2.6. Сила тяготения. Законы Кепплера.

(А)

1. g = n g_.

2. F_ = 800 н, F_m = 2.76 10 ^-3, F_s = 4.8 10 ^-1.

3. g(h)/ g_ = 1/4, g(3R)/ g_ = 1/9, g/g = -2h/R = 10 ^-3, h  3200м.

4. F = Mm/r ( 8/d² - 1/(d - R/2)² ).

5. x = 9/10 R.

6. v_I = (gR)^1/2 . v_II = ( 2gR)^1/2.

7. M_s = ² a_³/  = 2 10 ³⁰.

8. g_s = 2.7 10 ² м/c².

(B)

9. F = m( k² - 1)g R_²/ R².

10. v = (R_g / 3)^1/2, T = 6 (3 R_/ g) ^1/2.

11. R = 42 10 ³ км. ( рис. 8 - ки ).

12. x = 2 R.

13. E_ =  / a [ 1/2 ( M³ + m³ )/( M + m ) - mM ].

14. m = (v + V)² vT / 2, M = (v + V)² VT / 2, a = (v + V) T / 2.

15. T =  mM (r₂³ - r₁³) / (r₁ + r₂) r₁² r₂² .

16. E_ = - K, U = -2K.

17. v_II = (g R_ R_ / R)^1/2 ( [ 2 ( m R/ M_ r + 1 ) ] ^1/2 - 1 ). m - масса Луны, r - радиус Луны, R - радиус Лунной орбиты.

( C )

18. v = ( 3 g R )^1/2.

19. R  ( 3F x / 4 m ), R  10 ⁵.

20.  =  R² ( 1 + 2 M / Rv² ).

21. r = -2 m / v² + 2 ( ²m² / v⁴ + ² )^1/2.

22. t =  (R / g)^1/2 [ 1/2 ( 1 + R/r ) ] ^1/2.

23. t = T / 2 ( 8 )^1/2 = 65 сут.

24. В 2062 г.

25. v = ( g R ) ^1/2 /2, r_min = 2/3 R, r_max = 2R.

26. v_n² = 2 M_ n / R_(n + ) = 2 g R_ n / (n + 1). v² = 4 g R_ / (n² - 1).

27. v = 70 м/с.

a_min =  a_, a_max = (2- ) a_,  = 0.01. a = a_, b = ^1/2 a_, M / M_ = ^1/2 .

2.7. Вращение твёрдого тела

(A)

1. E = 1/2mR² ²

2. E = mR² ²

3. E = 1/2mR² ²

4. J = MR² + mr²

5.  = (2Fl / Mr²)^1/2 , а)  = 2(Fl / Mr²)^1/2 , б)  = (2Fl / J)^1/2 , в)  = (2Fl / (mr² + Mr²))^1/2

6. v = (gl( + m/2)/(m + M/2 + ))^1/2

7. h = v² / 2g , v = (gh) ^1/2 .

8. a = g /( J/MR² + 1) , a) a = g /(1 + m/M), б) a = g /(1 + m/2M).

(B)

9. A = a = g(M - m)/(M + m + J/ R²) ,  = A/R .

10. a = g sin/( J/mR² + 1) ,

a) a = g sin/2 , б) a = 2/3 g sin , в) a = 5/7 g sin.

11. n = ²/4 [(1 + ²)J/mgr(1 + )] ,

a) n = ²/4 [(1+²)/(1+)] r/g ,

б) n = ²/8 [(1+²)/(1+)] r/g ,

в) n = ²/10 [(1+²)/(1+)] r/g .

12. a = F /(M + m) ,  = F / mr .

13.  = (g sin - a cos)/(g cos + a sin) ,

F = 2Ma + ma + mg sin - ma cos .

14.  = g/R [ 2LM/ (F - g(m + M))]^1/2 , при F > g(m + M) ;  = 0 при F  g(m + M .

15. cos = 2g/² [(M - m)/(M + m)] если 2g/² [(M - m)/(M + m)] < 1 , иначе  =  /2 .

16.  = 2mvr/(MR² + 2mr²)

17.  =  mR²/J , E_f = E_i mR²/J , E_i = 1/2(mR² + J) ²

18.  = (k(R - l)/mR) ^1/2 , L = (4/3 nmR² + MR²) .

19.  = (j + J)/(j + J) , Q = j/2(² - ²) + J/2(² - ²) .

20. h = 7v² / 10g , h = v² / 2g.

21. v² = 2 (mgh - Mgr)/(M + m).

22.  = 100  = 25/6 (ч ^-1)

(С)

23. a) a = g/2 [1/(1 + h/l)] , б) a = g/2 [1/(1 - h/l)].

24. A = gr (mr - MR)/(J + mr² + mR²) , a = -gr (mr - MR)/(J + mr² + mR²), t = mg (J+MR(R+r))/ (J+mr²+mR²), T = Mg (J+mr(R+r))/ (J+mr²+mR²) .

25. (t) = ₀ - t mgR/J , v(t) = t g при t  t₀ , t₀ = ₀R/[g(1 + mR²/J)] ;  = ₀/(1 + mR²/J) , v = ₀R/(1 + mR²/J) при t > t₀ . Q/ E_i = mR²/J / (1 + mR²/J) . а) Q/ E_i = 1/2 , Q/ E_i = 2/3 , в) Q/ E_i = 5.7.

26. N = 4mMg/(m + M).

27. N = mg/4.

28. а) v = 2 (12/13gl)^1/2 , б) v = 2 (3gl)^1/2

29. Сила, дейст на горизонт пов.

30. v = 4g(R - r)/(3 + 2m/M) , F = mv²/R(1 + m/M)²

31. ₁ = ₃ = /3 , ₂ = - /3 , Q/ E_i = 8/9.

32. ₁ = ₀/4 , ₂ = - ₀/4 , ₃ = ₁ , ₄ = -₁ , Q/ E_i = 3/4 .

33. Nt/m = 2v₀ (cos + k sin)/(1 + k² + m(kR)²/J) , tg = (v₀sin - k Nt/m)/( Nt/m - v₀cos) , v² = v₀² - m (kR)²/J (Nt/m)² .

34. ₁ = (3 - )/4 , ₂ = (3 - )/4 .

35. v = ₀r/(1 + m/M + mr²/J) , V = - m/M v ,  = mr²/J ₀ /(1 + m/M + mr²/J) ; для обруча: v = ₀r/(2 + m/M) , Q/ E_i = (m/M + 1)/(m/M + 2) .

36. v = v₀ (m/M + mr²/J)/( 1 + m/M + mr²/J) , V = v₀ m/[M( 1 + m/M + mr²/J)] ,  = mr² v₀/[Jr(1 + m/M + mr²/J)] ; для обруча: v = v₀ (m/M + 1)/(m/M + 2) , V = v₀ m/M(2 + m/M) ,  = v₀/r(2 + m/M) , Q/ E_i = 1/(2 + m/M).

37. u = m/M v ,  = 2 mvh/ mR²

38. cos = 1 - 3m² v²/gl(4M + 3m)(M + m).

39.  = m/M = 18/13.

40. На расстоянии 2/3l отр руки.

41. На запад. В северном полушарии дуют северо-восточные пассаты.