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 r 2 = d2 r 0 2 + d2 r 2 (13.5) (13.4) m d2 r 2 = F + F, F = m d2 r 0 2 (13.6) S (13.6) F F F S 0 S S dr 0 = v 0 =, d 2 r 0 2 0, F = m d2 r 0 2 = 0 (13.7) S m d2 r 2 = F (13.8)
13.1. 147 13.1.2 ω a m 13.2 xy y y y F F x F O x O x 13.2: x = a cos ωt, y = a sin ωt (13.9) F x = m d2 x 2 = mω2 x, F y = m d2 y 2 = mω2 y (13.10) F = mω 2 r (13.11) ω S 13.2 F F = F = mω 2 r (13.12) v x 13.3 ω
148 13 Δt Δx = v Δt ω Δθ = ωδt Δy = Δx Δθ = vω(δt) 2 2vω 2 mωv y y y x O x O x 13.3: S z S z ω S S m d2 x 2 = F x, m d2 y 2 = F y (13.13) S S (x, y) S (x,y ) x = x cos ωt y sin ωt, y = x sin ωt + y cos ωt (13.14) 13.4 t =0 (13.14) d 2 x 2 = d 2 y 2 = ( d 2 x ( 2 d 2 x 2 ) dy 2ω ω2 x dy 2ω ω2 x ( d 2 y cos ωt ) ( 2 d 2 y sin ωt + 2 ) dx +2ω ω2 y dx +2ω ω2 y ) sin ωt cos ωt (13.15) F S (F x,f y ) S (F x,f y ) F x = F x cos ωt F y sin ωt, F y = F x sin ωt + F y cos ωt (13.16) 13.4 (13.15) (13.16) S
13.2. 149 (13.13) m d2 x 2 = F x +2mω dy + mω2 x m d2 y 2 = F y 2 mω dx + mω2 y (13.17) S S y y y y y F y y r x x F y F F x x O ωt x x O ωt F x x 13.4: F (C) F (C) x =2mωv y, F (C) y = 2 mωv x (13.18) v x v y S v F (C) v = F (C) x v x + F (C) y v y = 0 (13.19) S v mω 2 r 13.2 13.2.1 z
150 13 S S S S S S S P S P r S S P S P S 13.5 t S S O l ω ω l Q dr r r +dr θ O 13.5: ω (13.20) P S P S dr = ω r (13.21) P Q t t + P r r +dr dr ω Q r sin θ θ P r dr dr = ω rsin θ P dr = ωrsin θ. (13.22) r l P r ω ω r sin θ ω r (13.21) (13.21) (13.21) P r S r = x e x + y e y + z e z (13.23)
13.2. 151 P S x, y, z e x, e y, e z P r dr = de x x + y de y + z de z (13.24) S ω ij de x de y de z = ω 11 e x + ω 12 e y + ω 13 e z = ω 21 e x + ω 22 e y + ω 23 e z = ω 31 e x + ω 32 e y + ω 33 e z (13.25) e x e x =1 e x e y =0 e x de x =0, e x de y + de x ω ij e y = 0 (13.26) ω 11 = ω 22 = ω 33 =0 ω 12 + ω 21 =0, ω 23 + ω 32 =0, ω 31 + ω 13 =0. (13.27) ω 1 = ω 23 = ω 32, ω 2 = ω 31 = ω 13, ω 3 = ω 12 = ω 21 (13.28) S (13.25) de x de y de z = ω 3 e y ω 2 e z = ω 3 e x +ω 1 e z = ω 2 e x ω 1 e y P (13.24) (13.29) dr =(ω 2z ω 3 y ) e x +(ω 3 x ω 1 z ) e y +(ω 1 y ω 2 x ) e z (13.30) ω 1, ω 2, ω 3 r (13.30) x ω = ω 1 e x + ω 2 e y + ω 3 e z. (13.31) ω 2 z ω 3 y =(ω r ) x
152 13 y z r (13.30) ω r (13.21) (13.31) ω S ω ω S 13.2.2 S S S r = x e x + y e y + z e z (13.32) S d r = dx e x + dy e y + dz e z. (13.33) (13.32) e x, e y, e z x, y, z S x y z S S S S S (13.21) S (13.33) dr = d r + ω r (13.34) (13.34) r A S S A = A x e x + A y e y + A z e z = A x e x + A y e y + A z e z (13.35) da d A = da x e x + da y e y + da z e z (13.36) = da x e x + da y e y + da z e z (13.37) S A da = d A + ω A. (13.38)
13.2. 153 da/ A(t) d A/ S ω A S S 13.2.3 S S S 13.6 S (13.21) (13.38) S S r 0 S z z r S r O y r = r 0 + r (13.39) r 0 S v x v = dr = dr 0 + dr (13.40) O y S (13.38) x 13.6: dr = d r + ω r. (13.41) v = dr = dr 0 + d r + ω r (13.42) dv = d2 r 0 2 = d2 r 0 2 + d ( d r ) + ω r ( + d d r ) ( d + ω r r ) + ω + ω r = d2 r 0 2 + d 2 r 2 + ω d r + ω (ω r )+ d ω r (13.43)
154 13 dω = d ω + ω ω = d ω (13.44) ω ω =0 S m dv = F (13.45) (13.43) S m d 2 r 2 = F m d2 r 0 2 2m (ω d r ) m ω (ω r ) (13.46) m dω r S S S m ω (ω r )= m(ω r ) ω + mω 2 r (13.47) ω () ω ω 2 r (13.46) S S 13.6
13.3. 155 13.3 13.3.1 (13.46) (13.46) z y S O r 0 λ x 13.7 λ S 13.7: z x y S O (13.46) (13.46) 0 S (13.46) r 0 O O r 0 =6.4 10 6 m. (13.48) ω ω = 2π 24 3600 =9.3 10 5 s 1 (13.49)
156 13 (13.46) S S O (13.38) dr 0 = ω r 0 (13.38) d 2 r 0 2 = d ( ) dr0 = ω dr 0 = ω (ω r 0 ) S (13.46) m d2 r 0 2 = m ω (ω r 0) (13.50) (13.46) O λ 13.7 m ω (ω r 0 )=mω 2 r 0 cos λ (sin λ e x + cos λ e z ) (13.51) () r 0 cos λ O (13.46) m ω (ω r ) O r r r 0 O (13.51) (13.51) (λ =0) mω 2 r 0 ω 2 r 0 =3.4 10 2 ms 2 g =9.8 ms 2 1/300 (13.46) F (C) = 2m (ω d r ). (13.52)
13.3. 157 λ 13.7 ω x = ω cos λ, ω y =0, ω z = ω sin λ (13.53) 13.8 (13.52) F (C) F (C) x = 2mωsin λ dy F (C) y = 2mω ( sin λ dx F (C) z = 2mωcos λ dy ) dz + cos λ (13.54) ω z d r = ( ) dx, dy, dz. x z y y x z λ 13.8: x 13.7 m d2 x 2 = F x +2mωsin λ dy m d2 y 2 = F y 2 mω ( sin λ dx m d2 z 2 = F z mg+2mωcos λ dy ) dz + cos λ (13.55) mg F m v mωv ωv v =1ms 1 ωv 10 4 ms 2 10 1
158 13 13.3.2 13.7 m L S xy m d2 x 2 = S x dy +2mωsin λ L m d2 y 2 = S y (13.56) dx 2 mωsin λ L z L x y 13.9 (13.56) x y ( d x dy y dx ) = ω sin λ d (x2 + y 2 ) 13.9: x dy y dx = ω sin λ (x2 + y 2 ) (13.57) x = y =0 (= 0) xy (13.57) x = r cos ϕ, y = r sin ϕ (13.58) dϕ = ω sin λ (13.59)
13.3. 159 ϕ ω sin λ 13.9 (λ >0) (λ <0) ω sin λ λ = 1 sin λ x = y =0 13.3.3 V F P F C 13.10: 13.10 13.10 F P F C
160 13 13.11 F f 30-40 15-25 V F C F P F f 13.11: 13.3.4 (13.55) F x = F y = F z =0 13.7 x y z z z x y m d2 x 2 =0, m d2 x 2 dz = 2mω cos λ, t =0 x = y =0, z = h, m d2 z = mg (13.60) 2 dx = dy = dz = 0 (13.61) h x z y x =0, z = h 1 2 gt2 (13.62) d 2 y =2ωgcos λt (13.63) 2 y = 1 3 ωgcos λt3 = 1 [ ] 2(h z) 3/2 3 ωgcos λ (13.64) g t y >0