sec13.dvi

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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

2 1: ˆn (x, y, z) (x,y,z ) m O r m u O L L = r m u = m r (ω r )= ] m [rω 2 r (ω r ) L i = [ ] m rω 2 i r,i r,j ω j = m [ δ ij r 2 r,i r,j ] ω j = I ij ω j. I ij = m r 2 δ ij m r,i r,j I xx = m r 2 δ xx m x 2 = m (y 2 + z 2 ), I yy = m (x 2 + z 2 ), I zz = m (x 2 + y 2 ), I xy = m x a y,... (x,y,z ) I x x,i y y,i z z I ij = I i δ ij I : I = L i = I i δ ij ω j = I i ω i L = I ω. (13.3) I x I y I z 2

3 (x, y, z) I xx,i yy,i zz I x,i y,i z 13.3 ω T T = 1 m u 2 = 1 m u (ω r )= ω m r a u = 1 2 ω L = 1 2 ω I ω (13.4) T = 1 2 ( ) I x ωx 2 + I y ωy 2 + I z ωz m l m = kg l = cm=38.3 Ȧ I x = I y = 1 2 ml2 = kg m 2, I z = a M I = ρr 2 2πrdr = 2π 4 ρa4 = 1 2 Ma2. I =(1/4)Ma 2 3

4 a M a/2 a/2 I = dx dyρ(x 2 + y 2 )= ρa4 a/2 a/2 6 = 1 6 Ma a M I = dx dy dzρ(x 2 + y 2 )dxdydz = 2 dx dy dzρ(x 2 + y 2 + z 2 )dxdydz 3 = 2ρ 4πr 2 drr 2 = 4πρa5 = Ma2 2: l (x, y, z) (X, Y, Z) X = x, Y = y + l, Z = z. Z I = ρ dx dy dz(x 2 + Y 2 )=ρ dx dy dz(x 2 +(y + l) 2 ) = ρ dx dy dz(x 2 + y 2 )+l 2 ρ dx dy dz +2lρ dx dyy dz = I G + Ml 2 l Ml 2 4

5 13.5 : v ω M a I =(2/5)Ma 2 F N v L v L M dv = F, dt (13.5) dl dt = N I dω dt = N (13.6) F F P P v P = v aω v P v P 3: F h a h F N = (h a)f (13.5,13.6) Δt Δt dv Δt M dt dt = F (t)dt v(δt) = S M = v. S I Δt dω Δt dt dt =(h a) F (t)dt ω(δt) = (h a)s I = ω. 5

6 4: v >aω Δt Δt v,ω P v aω = S ( 1 M ) (h a)am = 5S ( 7 I 2M 5 h ). a I =(2/5)Ma 2 h =(7/5)a v = aω v >aω M dv dt = μmg, I dω dt = μmga. v,ω v(t) =v μgt, ω(t) =ω + Ma I μgt = ω + 5 2a μgt. v(t),aω(t) t 4 t =(2/7)(v aω )/(μg) v(t) =aω(t) 1 v <aω 6

7 2 7/5 3 (1) (2) (3) O O θ I d2 θ = Mglsin θ. dt2 I O I I = I + Ml 2 : d 2 θ dt 2 = Mgl sin θ. I + Ml2 5: I = 13.7 m 2a x x ω L = r 1 m(ω r 1 )+r 2 m(ω r 2 ). r 1 = a cos θˆx + a sin θ(cos ωtŷ +sinωtẑ), r 2 = r 1, ω = ωˆx 6: ] [ ] L =2m [ωr1 2 (ω r 1 )r 1 =2ma 2 ω sin 2 θˆx sin θ cos θ(cos ωtŷ +sinωtẑ). 7

8 L ω ω = const., L const. dl dt =2ma2 ω 2 sin θ cos θ[sin ωtŷ cos ωtẑ] =N. N L ω N = ω L : (a) (b) ω 7(b) (x, y, z) ω ω ω = ω xˆx + ω y ŷ + ω z ẑ. (13.7) L L = I ω = I x ω xˆx + I y ω x ŷ + I z ω x ẑ (13.8) dl dt = N 8

9 (ˆx, ŷ, ẑ) (ω x,ω y,ω z ) dl dt = I xω xˆx + I x ω x ˆx + Iy ω y ŷ + I y ω y ŷ + Iz ω z ẑ + I z ω z ẑ. r ω ω r r ˆx, ŷ, ẑ dr dt = ω r. dˆx dt = ω ˆx, dŷ dt = ω ŷ, dẑ dt = ω ẑ. I x ω x ˆx + Iy ω y ŷ + Iz ω z ẑ = Lx (ω ˆx)+L y (ω ŷ)+l z (ω ẑ) =ω L dl dt = d L + ω L = N. (13.9) dt d L dt = I xω xˆx + I y ω y ŷ + I z ω z ẑ L = I ω dω x I x dt (I y I z )ω y ω z = N x, (13.1) dω y I y dt (I z I x )ω z ω x = N y, (13.11) dω z I z dt (I x I y )ω x ω y = N z. (13.12) N = z ω z =Ω,ω x = ω y = ω x,ω y,ω z I x ω x = (I y I z )Ωω y, I y ω x = (I z I x )Ωω x, I z ω z. 9

10 ω x,ω y e σt I z >I x,i y I z <I x,i y σ 2 = (I y I z )(I z I x ) I x I y Ω 2. σ 2 = AΩ 2 (A>) σ = ±i AΩ ω x,ω y I z σ = ± AΩ θ ω x z I x = I y = ma2 4, I z = ma2 2 ω x = ω sin θ, ω y =, ω z = ω cos θ. L = I ω = ma2 4 ω L ma2 ω sin θˆx + ω cos θẑ 2 8: θ ω x z cos = ω L ωl = 1+cos2 θ 1+3cos 2 θ. ω L θ =, or π/2 z x y X, Y, Z x, y, z ω ω = θˆx + ϕẑ + Sẑ Ẑ Ẑ = sin θŷ +cosθẑ 9: 1

11 ω = θˆx sin θ ϕŷ +( ϕ cos θ + S)ẑ. I z = 1 2 Ma2 A, I x = I y = 1 4 Ma2 + Ml 2 C L = C( θˆx sin θ ϕŷ)+a( ϕ cos θ + S)ẑ. N = Mglsin θˆx. (13.13) θ θ = S ϕ L = ASẑ Z ω = ϕ d L dt + ω L = N (13.14) L d L/dt = (13.14) ω L = N (13.15) ω L ω L = ϕẑ ASẑ = AS ϕ sin θˆx (13.15) x AS ϕ sin θ = Mglsin θ ϕ = Mgl AS = 2gl a 2 S (Vega) 11

12 O L z ξ θ x R dv df = GMρdV R 2 R R 1: R =(x, y + L cos θ, z + L sin θ), R 2 = x 2 + y 2 + z 2 + L 2 +2L (y cos θ + z sin θ), R L [1+ 1 ] (y cos θ + z sin θ), L 1 R 3 1 [ L ] (y cos θ + z sin θ). L dn dn = R df = GMρdV R 3 L y sin θ z cos θ x sin θ x cos θ. N = 3GMρ L 3 = 3GMρ L 3 dv (y cos θ + z sin θ) dv (y cos θ + z sin θ) y sin θ z cos θ x sin θ x cos θ y sin θ z cos θ. N x N x = 3GMρ L 3 cos θ sin θ dv (y 2 z 2 ). z 2 a 2 + x2 + y 2 b 2 =1, (b >a) 12

13 dv (y 2 z 2 )=(b 2 a 2 ) 4π 15 ab2 N x = 3 5 ω L = N GmM L 3 (b 2 a 2 )cosθsin θ. ω L = 2 5 ma2 S ϕ sin θ ϕ = 3 GM b 2 a 2 2 L 3 S a 2 cos θ. ω (GMm)/L 2 = ml ω 2, (GM)/L 3 = ω2 ϕ = 3 ω 2 2 S 2π ϕ = 2 3 ( 2π ω b 2 a 2 a 2 cos θ, ) 2 S a 2 2π = a 2 b 2 a 2 cos 1 θ = 2 1yr 2 a 2 3 1day b 2 a 2 cos 1 θ b 2 a 2 cos 1 θ [yr]. b/(b a) = π ϕ = /(2 cos θ) [yr] = 4 [yr]

Gmech08.dvi

Gmech08.dvi 51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r