 0 κq 0 14.3 κ 1 χ χ = (κ 1)ɛ 0 (14.7) ɛ κɛ 0 ɛ 0 ɛ = ɛ 0 + χ (14.8) ɛ r = ɛ ɛ 0 (14.9) Q(= κq 0 ) Q (1 1/κ) Q/κ E E κ = ɛ 0 ɛ E (14.10) (14.11)

202 14 14.3 D D = ɛe (14.12) (14.8) D = (ɛ 0 + χ)e (13.6) D ɛ 0 E n ds = D n ds = Q (14.13) D nds = Q (14.14) q D = q 4πr 3 r (14.15) E = q 4πɛr 3 r (14.16) σ P = P n (14.17) σ P P 14.3

14.2 203 P E σ σ P σ σ P ɛ 0 = n E (14.18) (13.16) σ = σ P + ɛ 0 n E = n (P + ɛ 0 E) (14.19) (14.14) σ = n D D = ɛe D = ɛ 0 E + P (14.20) P = (ɛ ɛ 0 )E = χe (14.21) 14.1 d d 0 C = S (d d 0 )/ɛ 0 + d 0 /ɛ (14.22) Q E = Q/ɛ 0 S (13.31) E E 1/κ Q/ɛS V = E(d d 0) + E d 0 = Q ɛ 0S d d 0 + d 0 ɛ 0 ɛ (14.23) Q = CV C ɛe = ɛe/κ = ɛ 0 E 14 A 1. E 0 σ E 2. R ɛ Q r 13.2

204 14 (a) (b) (c) r > R r < R 0 r

15 I [C/s] [A] = C/s (15.1) j 15.1 n v j = nev (15.2) I j ds S t S cos θ nv t θ S Z t nv ds Z Z I = ( e)nv ds = j ds 15.1 S j(r) n(r)ds n(r) S ( )

206 15 0 j(r) n(r)ds = 0 (15.3) S 15.1 S j(r) n(r)ds Q j(r) n(r)ds = Q t S (15.4) m dv mv = ee (15.5) dt τ τ v m v τ = ee (15.6) 15.2 j = n( e)v j = σe, σ = ne2 τ m (15.7) σ σ

15.1 207 R ρ l S R = ρ l S (15.8) Ω Ω m Ω 1 m 1 0 1. 0 2. 2 1 W W W = IV = I 2 R = V 2 (15.9) R 1310 W 100 V 13.1 A [Wh] ), [kwh] 1 [W] 1 1 W h = 3.6 10 3 J (15.10) 1 1310 3600 J kwh 1.31 kwh 1 1 kwh 20 1 26 (15.7) Ω 1 m 1 = m 3 C 2 s kg Ω = m 2 kg s 3 A 2 MKSA m, kg, s, A 15.2 40W 1 1 100V 0.4 A 1 0.4 C 0.4C = 0.4 1.6 10 19 = 2.5 1018

208 15 LED CO 2 10 W 50 W LED LED V W W W W = W + W (15.11) W = I V W = I 2 R I V V 100 V 15.2 1 B B T q F = q(e + v B) (15.12)

15.2 209 : v 0 v 0 v 0 v 15.12 15.3 z B x-y 1. m e ω c ω c = eb m e (15.13) 2. 0.01 T 1. evb mr c ωc 2 = m e vω c eb = m e ω c ω c = eb m e 2. 12 m e = 9.1 10 31 kg 1.8 10 9 Hz=1.8 GHz q v 1 qvb S L n L nsl F = nsl (qvb) F = S qnv B = IB (15.14) L (15.2) qnv S S (qnv) I I

210 15 q, n q = e µ 0 ɛ 0 q v B E H qe H = qvb E H = vb (15.15) j j = qnv E H = jb (15.16) qn q n 15.3 B nds = 0 (15.17) 0 B dr = µ 0 I (15.18) C C B r B r 15.2µ 0 15.4 µ 0 = 4π 10 7 N A 2 (15.19) I r r (15.18) Z B dr = 2πrB C B = µ 0I 2πr (15.20)

15.3 211 15.2 C I 2 2 I 1 I 2 2 d I 1 I 2 (15.20) B = µ 0I 1 2πd I 2 (15.14) - = µ 0I 1 I 2 2πd (15.21) 13-1 1 - - I ds db db = µ 0 4π Ids r r 3 (15.22) 1 m 2 1 A 2 10 7 N 1 A µ 0 ( (15.19))

212 15 15.3 - B = µ 0I dr ˆR 4π R 2 (15.23) 15.4 r x y z ( ) r R 2 = r 2 + z 2, dr = ẑdz = (0, 0.dz z ), ˆR =, 0, R R (15.24) y ŷ = (0, 1, 0) ( (7.32)) B(r) = µ 0I 4π ŷ dr ˆR = ŷ r R dz (15.25) rdz (r 2 + z 2 ) = µ [ 0I 3/2 4π ŷ z ] z = r(r 2 + z 2 ) 1/2 z = (15.26) B(r) = µ 0I 2πr ŷ (15.27) (15.20) 15.5 a I µ 0 Ia 2 B z = (15.28) 2(a 2 + z 2 ) 3/2 (15.22) θ s s = a θ( sin θ, cos θ, 0)

15.3 213 15.4 15.5 z r = (0, 0, z) (a cos θ, a sin θ, 0) s r = a θ(z cos θ, z sin θ, a) B = µ 0 a( z cos θ, z sin θ, a) I a θ (15.29) 4π r 3 θ 0 2π x, y 0 r = p a 2 + z 2 B z = µ 0 4π I 2πa2 r 3 = µ 0 2r 3 Ia2 (15.30) z a z µ 0 Ia 2 B z = (15.31) 2[a 2 + (z z) 2 ] 3/2 N L dz NIdz /L db z = L/2 L/2 B z = µ 0NIa 2 2L = µ 0NI 2L µ 0 NIa 2 dz 2L[a 2 + (z z) 2 ] 3/2 (15.32) L/2 L/2 dz [a 2 + (z z) 2 ] 3/2 (15.33) z z [a 2 + (z z) 2 ] 1/2 L/2 L/2

214 15 0.8 0.6 0.4 0.2-7.5-5 -2.5 2.5 5 7.5 15.6 B z/µ 0nI z/a L/a = 8 l B z = µ 0NI 2L { } L/2 z [a 2 + (L/2 z) 2 ] + L/2 + z 1/2 [a 2 + (L/2 + z) 2 ] 1/2 L a, z (15.34) B z = µ 0NI L = µ 0 ni (15.35) n = N/L 15.6 l Z B dx = B l + 0 + 0 + 0 = µ 0nlI (15.36) B = µ 0 ni

15.4 215 15.4 = (15.37) ( 15.7) 15.7 D = ɛ 0 E + P = ɛe E H (14.20) B = µ 0 H + M = µh (15.38) H M M = 0 B = µ 0 H (15.39) ( ) M M = χh (15.40)

216 15 χ (15.38) B = (µ 0 + χ)h = µh, µ = µ 0 + χ (15.41) N S PC 15 A 1. Q ω 2. l v 3. (15.28) 4. B z dz a I r > a r < a 5. a I r 6. a Q ω (a) (b)

16 20 X TV 16.1 ( ) 1. 2. 3. = (16.1)

218 16 θ = cos θ (16.2) 16.1 Φ S B Φ = B i S = dsn B (16.3) Φ Φ n 16.1 T=Wb/m 2 T m 2 Wb ( ) V ind Φ V ind = dφ dt (16.4) V = L di dt L [L] = [Φ] [I] = W b A H (16.5) = W b/a (16.6)

16.2 219 2 A, B A V B = M di A (16.7) dt M S (15.35) Φ = S µ 0 ni (16.8) a n Φ = πa 2 µ 0 n 2 I = LI, L = πa 2 µ 0 n 2 (16.9) L 16.1 L n = 10 4 m 1 a = 1cm L = πµ 0 a 2 n 2 = 2.5 10 4 H n n A N B 16.2 A I A Φ = N B Sµ 0 n A I A (16.10) S M = µ 0 n A N B S (16.11) A B 16.2 16.2 (13.40) ɛ 0 E 2 /2 ɛe 2 /2 D = ɛe U = E D 2 (16.12) 15.1 100 V

220 16 16.2 I C Q -Q L 16.3 LC C L dq dt = I, L di dt + Q C = 0 (16.13) L d2 Q dt 2 = Q C (16.14) ω 0 = 1/ LC 5 (5.8) (5.10) t = 0 Q 0 Q(t) = Q 0 cos ω 0 t, I = Q(t) = Q 0 ω 0 sin ω 0 t (16.15) Q 2 2C + LI2 = Q2 0 2 2C 2 (16.16)

16.2 221 16.4 RC I C Q -Q R V 0 B = µ 0 ni, L = (πa 2 l)µ 0 n 2 (16.9) l LI 2 2 = V µ 0n 2 I 2 2 = V B2 2µ 0 (16.17) V = πa 2 l u = ɛ 0E 2 ( (13.40)) u = B2 (16.18) 2 2µ 0 u = H B 2 (16.19) C R CR L LR 16.3 LC 16.2 CR V 0 CR Q(t) t = 0 RI + Q C = V 0, I = dq dt R dq dt = V 0 Q C

222 16 (4.21) m dv dt = mg C v m R, v Q, mg V 0, C 1 C, v = mg/c CV 0 (4.30) t = 0 v = 0 t = 0 Q = 0 (4.30) v = v (1 e (C /m)t ) Q = CV 0 (1 e RC t ) RC R 1 C pf 10 12 16.3 LCR LC LCR V L d2 Q dt 2 + Q C + R dq dt = V (t) = V 0 cos ωt (16.20) (16.20) Q = A cos(ωt + φ) A, φ Q = Re[Q 0e iωt ] «1 Re C Lω2 + iωr Q 0 e iωt = ReV 0 e iωt Q 0 = CV 0 1 ω 2 /ω 2 0 + iωτ, Q 0 = ω 0 = 1/ LC, τ = RC Q 0 = Q 0 e iφ Q = Re[Q 0 e iωt ] = Q 0 cos(ωt + φ) CV 0 p (1 ω2 /ω 2 0 )2 + (ωτ) 2 I = Q 0 ω sin(ωt + φ) I CV 0ω 16.3 LC 16.3 I = dq/dt 0 0

16.3 223 Q /CV 0 0 20 18 16 ωτ = 0.05 14 12 10 8 6 4 2 0 0 0.5 1 1.5 ω /ω 2 0 16.5 LCR ω L, C ω 0 ω 0 ω E(t) Q ( (13.16)) I = dq = ɛ 0 S de de j = ɛ 0 (16.21) dt dt dt j = ɛ 0 de dt = dd dt (16.22) D (14 ) (14.12) j 16.6 1 E(x + x) l E(x) l (l x B(x + x/2, t)) t B(x, t) t ( ) E(x + x) E(x) = x (16.23)

224 16 B(x, t) t = E(x, t) x (16.24) 16.6 ( 16.7) h B(x) h B(x + x) h = µ 0 I B x = ɛ E 0µ 0 t = µ 0 (h x) j = ɛ 0 µ 0 (h x) E t (16.25) (16.26) 16.7

16.3 225 (16.24) (16.26) 2 E t 2 = 1 ɛ 0 µ 0 2 E x 2, 2 B t 2 = 1 ɛ 0 µ 0 2 B x 2 (16.27) (16.27) 2 2 λ f(x, t) = A sin 2π λ x (16.28) v f(x, t) = A sin 2π λ (x vt) (16.29) k = 2π λ ω = 2πv λ (16.30) (16.31) v f f(x, t) f(x, t) = A sin(kx ωt) (16.32) f(x, t) = f(x vt) (16.33) 2 f(x, t) t 2 = v 2 2 f(x, t) x 2 (16.34) 2 2 v 2 16.27 1 ɛ 0 µ 0 1 = 2.99792458 10 8 m/s (16.35) ɛ 0 µ 0 17 2 10 8 m/s (2.98 ± 0.02) 10 8 m/s