) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
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- けいざぶろう やまがた
- 5 years ago
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1 4 4
2 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7 ˆr = xi + yj + z x + y + z x = x e)e x = x x e)e.. 4 F = fi µ Ni + Nj j F = lj j 6 F = ρv j + bvj j 7 = i + 6j + 46 ) F = N sin θi µ N cos θi +N cos θj +µ N sin θi j ) F = µ Ni + Nj + sin θi cos θj 8 ) F = πε 0 ) F = xi + yj + z x + y + z ) / 6 / πε 0 i 9 F = 4 0 ) ) i j ) = ) i = j ) ) cos θ sin θ i sin θ cos θ j ) ) cos θ sin θ i sin θ cos θ j ) F = F x cos θ + F y sin θ F x sin θ + F y cos θ ) i j ) 4) F x cos θ + F y sin θ) + F x sin θ + F y cos θ). ) F = i ) F = i + j ) F = i + j, F = 7 ) ) i i ) 4) j = j ) F = ) i j ) F = F x i + F y j i, j F = F x i + F y j + F z i, j,. 4 N 0 = [N ] N A = 0.7 [N ], N B = 0.4 [N ], N C = 0.0 [N ]. = 9.80[/s ] N A =.6 [N ], N B =.7[N ], N C =.9 0 [N ] ) N = RF y cos θ RF x sin θ) ) N = R F y cos θ F x sin θ 9 ) N 0 = ) N 0 = 0
3 ) N 0 = 0 4) N 0 = 0 ) N A =, N B =. ) N 0 = x + y) ) x, y) =, 4 ) ) N 0P = j 4i ) x, y) =, 4) ) N 0P + N 0Q = 8j 7i ) x, y) = 8, 7 ).4 ) ) = 0 [N/]. = 0 [N/]. ) B = 4 0 [] 4 ) A = + ) B = + T = 6 F = M Rh h 7 F M R h 8 x = []. cos ϕ sinθ + ϕ), T = cos θ sinθ + ϕ). 9 0%. 0 cos θ = R R + l) θ θ x = L µ ) + M) M ) A BC M : ) M F = [N]. 4 T = [N]. = 9.80[/s ] T =. 0 [N]. T = T = 6 + cos θ, N = sin θ + cos θ. ) i, j BG = ) 4 li + j ) ) x = l 7 R = l d ) W 8 F = 4 [N], F = 6 [N]. F = 0 [N], F = 4 0 [N]. 9 F = 00 [N]. = 9.80[/s ] F = [N] F = [N]. sin sin = 0. F = [N]. 4 x = []. 4 F = 0 [N]. 4 ) ) ) L 4 L L 44 R = 4 tan θ ) 6 [N], [N]. = 9.80[/s ].88 0 [N],.94 0 [N]. ) F L = 6 + 4M, F R = + M. ) 0. [] 46 M < 9 []. ) N L = L L + L N R = L L + L.
4 L ) f L = µ L f L + L R = µ. L + L ). ) 47 ) F = µ cos θ + µ sin θ ) tan θ = µ θ ) F = µ cos θ + µ sin θ 4) tan θ = µ θ 48 N r = h l F cos θ, N f = h F cos θ F sin θ + l, µ = F cos θ F sin θ ) 0 a + ) tan θ = a tan θ = a ) N R + N L = ) N R N L = h l a ) N R = l h. + h l a ), N L = h l a ), a =.7 4) 4 n = 0 [rps]. n = [rps]. π v > 0.6[/s]. = 9.80[/s ] v >.4[/s]. 6 v > r 7 ) n = π l cos θ ) N = l cos θ)πn) 8 cos θ = 9 rπn) ) v < [/s]. v < [/s]. ) v < 0 [/s] 60 v < r 6 F = lω sin θ cos θ, T = lω cos θ. 6 ) v = 0 4 [/s]. = 9.80[/s ] v =.0 0 [/s], v =.4 0 [/h]. + µ ) v = 40 [/s]. µ = 0. µ 6 ) v v =.6 0 [/s]. v = [/h]. dy dx = xω ) y = ω x + C 64 ) dy dr = A r ) y = A r + C 6 h = [] = + h ). h R = 8.7 [/s ]. 66 h = R + h)/ [] T = π. GM T = [s]. T =. 0 [h]. 67 h = GM ω R, v = R+h)ω. h = [], v =.07 0 [/s]. 68 = Rω R ω ) cos θ, Rω. Rω 0.46[%].
5 F = F x i + F y j + F z 74 r F rf sin θ θ r F P
6 .. ) F = j ) v x, v y ) = 0, ) ) v x, v y ) = v 0 cos θ, t + v 0 sin θ) 4) x, y) = v 0 cos θt, t + v 0 sin θt) ) y = 6) x = v 0 7) θ = π 4 v 0 cos θ x + tan θx sin θ ) F = i bvi ) v = bv ) vt) = b e b/)t ). 4) t vt) v = b ) F = xt)i ) F = + xt) i = xt)i ) ẍ = x 4 ) T = n, ω = πn ) v = πrn ) θt) = πnt 4) rt) = R cos πnti + R sin sin πntj ) aa) =, ab) = 0, ac) = 4, ad) =, ae) = [/s ] ) = 9[]. ) = 7[]. 6 ) T = 6 [s], ν = 6 [Hz]. ) ω = π [rad/s], A = [c] ) ) xt) = sin π t 4) vt) = π cos π t ) ) at) = π 9 sin π t ) 7 ) T = 4 [s], ν = 4 [Hz]. ) vt) = sin π t ) ) at) = π 4 cos π t ) 4) xt) = π cos π t ) + π 8 ) rt) = ti + tj, vt) = i + j, at) = 0, r0) = 0, v0) = i + j, vt) =, at) = 0. ) rt) = ti t j, vt) = i tj, at) = j, r0) = 0, v0) = i, vt) = + t, at) =. ) rt) = ti + t j + 4, vt) = i + tj, at) = j, r0) = 4, v0) = i, vt) = + 4t, at) =. 4) rt) = cos t i + sin t j, vt) = 6 sin t i + 6 cos t j, at) = cos t i sin t j, r0) = i, v0) = 6j, vt) = 6, at) =. ) rt) = cos t i + sin t j + t, vt) = 6 sin t i + 6 cos t j+, at) = cos t i sin t j, r0) = i, v0) = 6j +, vt) = 7, at) =. 9
7 ) at) = 0, rt) = v x t)i + v y t)j, ) at) = ai + αj, ) ) rt) = a t + bt i + α t + βt j 0 t = t = v Al + v B h va + v B rt) = h l = h. v A v B i + hj, vt) = h θ tan θ ) OP = R + a cos θ)j a sin θi. ) yt) = R + a cos ωt.. sin θ i. ) v = Rω sin ωti + Rω cos ωtj v r = 0 ) a = Rω cos ωti Rω sin ωtj = ω r ) Rω, 4 ) r = R cos θi + R sin θj ) v = R θ sin θi + R θ cos θj v = R θ = Rω ) a = R θ) cos θi R θ) sin θj R θ sin θi +R θ cos θj. a = R θ) = Rω, a = R θ = R dω. 4) F = Rω, F = R dω. ) F = 0 ω. ) F = RΩ F = 0 ) F = Rα t F = Rα 6 ) yt) = 0. sin ) vt) = 0. π T π T t ) at) = 0. π) T ) = 0. π) T ) cos π T t, sin π T t ). ) sin π T t 4) T > π 0 [s]. T > 6 0 [s].. 7 dv = 8 dv = cv t) 9 dv = f µ 0 dv = sin θ µ cos θ d y = yt) yt) = xt) + l d x dv 4 dv dv = ρv bvt) = j = j bvt) 6 d r = GM r r 7 d r = qq 0 4πε 0 r r 0 r r 0 8 dv x = v y B dv y = xt) = v x B dv z = 0..4 ) 9 dv = sin θ µ cos θ, vt) = sin θ µ cos θ)t, xt) = sin θ µ cos θ)t. 0 v ) T = 0 sin θ + µ cos θ), L = v0 sin θ + µ cos θ). ) tan θ > µ ) T = vt) = qv d t v 0 sin θ µ ) cos θ. ) ẍ, ÿ) = 0, ), x, y) = v 0 cos θt, ) t + v 0 sin θ t, y = ) t = v 0 sin θ ) y = v 0 sin θ v 0 cos θ x + tan θ x. 6
8 ) t = v 0 sin θ 4) x = v 0 sin θ H + L tan ϕ) ) T =, + H/L + tan ϕ) v 0 = L, H/L + tan ϕ) 4 θ tan θ = H/L + tan ϕ. ) t = v 0 sin θ cos ϕ ) L = v 0 sinϕ + θ) sin ϕ cos ϕ ) θ = π 4 ϕ, L = v 0 + sin ϕ). ) v 0 R v0 ) r = v 0 ) R.6 ) 6 ) ρv dv = ρv ρ 0V bv ) vt) = b ρ ρ 0) b/ρv e )t) 7 ρv dv = ρ 0 V ρv bv, vt) = b ρ 0 ρ) e b/ρv )t). 8 dv = cv t), v = vt) = c v tanh c t..7 4) 9 ) ẍt) = 4xt) ) c) xt) = cos t) f) xt) = cos t)+ sin t) ) T = π [s], ν = [Hz], ω = [rad/s]. π t ) 4) xt) = sin + π..0,.6[rad]. ) xt) = sin t), 4[/s] 40 xt) t = π + π. ) ẍt) = xt) ) xt) = A sin ωt, A = v 0 ω, ω. ). 4 T = π 0 T = 0 [s]., [s]. 4 ) = 00[N/] ) = 40 π 4 [] ) y ÿt) = yt) ) ω = yt) = l cos ωt ) vt) = ωl sin ωt 4) T = π, ν = π 44 B = A 4 4 ) j F = yt)j. ) ÿt) = yt) ) ω = yt) = A sinωt + ϕ) 46 ) = 4π 0 [N/]. = [N/]. ) ν = 0 [Hz]. ν =.8 [Hz]. ) =.π 0 [N/]. = [N/]. ) l : l = : 7
9 ) i F = + )xi ) ẍt) = + )xt) xt) = ) + A sin t + ϕ 4) ν = + π 48 ) l = ρs ) F = ρsyt)j ) ÿt) = ρsyt) 4) ν = ρs π 49 ) i F = T x l + x i ) ẍt) = T x l + x ) T = π ω = π l T 0 ) T = l + h h ) j F = + y ) ) + h + y) /l h + h /l j ) ν = π h ) ω ) K M b M e b/m)t C sin ωt + D cos ωt) K M xt) = ) x = v 0 ω e b/m)t sin ωt, v = b v 0 M e b/m)t ω sin ωt + e b/m)t v 0 cos ωt..8 ) ÿt) = yt) + f 0 sin Ωt, ) f yt) = 0 sin Ωt + A sin Ω t + ϕ. ) j F = yt)j + y 0 t)j ) ÿt) = yt) + A sin Ωt ) Ω = 4 ) T = λ v, Ω = π λ v. ) v = λ π x p t) = A sinωt + ϕ) A = F K MΩ) + bω), tan ϕ = bω K MΩ..9 6) 6 dv v = f R = T 7 F = 08 0 [N], f = 04 [N]. 8 ) t = v ax [s]. ) x = [] ) 9 = [s]. t = π ) v = l dθ ) l θ = sin θ l θ) = T cos θ. ) ) l θt) θt) θt) = α cos l t, T n = π l. T = α sin ωt ) cos ωt = π [/s ]. = 9.87 [/s ]. 6 ) R θt) = sin θt) θt) ) T = π R 6 ) F = Lθ sin θ, F = T cos θ. ) L θ = L + )θ. ) ν = π + L 8
10 6 ) L θ = sin θ, L θ) = T cos θ. ) L θ) L cos θ = v 0 L. ) T = v 0 L + cos θ 4) v 0 > L 64 ) L θ = sin θ, L θ) = T + cos θ. ) L θ) + L cos θ = v 0 + L ) T = v 0 L + cos θ 4) T = v 0 L +, v = v0 + 4L 9
11 . b). a)c) ) W a = W b =. ) W a = W b =. ) W a = W b =. 4) W a = W b =. ) W a = W b = 0. 6) W a = W b =. ) W = h ) W = h ) )) 4 ) W = µ l ) W = 7µ l ) ) N = 0 [N]. = 9.80[/s ] N = [N]. ) h =. [J]. h =. 0 [J]. ) J = 4.9[J/cal] [cal] J. [] [cal] 9.0J 0 4 []..8 []. 6 h =.[J] h =.[J].. 0 [J]. 7 ) P Sdl ) W = c lo V V 0. 8 F x = U x F y = U y F x = U y x y F y x = U x, y y x 9 F x, F y ) = x, y), W = 0. 0 W = 7 ). U = ). U = xy. ). 4). U = x y. ). U = x y. x + y ). 6) U = x y. F = ) x x i. U = 0.[J]. 4 F = z). z U x = GM x. y, z. r. 6 ) 00 [J] ) µ [J]. µ = [J]. ) K = 00 µ [J]. µ = 0. K K = 4. 0 [J]. 4) v = K [/s]. v =. 0 [/s]. 0
12 7 v > v d 0 a 8 W = 4 0 [J] E = W 0. = [MJ]. 4.6[MJ/l] [l]. 9 ) F = tan θ. ) W = L cos α) 0 ) ) ) µ = ) x 0 =. d L. ) x = x 0 =. ) v = ) 4) ) )..4 µ P = + µ tan θ v. 4 v = 00 [/s]. = 9.80[/s ] v =.0 0 [/s]. P = 7[W]. = 9.80[/s ] P = 7. 0 [W]. 6 v = P F = 4 [/s]. 7 ) vt) = ) xt) = P t P t/ ) )). 8 ) dv = R vt) v vt) ) t vt) v = P R ) t = ) R v lo v P R. 9 d = 0 h ) = L L l). ) = 0 [N/]. 7 = 9.80[/s ] =.80 0 [N/]. ) l sin θ,. ) l sin θ. ). v = l cos θ), T = cos θ). ) L θ) = T cos θ ) L θ) + L cos θ) = v 0 ) v 0 > L. 4 θ 0 > π ) R θ) = cos θ N ) v 0 + R cos θ) = R θ) ) cos α = + v 0 R α 6 v 0 > 0 4 [/s]. 7 ) v 0 = GM R ) v = GM R GM R. v 0 >.
13 8 ) v = GM r ) E = G M r. ) E r E r..6 9 h = v t) + yt). 40 y v t) + y t) yt) = E. 4 ) d = L a ) U = d + a θ ) K = L θ) 4) ω = 4 a L. ) U = aθ) L cos θ) a θ Lθ ) K = L θ) ) a > L 4 ) U = ρsy ) K = ρslẏ) ) ω = 44 l ) U = σy ) K = σlẏ) ) y = A e /l t + B e /l t dv = F dr a). b). 0 a). b). c). ) h, h. ).. ) h. h. ) F l = µ l cos θ + l sin θ, l sin θ, µ l cos θ. ).. ) l sin θ..
14 ) A a = A T B a = T B. ) a = A B A + B ) A a = A T, B a = T µ B. ) a = A µ B A + B ) a = sin θ A T, a = T sin θ B. ) a = sin θ A sin θ B ) 4 a = sin θ, A = sin θ cos θ. M ) ẍ = sin θ x x d), ẍ = sin θ + x x d). ) ẍ G = sin θ, ẍ r = x r d). ) x G = sin θt + Ct + D, ) x r = d + A sin t + ϕ. 6 ) ẍ = x x ) x, ẍ = x x ) x. ) ẍ G = x G, ẍ r = x r., 4. 7 ) x G 0) = Ml + M ) + M)ẍ G = 0 x G t) = x G 0) ) X = M M + l ) x G 0) = Ml + M ) + M)ẍ G = + M) sin θ x G t) = x G 0) + sin θt ) T = l + M sin θ 4) f = M sin θ 4. 9 v 0 cos θ, ev 0 sin θ) 0 ) v = 0[/s].. = 9.80[/s ] v =.4 0 [/s]. ) f =.[N], f = 0.[N]. f = [N], f = 0 [N]. ) v 0 = 0 [/s]. = 9.80[/s ] v 0 =.4 0 [/s]. ) v = 60 [/s] f t =.6 0. { 60 + v ) i + } 60j. f t = v 60 + v. f t =.09 0 [N s]. ) t =.0 0 [s] f =. 0 4 [N]. ) f t = e)v ) ρsv t ) f total = ρs e)v 4.4 ) V = [/s].
15 ) f = [N]. f = 0 4 [N]. 4 v = 0[/s] v 0 = + M h 6 v = [/s]. v = 0 [/s]. ) 7 V, v ) = M M + V, M M + V. 8 0, 9 V, 4 9 V, 4 ) V 9 v A sin α = v B sin β v A + v B = v. 0 K = M p + p + p p cos θ) ) v = + d)v + dv) dv V ). ) d = b, = 0 bt. ) vt) = v 0 V lo b t 4) vt ) = v 0 + V lo 4. dv F. = F v = j e = v j v j 4 e = v v v v Ft), Ft) v = v = F. 6 ) ) ) e =, 0 e <, e = 0 F 4
16 ) dl o = τ o = 0 L o., 0[s], θ total = 0. α = π [rad/s ], θ total = 7. ) L = 0 ) L = ) L = ωr 4 L = l ω L = Ml ω, L = 4 Ml ω. 6 ω = v + M l 7 ω = va + M) R 8 Ω = r r + MR ) ω. 9 ) vl = v l + I o ω ) V = lω ) M = I o l 0 ) v = v + MV ) vl = v l + I G Ω ) V = V + lω 4) M = M + l I G ω = l a) a + l a) v ) R ω MR Ω = 0 ) Φ = + M π ) L o = R ω ω ) ω = GM R 4) T = π 4 R GM ) r = rω, r = l cosh ωt. ) N = lω sinh ωt ). ) ) τ = π ) 0 [N ] 6 ) t = [in] ) τ = π ) 0 7 [N ] ) 00[N]. ) 7 [N ]. 0 [N ]. ) t = 00π 7 [s]. t = 0 [s]. 8 τ = 00 [N ] 9 ) I dω = τ fr ) J dω = fr ) rθ = RΘ 4) Ω = τ R r I + r R J.
17 . ) 0 ) dv = T ) ) a = M dv = T + M 4) T = M + M a = τ R + M ) dv = T dv = T. ) ) a = Ma = T T + + M < + ) M dv = F f, M dv = f. ) a = F M ) f = F 4) µ > F M 4 ) M dv = M sin α f, M dv = f. ) a = 7 sin α ) f = M sin α 7 4) µ > 7 tan α a = sin α, a = sin α. 7 6 I G = MR I G = MR 7 a = 8 ) MR dω = F f, MR dω = f. ) f = 7 F ) µ > 7 9 F M ) MR dω = F f, ) f = 7 ) h = R h R ) F MR dω = h R F + f..4 ) 0 ML θ MLθ, ω n = L. M h + ) θ L Mhθ, h ω n = h +. L M l + R) + ) θ R Ml + R)θ, l + R) ω n = l + R) +. R M a + ) θ R Maθ, a ω n = a +. R 4 ) I o = 4 l ) τ o = l sin θ ) ω n = ) d = ) l M 4 Ml θ = l θ, ω n = 6 ω n = a L 7 a > L, ω n = 8 ) d = 4 M. a L L. ) M dv = T x + d) ) ω n = + M 9 M dv = x f, M dv = f, ω n = M. 6
18 . 4) 40 ) ρ = M Sl. l ) ω n = a ) ω n = l 4 ) Mÿ = y, ω n = ) ω n = 6 M M. ) T n = π M, T n = π M 6. T n =. 0 [s], T n =.8 0 [s]. 4 ) ÿ = y, ω n = ) ω n =. ) ω n = 4 + M ω n = + M. ) l θ = a θ θ ) l sin θ l θ = a θ θ ) l sin θ. ) l Θ lθ ωn = L. l Φ l + a a Φ lφ ω n = l. 48 h = v = V 0 h + M 0 h = + M h R ω.7 dv = F... 4 N o = r F = 0. dl o = 0. r L = 0. 6 L z = r + R )ω i, r i, ω i L = i ir i ω i. ω L = i ir i ) ω. I o. dω 60 L = I o ω I o o = τ o L o = I o ω. ω ω = l cos θ 4 ω = l 4h 46 v = 47 I G = MR I G = MR 7
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dynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
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9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
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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
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
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K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
II 2 II
II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
4‐E ) キュリー温度を利用した消磁：熱消磁
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1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
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1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t
1 1 2 2 2r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t) V (x, t) I(x, t) V in x t 3 4 1 L R 2 C G L 0 R 0
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
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20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
(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
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
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )
http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b
Gmech08.dvi
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
2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ
1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m 2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t)
K E N Z OU
K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
Gmech08.dvi
63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
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1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
![c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l](/thumbs/85/91372642.jpg "c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l")
c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
23 7 28 i i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI..............................
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
![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](/thumbs/85/92098187.jpg "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")
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)
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
曲面のパラメタ表示と接線ベクトル
L11(2011-07-06 Wed) :Time-stamp: 2011-07-06 Wed 13:08 JST hig 1,,. 2. http://hig3.net () (L11) 2011-07-06 Wed 1 / 18 ( ) 1 V = (xy2 ) x + (2y) y = y 2 + 2. 2 V = 4y., D V ds = 2 2 ( ) 4 x 2 4y dy dx =
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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1 17 () BAC9ABC6ACB3 1 tan 6 = 3, cos 6 = AB=1 BC=2, AC= 3 2 A BC D 2 BDBD=BA 1 2 ABD BADBDA ABC6 BAD = (18 6 ) / 2 = 6 θ = 18 BAD = 12 () AD AD=BADCAD9 ABD ACD A 1 1 1 1 dsinαsinα = d 3 sin β 3 sin β
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π
![1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π](/thumbs/101/149329485.jpg "1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π")
. 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
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128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
77
O r r r, F F r,r r = r r F = F (. ) r = r r 76 77 d r = F d r = F (. ) F + F = 0 d ( ) r + r = 0 (. 3) M = + MR = r + r (. 4) P G P MX = + MY = + MZ = z + z PG / PG = / M d R = 0 (. 5) 78 79 d r = F d
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
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06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 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 ),
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
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: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
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V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
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2007 04 02 12 1 2 2 3 2.1............................ 4 3 6 3.1............................. 7 3.2....................... 9 3.3............................. 10 4 Frenet 12 5 14 6 Frenet-Serret 15 6.1 Frenet-Serret.......................
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数学演習：微分方程式
( ) 1 / 21 1 2 3 4 ( ) 2 / 21 x(t)? ẋ + 5x = 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ
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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
1 [ 1] (1) MKS? (2) MKS? [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0 10 ( 1 velocity [/s] 8 4 O
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: 2014 4 10 1 2 2 3 2.1...................................... 3 2.2....................................... 4 2.3....................................... 4 2.4................................ 5 2.5 Free-Body
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
1 1.1 [ 1] velocity [/s] 8 4 (1) MKS? (2) MKS? 1.2 [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0
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9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
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m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
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24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
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24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),