2{3. 53 2{3 [ ] 4 2 1 2 10,15 m 10,10 m 2 2
54 2 III 1{I U 2.4 U r (2.16 F U F =, du dt du dr > 0 du dr < 0 O r 0 r 2.4: 1 m =1:00 10 kg 1:20 10 kgf 8:0 kgf g =9:8 m=s 2 (a) x N mg 2.5: N
2{3. 55 (b) x 1 kgf 1 g = 9.80665 m=s 2 1kg 1 kgf 1 kgf = 1 kg 9:80665 m=s 2 =9:80665 N 2 3 N kgf 2 3 (i) 2.5 mg N N 1 kgw kg. 2 kg
56 2 (ii a ma = N, mg (iii) a a = N m, g (a) ; N =1:20 10 kgf = 1:20 10 kg g; m =1:00 10 kg N ( )+( ) 1:20 10 kg g a =, g 1:00 10 kg =0:20g =0:20 9:8 m=s 2 2:0m=s 2 (b) ; N =8:0 kgf=8:0 kg g; m =1:00 10 kg a = 8:0 kg g 1:00 10 kg, g =,0:20g,2:0m=s 2 a <0 a =0 N = mg a =,g N =0 0 kgf 1 a C M m N C N C g (1)(a) (b) (2)(a) (b) (3) N ; N C M;m;g;a 3 3 1 (1)(a) N (b) mg (2)(a) N C (b) N ;Mg (3) N = m(g+a); N C =(m+m)(g+a)
2{3. 57 a b c C D 2.6: 2 2 2.4 0 2.6 a (a) (b) b c C 2.7: a b C c D 1 2 2.7 (a) 2 (b) (a) 2 (b)
58 2 0 4 1 N R 05 R< 0 N (2.29) 0 N N 2 2 R R = N (2.30) 2 6 I{2{1 3 < 0 (2.31) 4 5 F F Reibung 6
2{3. 59 2 2.8 m N = c R g (1) =0 mg sin (2) < c mg cos mg (3) 2.8: (4) > c 4 mg N R 2.8 x y x +mg sin y,mg cos (1),(2) 0 ( x :0=mg sin, R (2:32) y :0=N, mg cos (2:33) (2.32) (2) R = mg sin =0 (1) R =0 (3) 0 (2.33) N = mg cos (2.29) R 0 N; mg sin <mgcos! c mg sin c! mg cos c 0 =tan c (4) R = N = mg cos x ma = mg sin, mg cos
60 2 a = g(sin, cos ) tan >tan c (3) sin cos > 0 (2.31) sin > 0 cos >cos a>0 2 l = 176 cm h =93cm s = 168 cm t =2:0 s 7 2, 1 2 2 2.9: 2.9 1m 2 10 8 N 2.9 7 2s 2 = tan, =0:521 gt 2 cos sin = h l
2{3. 61 3 2.10 m m F F F g t =0 1 C D D E C D E 2.11 m g F N F C N C m g E N C x C 2.10: T CD N F m g D N C C N DF T DE F E D E N E y N E m g T DE T CD 2.11:
62 2 N E C N C CD D C T CD E N E D E T DE 4 1 D T CD T DE N DF C D E 2 2.10 x y O t =0 t x y y = x dy dx dt = dt d2 y dt 2 = d2 x dt 2 8 F v a C D E v a D 4 3 C D E F (x ) m a = N C (2.34) (y ) m 0=N F, m g (2.35) m a = m g, N E (2.36) C 0 a = T CD, N C (2.37) D(x ) 0 a cos 4 = N DF cos 4, T CD (2.38) D(y ) 0 a sin 4 = T DE, N DF sin 4 (2.39) E 0 a = N E, T DE (2.40) C E D 8
2{3. 63 4 4 (2.34) (2.40) (2.35) (m + m )a = m g (2.41) 1 a = m m + m g (2.42) (2.42) a (2.34)(2.40) N F = m g; N C = T CD = T DE = N E = p 1 N DF = 2 m m m + m 5 t =0 t t 0 t =0 x = y =0 v =0 t v = at; x = y = 1 2 at2 6 : v t x y I{1{2 a = dv (2.41) dt (m + m ) dv dt = m g v v = dy dt dy dt (m + m )v dv dt = m g dy dt
64 2 t v y t =0 t y =0 y (m + m ) Z v 0 vdv = m g 1 Z y 0 dy 2 (m + m )v 2, m gy =0 (2.43) y =0; v =0 t =0 0 U y =0 U =0 U =,m gy N C W W = N C x N F N F 0 N E W W =,N E y (2.37) (2.40) N E = N C W + W =0 C D E 1 3 2.12 m m 2.12: g t =0 1 [ ] L = 1 (2.41) 2 (m +m )v 2 +m gy d dt @L @v, @L @y =0
2{3. 65 y v 2 4 2.13 m m P P m C C g l 1 + l 2 C l 2 t =0 C m = M m =3M m C =4M C 2 t 1 C P 2.14 C P T 1 P P P 3 P l 1 + l 2 m g P l 2 P Q 2.13: T 1 T 2 T 1 T 1 P 2 2.14: Q T 2 T 1 P C Q Q 2 2.15 O x Q 2l 1 PC 2l 2 2 T 1 m g C C T 2 m C g 2 3 a = m, m sin m + m g; 1 2 (m + m )v 2, (m, m sin )gy =0
66 2 (x, x P )+(x, x P )=2l 1 (2.44) x C + x P =2l 2 (2.45) C P 4 4,2 =2P x C O C x P P x r = x,x P x P x r (2.24) (2.25) x P x P x = x P + x r (2.46) x =2l 1 + x P, (x, x P )=2l 1 + x P, x r (2.47) x C =2l 2, x P (2.48) x x C P v v v C v P ; a a a C a P (2.25) t 0 2.15: x C dt + x P dt =0 v C + v P =0 C v C P v P =,v C (2.24) t x dt, x P x + dt dt, x P =0 dt P v r = v, v P P,v r = v, v P P v r,v r v = v P + v r (2.49) v = v P, v r (2.50) v C =,v P (2.51) (2.49) (2.50) (2.51) t a = a P + a r (2.52) a = a P, a r (2.53) a C =,a P (2.54) 3 P P 0
2{3. 67 0 a P =0 g +2T 1, T 2 P C T 2 =2T 1 C m a = m g, T 1 (2.55) m a = m g, T 1 (2.56) C m C a C = m C g, 2T 1 (2.57) 4 (2.55) (2.56) (2.57) (2.52) (2.53) (2.54) m (a p + a r )=m g, T 1 (2.58) m (a p, a r )=m g, T 1 (2.59) C m C (,a P )=m C g, 2T 1 (2.60) T 1 (2.58),(2.59) (2.58)+(2:59), (2:60) (m, m )a P +(m + m )a r =(m, m )g (2.61) (m + m + m C )a P +(m, m )a r =(m + m, m C )g (2.62) 3 m = M m =3M m C =4M (2.61),(2.62) (,2Ma P +4Ma r =,2Mg 8Ma P, 2Ma r =0 a P =, 1 7 g; a r =, 4 7 g (2.52) (2.53) (2.54) a =, 5 7 g; a = 3 7 g; a C = 1 7 g 3 1 [ ] L = 2 m (v p + v r ) 2 + m (v p, v r ) 2 + 1 2 m Cv 2 P + f(m + m, m C )x P +(m, m )x r gg d @L, @L d @L =0;, @L =0 dt @v P @x P dt @v r @x r (2.61),(2.62)
68 2 C C (2.55) T 1 = 1 2 T 2 = m (g, a )= 12 7 Mg 5 t =0 x = x = l 1 + l 2 x C = l 2 v = v = v C =0 C v =, 5 7 gt; v = 3 7 gt; v C = 1 7 gt C t x = l 1 + l 2, 5 14 gt2 ; x = l 1 + l 2 + 3 14 gt2 ; x C = l 2 + 1 7 gt2 6 :(2.61) (2.62) a P = dv P dt a r = dv r dt v r v P t (v r =) dx r dt (v P =) dx P (2.61) 2 (2.62) 2 dt Z vr (m + m )v P 1dv r =(m + m )v P v r 0 Z vp (m + m )v r 1dv P =(m + m )v P v r 0 v P v r (2.61)+(2.62) Z vp Z vr v r (m + m + m C ) v P dv P +2(m + m )v P v r +(m, m ) 0 0 dv r Z xp Z xr =(m + m, m C )g 1dx P +(m, m )g l 2 0 x P = l 2 x r = x, x P =(l 1 + l 2 ), l 2 1 2 (m + m + m C )v 2 P +2(m + m )v P v r + 1 2 (m, m )v 2 r l 1 1dx r
2{3. 69 =(m + m, m C )g x P, l 2 )+(m, m )g(x r, l 1 ) 1 2 m (v P + v r ) 2 + 1 2 m (v P, v r ) 2 + 1 2 m Cv 2 P 1 2 m (v P + v r ) 2 + 1 2 m (v P, v r ) 2 + 1 2 m Cv 2 P, m g(x P + x r ), m g(2l 1 + x P, x r ), m C (2l 2, x P ) =,m g(l 1 + l 2 ), m g(l 1 + l 2 ), m C l 2 (2.63) (2.46)(2.51) 1 2 m v 2 + 1 2 m v 2 + 1 2 m Cv 2,m C gx,m gx,m C gx C =,m g(l 1 + l 2 ), m g(l 1 + l 2 ), m C gl 2 C (2.23) x =0 4 2.16 m P m P Q g 4 P Q 2.16: N N R R 3 4 5 C m m 0 2.17: 2.17 N 0 g 4 4 a = 1 2 a = 2m, m g; T =2T = 6m m g m +4m m +4m N C
70 2 1 2.18 N R N N m g N R C 2.18: C N C R N N m g 2 x y x v v ; a a R a R v = v ; a = a b R v 6= v ; a 6= a 3 a R a = a = a N C (x ) m a = N, R (2.63) (y ) 0=N, m g (2.64) (x ) m a = R (2.65) (y ) 0=N C, N, m g (2.66) b R (2.30) R = N y x (x ) m a = N N (2.67) (x ) m a = N (2.68) 4 a R (2.63)+(2.65) (m + m )a = N (2.69)
2{3. 71 N a = N m + m (2.70) N (2.64) (2.66) N = m g (2.71) N C = N + m g =(m + m )g (2.72) (2.65) (2.71) R = m a = m m + m N (2.73) (2.29) R < 0 N (2.70) R < 0 m g (2.74) (2.71) (2.73) m m + m N< 0 m g N< m (m + m ) 0 g m N b R m N < (m + m ) = 0 g m (2.71) R = m g (2.75) (2.67) (2.68) a = N m, g; a = m m g
72 2 5 O t =0 x =0 x =0 v = v =0 a b 6 a R (2.69) v dx dt x a = dv dt (m + m )v dv dt = N dx dt t v x 0 v 0 x (m + m ) Z v 0 vdv = N N =const. 1 Z x 0 1dx 2 (m + m )v 2, 0=Nx (2.76) N 0 ) b R (2.67) v dx dt (2.68) v dx dt m v dv dt + m v dv dt =(N, N ) dx dt + N dx dt (2.71) N = m g 1 2 m v 2 + 1 2 m v 2, 0=Nx, m g(x, x ) (2.77) x >x 2 N 100% 2 5 C m m N C 2.19:
2{3. 73 0 2.19 N 0 N N l g 5 3 4 2.4 3 4 2.20 f l 1 x 1 l 2 x 2 2.21 f > 0 5 5 N = 0 (m + m )g; m gl x 1 > 0 x 1 +x 2 > 0 f>0 l 1 l 2 f<0 x 1 < 0 x 1 +x 2 < 0 2.20:
74 2 f x 2 l 2 K f x 1 l 1 f = K x 1 l 1 = K x 2 l 2 n l 1 l 2 l n x 1 x 2 x n l = l 1 + l 2 + + l n f K x 2 l 2 x =x 1 +x 2 + +x n K x 1 l 1 f K x 1 = K x 2 = = K x n l 1 l 2 l n 2.21: = K x 1 +x 2 + +x n l 1 + l 2 + + l n K = k l x x >0 x <0 = K x l,t x T x,! T T x,kx (2.78),,! T,T y (,kx < 0 ;,kx > 0 ) 2.22: 6 k 6
2{3. 75 2.22 T 0 6 2.23 m x O x 2.23: k 1 O x x>0 x <0 x,kx 2 a x ma =,kx (2.79) a = d2 x dt 2 m d2 x dt 2 =,kx II 3 t =0 t =0 x v =0
76 2 4 (2.79) v dx dt a = dv dt mv dv dt =,kxdx dt t v x 0 v x m Z v 0 vdv =,k 1 Z x xdx 2 mv2 =, 1 2 kx2 + 1 2 k2 1 2 mv2 1 2 kx2 = 1 2 k2 (2.80) U (x) = 1 2 kx2 (2.81) x =0( ) ( 2.4 (2.80) v x v xy! x 2 a 2 + y2 b 2 =1, O x x = a cos ; y = b sin,! k m =!2 (2.80) x 2 + v2 2 (!) =1 2 2.24: x, v x v 2.24!
2{3. 77 t t =0 x = cos 0 = ; v =! sin0=0 x v (i) (ii) (iii) j,j (iv) (v) 7 =!t x = cos!t (2.82) T x = cos 2 = cos!t T = 2! =2 r m k (2.83), p v =! 2, x 2 =! sin!t!t < v<0, v =,!sin!t (2.84) 8 (2.80) a = k m x =,!2 x =,! 2 cos!t a, t v, t x, t 6 2.25 k m g (1) (2) O 1 2 kx2 (3) 9 7 8 v = dx dt = d cos!t =,! sin!t dt 9 6 (1)2r m mg mg,(2) (3) k k k 2.25: