1 180m g 10m/s v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =
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- いぶき いいはた
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1 I hidekazu/class.html 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)
2 1 180m g 10m/s v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 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
3 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
4 F = ma. (24) N( ) = kg m /s 2. (24) (25) p = mv. (25) d p = F. (26) (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
5 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 v t 3. ( v t 5
6 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
7 1. 2. gradient F x U x F y = U y. (39) F z U z F = grad U = U (40) ( ) 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 g = 9.8m/s km 1kg k 7
8 y e θ e r 4 r θ x 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
9 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 = 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
10 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) (r, θ) (43) (ṙ 0) ( vx v y ) ( cos θ sin θ = sin θ cos θ ) ( ṙ r θ ). (60) 10
11 ( 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
![(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.](/docs-images/45/20554733/images/page_11.jpg "(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.")
12 Y y a a 1- e 2 ae r θ x, X 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
13 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
14 F F 21 = F 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) 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
15 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
16 9 9.1 ( 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
17 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
18 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 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
19 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
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
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2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t)
1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0
A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
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2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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振動と波動
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- - m k F = kx ) kxt) =m d xt) dt ) ω = k/m ) ) d dt + ω xt) = 0 3) ) ) d d dt iω dt + iω xt) = 0 4) ω d/dt iω) d/dt + iω) 4) ) d dt iω xt) = 0 5) ) d dt + iω xt) = 0 6) 5) 6) a expiωt) b exp iωt) ) )
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
7 4.., ], ], ydy, ], 3], y + y dy 3, ], ], + y + ydy 4, ], ], y ydy ydy y y ] 3 3 ] 3 y + y dy y + 3 y3 5 + 9 3 ] 3 + y + ydy 5 6 3 + 9 ] 3 73 6 y + y + y ] 3 + 3 + 3 3 + 3 + 3 ] 4 y y dy y ] 3 y3 83 3
5 36 5................................................... 36 5................................................... 36 5.3..............................
9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
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( ) c a sin b c b c a cos a c b c a tan b a b cos sin a c b c a ccos b csin (4) Ma k Mg a (Gal) g(98gal) (Gal) a max (K-E) kh Zck.85.6. 4 Ma g a k a g k D τ f c + σ tanφ σ 3 3 /A τ f3 S S τ A σ /A σ /A
II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1
II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4...........................
11 2 5 1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4................................. 3 2.5...............................
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3 ............................................2 2...........................................2....................................2.2...................................2.3..............................