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 1 v 0 = t 0 1 = a, (4) t1 t 0 a(t). (5)

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 = dr (8) v = v x e x + v y e y + v z e z, (9) dr = dx e x + dy e y + dz e z. (10) a = dv = d2 r 2 (11) d 2 r 2 = d2 x 2 e x + d2 y 2 e y + d2 z 2 e z. (12) 1 2

x v x a x = 0, (13) v x = v x0, (14) x = v x0 t + x 0. (15) z a z = a z0 = g, (16) v z = a z0 t + v z0, (17) z = 1 2 a z0t 2 + v z0 t + z 0. (18) θ v 0 v x0 = v 0 cos θ, v z0 = v 0 sin θ. (19) (x max, z max ) t max 4 (23) x max = x 0 + v x0v z0, (20) g z max = z 0 + v2 z0 2g, (21) t max = v z0 g. (22) z = z max g 2v 2 x0 (x x max ) 2 (23) 5 x 1 z 1 θ tan α = z 1 /x 1 α θ = α/2 + π/4 3

2 3 2.1 1 2.2 2 F = ma. (24) N( ) = kg m /s 2. (24) (25) p = mv. (25) d p = F. (26) 2.3 3 (m 1, x 1, v 1 ) (m 2, x 2,v 2 ) m 1 d v 1 = F, m 2 d v 2 = F. (27) d (p 1 + p 2 ) = 0. (28) 4

p = m 0 v/ 1 (v 2 /c 2 ) p = h/λ p 1 p 2 p 1 p 2 t2 F (t) (= ), (29) p 1 p 1 = t 1 t2 p 2 p 2 = F (t). (30) t 1 p 1 + p 2 = p 1 + p 2. (31) 1 6 kv k g 1. 2. v t 3. ( v t 5

3 3.1. dr F dw dw = F dx cos θ = F dr. (32) W = r2 r 1 F dr. (33) (32) cos θ (33) dw = mgdz, W = mg(z 2 z 1 ). P = dw = F v. (34) 3.2 : K = 1 2 mv2 (35) ( ) d 1 2 mv2 = v d (mv) = v F (= ). (36) 1 2 mv(t)2 1 2 mv(0)2 = W (t) (37) du = F dr, U(r) = r F dr (38) 6

1. 2. gradient F x U x F y = U y. (39) F z U z F = grad U = U (40) 3. 3.3 ( ) d 1 2 mv2 = v d (mv) = v F = v gradu = d U(r(t)). (41) d (K + U) = 0. (42) 7 F = kx x k.) 8 k/ r k 1. 2. g = 9.8m/s 2 6400km 1kg k 7

y e θ e r 4 r θ x 4.1 2 2 2 r = θ = ωt (ω ( )). x = r cos θ, y = r sin θ. (43) r = r cos θe x + r sin θe y. (44) v = ωre θ, a = ω 2 r. (45) F = mω 2 r. (46) (45) v = dr = r sin θ dθ e x + r cos θ dθ e y e θ = sin θe x + cos θe y = ωr( sin θe x + cos θe y ). (47) v = ωre θ. (48) a = dv = ω2 r( cos θe x sin θe y ) = ω 2 r. (49) 4.2 k m : m d2 x = kx. (50) 2 8

x d 2 x/ 2 = ω 2 x x = A cos ωt, ω = k/m. (51) v = ωa sin ωt, K = 1 2 mv2 = 1 2 mω2 A 2 sin 2 ωt, U = 1 2 kx2 = 1 2 ka2 cos 2 ωt. K + U =. (52) 4.3 F = Gm 1m 2 r 2 r 1 2, U = Gm 1m 2 r 2 r 1. (53) (G = 6.7 10 11 N m 2 /kg 2 ) ρ(x, y, z) U(r 2 ) = V (ρ = ρ(r)) Gm 2 ρ(x, y, z) dxdydz. (54) r 2 r r, r 9 R m 0 1. r G M 2. r t 3. r GmM/R 9

5 A B A B sin θ (θ A B ) A B A A = 0, B A = A B. (55) e x e y = e z, e y e z = e x, e z e x = e y. (56) A B = (A y B z A z B y )e x + (A z B x A x B z )e y + (A x B y A y B z )e z. (57) L r p. dl = dr p + r dp = r F ( N). (58) (N r F ) (F (r) = f(r)e r ) N = 0, L =. (59) 10 (57) 6 4 2 (r, θ) (43) (ṙ 0) ( vx v y ) ( cos θ sin θ = sin θ cos θ ) ( ṙ r θ ). (60) 10

( ax a y ) ( cos θ sin θ = sin θ cos θ ) ( r r θ 2 r θ + 2ṙ θ ). (61) { er = cos θ e x + sin θ e y, e θ = sin θ e x + cos θ e y. r θ ( ) ( ) ( ) ( Ar A er cos θ sin θ Ax = = A θ A e θ sin θ cos θ A y (60) (61) ( ) ( ) vr ṙ = r θ, v θ ( ar a θ (62) ). (63) ) ( ) r r θ2 = r θ + 2ṙ θ. (64) F = f(r)e r m r = f(r)e r (64) θ ([66] ) m( r r θ 2 ) = f(r), (65) m(r θ + 2ṙ θ) = 0. (66) d dl z (mr2 θ) = = 0. (67) L z U(r) (65) f(r) = (du/dr) m r L2 z mr + du = 0. (68) 3 dr ṙ d ( ) 1 2 mṙ2 + L2 z 2mr + U(r) = 0. (69) 2 E = 1 ( 2 mṙ2 + L2 z 2mr + U(r) = 1 ) 2 2 m(v2 r + vθ) 2 + U(r) 11 L z = mr 2 θ (69) (70) 11

Y y a a 1- e 2 ae r θ x, X 7 7.1 3 1 x-y x (x + ae) 2 y 2 + a 2 a 2 (1 e 2 ) = 1. (71) a e a(1 e) a(1 + e) 3 (43) (71) r 12 (72) (71) r = a(1 e2 ) 1 + e cos θ. (72) 7.2 U = GMm/r 2 mr 2 θ = E, L z a, e U = GMm/r E = 1 2 mṙ2 + L2 z 2mr GMm 2 r (73) ṙ = 0 r 2 + GMm E r L2 z 2mE = 0. (74) 2 a(1 e) a(1 + e) E = GMm 2a, (75) L z = m GMa(1 e 2 ). (76) 12

T r 2 θ/2 = Lz /(2m) T = πa2 1 e 2 L z /(2m) = 2π a 3 GM. (77) (76) 3/2 (72) (68) r = θ u 1/r L2 z m 2 r GM 3 r. (78) 2 ṙ = dr θ dθ = L z dr mr 2 dθ = L z m du dθ. (79) r = d ( L z m (78) u = u GMm 2 /L 2 z ) du = L z dθ m θ d2 u dθ = L2 z d 2 u 2 m 2 r 2 dθ. (80) 2 d 2 u GMm2 = u +. (81) dθ2 L 2 z d 2 u dθ 2 = u. (82) u = u GMm2 L 2 z = A cos(θ θ 0 ). (83) L z (76) A = e/[a(1 e 2 )] θ 0 = 0 (72) 13 (75) (76) 14 (73) (72) (73) (79) (75) (76) (du/dθ) u a e (72) 13

8 8.1 2 2 2 F 12 2 1 F 21 = F 12. 2 r = r 2 r 1 m 1 d 2 r 1 2 = F 12, m 2 d 2 r 2 2 = F 21. (84) (84) 2 2 ( ) d dr 1 m 1 + m dr 2 2 = 0. (85) r G r G m 1r 1 + m 2 r 2 m 1 + m2. (86) M = m 1 + m 2 (85) d (Mṙ G) = 0. (87) r = r 2 r 1 (84) 2 2 1 F 21 = F 12 ( d 2 1 r = + 1 ) F 2 21 (r), µ d2 m 1 m 2 r = F 21(r). (88) 2 ( 1 µ = + 1 ) m 1 m 2 = m 1m 2 m 1 + m 2. (89) 15 ( M) ( m) U = GMm/r 2 14

8.2 N ( ) F ij f i (i, j = 1 N.) m i d 2 r i 2 N = F ij + f i. (90) j=1 j i r G = 1 M m i r i. m i ) (91) ( M = i i P (= i p i ) dp = M d2 r G 2 = i f i. (92) L (= i l i ) l i = r i p i dl i N = r i F ij + r i f i. (93) j i i (F ij / (r i r j )) r i F ij + r j F ji = (r i r j ) F ij = 0 (94) dl = i r i f i (95) E U ij (r ij ) U i (r i ) r ij = r i r j [ de = d 1 2 m i v i 2 + i i j<i U ij (r ij ) + i U i (r i ) ] = 0. (96) 16 (90) (96) 15

9 9.1 ( 2 9.2 z Ω v = Ω r e θ, v x = Ω y, v y = Ω x, v z = 0. (97) Ω = Ωe z v = Ω r. (98) i m i r i ( 8.2) L = i m i r i v i. (99) L = ( ) m i r i (Ω r i ) = m i ri 2 Ω e z i i = I z Ω e z. (100) Ω I z z I z = i m i r 2 i. (101) 16

E = i 1 2 m i v i 2 = i 1 2 m i (Ω r i ) 2 = 1 2 I z Ω 2. (102) 2 r G I G,z h I G,z + h 2 M M (101) ρ I z = (x 2 + y 2 )ρ(x, y, z)dxdydz (103) M = ρ dxdydz 2a I G,z = 1 3 a2 M (104) a I G,z = a 2 M (105) a I G,z = 1 2 a2 M (106) a I G,z = 2 5 a2 M (107) (100) (95) Ω N z N z 17 I z dω = N z (108) 18 (103) 19 17

9.3 2 9.2. r i = r G + r i, v i = v G + v i. (109) m i r i = 0, m i v i = 0. (110) i v i i v i = Ω r i = Ω r i e θ. (111) P P = Mv G (112) L L = i m i r i v i = Mr G v G + I G,z Ωe z. (113) E E = i 1 2 m i v i 2 = 1 2 M v G 2 + 1 2 I G,z Ω 2. (114) 2 2 Ω (108) M d2 x G 2 = F x, M d2 y G 2 = F y. (115) F x, F y F x y 18

20 α M I G a x y 1. ( Mg, F ) N) Ω 2. v G,x = aω 3. x F M I G a g α F 1. M d2 x G 2 = Mg sin α F, (116) M d2 y G 2 = Mg cos α N. (117) I g dω = af. (118) 2. θ aθ x v G,x = a θ = aω. (119) 3. (116) (118) (119) F Ω d 2 x G 2 = a2 M a 2 M + I G g sin α (120) F F = I G a 2 M + I G g sin α (121) I G 19