1.

1. , . , D C , A D . . 148

o d e o b c ' . , , , t A B C > 1A D C -

:

t A D - V Y vd ^ 2sK

g r n r sin a D C = ft = 0D*DC + ~ ~ ^ 2 ------

TOD 1 / D , 2fl

g sin a V g 2 sin2 a ' g sin a '132

,

ADC AD> + ' ~ 5 ^ - ( K l + s i n a + s i n a - l ) .

t A B Q :

t , B = ^ , o R = \ r 2 g h sin a, g s i n a B

BC

i ,ABC

- v* + V g2

= / 2 -1 sin a Y sin a

+ V\ + sin a j .

( ', f 2 h i sin a

' 1a d c ~ V g sin a ( V sin a + 1 V 1 + sin a) > 0,

K s i n a + l > y ^ s i n a + l , , , .

2. t

, L . , g cos a, L = D cos a, D . ,

g

.3. h ,

a = g , t = j / ^ - = 2 , t = t ,

= At - 8 / 4 -

2 ?4.

h _ gt* g L \ 2 2 v 2 t , v :

, 2 2 W Nv m ~

133

m , N 6,02 1023 \ / .

=1 ~ 112,1(5-. t

at2H = + v t sin a;

*' sin a (. 149). L :

L = v t cos a.

t,

, 2 cos a sin a , 2g HL ~ i IK 1 +

6. , (. 150).

, , . 2 sin2 a --------'

sin a . ,, 2, 3 , a = 60, 45 30,

( : 2 : 3 = (sin 60)2 : (sin 45)2 : (sin 30)2 = 3 : 2 : 1.

t :

L :X)

L vt cos a = sin 2a,

L t : L 2 : 3 ** sin 120: sin 90; sin 60 V 3 2 *134

7. t (. 151)

v t L cos .

g t 2 , .

-Sg = L sin a .

, 22 sin a

~ g eo s2a .

8. v

v - V ^ g h . (1) ,

. , , , , , , . , (. 152).

. , , , (. . 152). , , 0. ,

~tgct. (2)

t ,

e t 2y = v l cos 2 a -----, (3)

g t 2 = v l cos 2 a ---- g,

135

: = v t sin 2. (4)

(3) (4) t , :

= ctg 2 - j j h r . (5) 22 sin2 2

, (5) 0. (2) (5) 0, :

= 4Atg2a s in 2 2a ^1 + )

, sin 2a = 2 sin a cos cos 2a = cos2 a sin2 a,

8 sin2 a.

S, , ,

S = 1 1 = 8 sin a = 4/i = 8 . sin a

, S g 0. .

9. h i ,

2hi = v T ~ l ~ ,

v . ( i ) ,

B ( T - t Yh2 v(T t)------ ----- .

( = 2,

v

10. , , . . , .

, ,

,1= 2 = 2 "135

t (

/ _L + 1/ 2 ( H i - H t )1 8 g

/2 t 2 v 2/ g . ,

to = vl v2 + V1 ~ v2 54 .

11. (. 153)

L .

v x = v cos , V y v sin

( ,

^ = l v t cos . , , .

t , = v t cos , ]

=* v t sin v t 2 COS I 2l

) v t cos ,

t = , = 0, ~ L , ,

L = v T cos ,u.vT2 cos

0=07" cos --------- --------- ;

s i n c p u J 2 v = L / v cos . , = 30, 35 . , ^ 2 v , .

12. , , ( , . 154). , , , ; (

137

). , , 0 \ ( 0 1 _L ) . , , , , -

. O D - , ( O D _L ) , B O D , ( ). ,

cos =

. 154.

(1) (2),

v ..= 583 / .

O D

O D

c tIT'c t _

lit

( 1)

(2)

13, ;V h2

// =

2

c tY~- 10 kmv ( 12).

14.

(. 155) :

sin = T L cos ,

L ~ ,

sin = cos ,

- 2 2) = 2 - -

^2 j == sin 2.

, sin 2 sin 2 4 cos

15. , L \ . ,

L i L 0 = , (1)

138

L o , . , Z.j/2. cti ,

Li La2

ctP2 = , ,

,= 2 ' ( 2 )

, . L2:

L2 = (~ + )+ ( " + '2)' ()

, . (2) (1) (3) ,

L a 3 L 1 2L2.

16. , , . 156: , N . :

2 m g = 2 N cos f5. (1)

m g + N cos = cos , (2)N sin (3 = sin . (3)

(I) (2) (3) (2),

tgP = 2 i g a . . 156.

, . .

17. . 157 , : , , / , . , N , . , / . ,

/. (139

, / + ? cos 30 = F cos 60 (2)

( ). F Tp = k F . (1) (2) (1) (2) f Tp,

k = cos 60 = 11 + cos 30 2 + '

.18. L /4

.

, :

F h + M g H sin ^ M g x c o s , (

F ,

F = m g sin a, M g = , m g = .

(1) :

g ( m h + ) sin ^ cos ,

tg > M L / Am L + M H

0,4,

( 2 )

. . =10 .19. ,

( . 158), , . . . , , R / 2 . , a in

140

,

( 2 ) = ^ jtj.

. , 1 ,, (

),

R_6 20. .

(. 159) = ( R r ) ,

( ). , . .

, .

21. ,

. 160.

f , (. 160), . .

R F Tp = r f . (1)

, F Tp = m g k , a f = m g k + , . (1),

k

(R ~) g ' (2) . ,

, , . (, , 14), , .

141

22. (. 161) , . , . cti ,

, P t , , ,

( + ,) Sin ) == k ( + P i ) cosaj. (1)

t g a i = k . (2)

= 2, ( ) . ,

, , , . . ( + P i ) sin a2. , , ); . , a = a 2

R ( + ,) s i n a 2 = - |p P , . (3)

, P t = = 5/8, (3)

sin 2 = 1039 1

tg a 2 0,26 < tg ,.

, .

, , , , , . (. 21.)

23. sin a = 1 /6 ( 22).24. , , . 162.

, m g .

142

L , (. 21)

3 , 1 ,FA-4L = m8 j L

(F , ). , , S.

Fa = S T P*?' mg = S L p g ,

' -

25. !

V = Pil7i + 2 2

V i 2 , ; , V = V \ + V 2,

V. = V ~ - - P l - P :

26. V , , , . ,

V t'Pa- .(1) , v h

v = S h . (2)

, , .

p = ^ + Z 1, (3)

. ,

= Peg ( / / + /;) (4)

(3) (4)

- + = PbS ( H + h ) .

143

, , = p Bg S H , (2). :

//] = / / + 4 = - = 15,25 .

27. , F x F 2

,

P + F 2 = F u

-^-rtD2Apg + -^-n(D2- r f 2) H p Bg =

= ~ rtD2 ( + h ) p Bg .

- - _

.

28. , , . , , -, , -, Mg sin (. 163):

N x + N 2 = M g cos ,

N x ~ = N 2 - ^ + F TpH ,

FTp = M g sin .

N x = M g + . sin j o,56 T,

N 2 = M g -- sin a j 0,44 T.

29. .

F T N , FTp ^ k N . , .

Nk N '

= ctg a.

144

30. , , , . , . . 164, , :

= N \ cos ,0 = m g N i sin N .

, sin = ( R H ) / R N 0,

(2R - H ) S '

31. , . A'j m g (. 165)

= N 1 cos N 2 cos , m g = 71 sin + N 2 sin ,

cos = R / 2 R , = 60. ,

N , = ( w + )'

N 2 \ = \

> g ! V 3 ?32. , ,

, . D B (. 166) .

: = p g h ,

2 L S a = g h S , h 2 h a g

10 . . . 145

, h . . > 4 ^ - . -

=- + , I

2 - 2 l ^ + S (/ + A i) + + pgfti],

h + 2 h { = ;

+ 8'33.

= k m g ,

, :2

,2 /

1 =2 g k

100 .

34. ( \ + t n 2) = F . . , , , 2, , , = / 2,

n t i + 2F = P 2

2, n i i + 2

= mi35.

F ( . + _ + ) g 188,8 16 9,8 / 2 = ------ -------------------- = ----- :---------- ------------------- = 2 / \

+ + 16

/ = ( + ) ( + g ) = 106,2 ,

/ = [ + Y ) ( + ) 82,6 .

36. k . k - M g \ -

~ M a i = k ~ M g , , = k g ,

146

, v v 0 k g t . , ,

*1 = 2k g '

. F T : F T = k M g . 2 :

2 = F F - : k M g - ~ k M g = ~ k M g ,

= ~2 kS

v 2 = v 0 + j k g t ,

t \ v2 = v0+ j k g Vq.2 k g 4

37. , , , . \ 2 (. 167):

1 = n i j g , 2 = m 2g ,

. 2.

2 = jin lin2g 111 1 + in2nit m2 m i + m2

= 4,7 , T = 2,35 h,

g = l,9 6 M l 2.

38. 37

-nil + ; 3,

, a m i m 2 (mi > 2). t /2. ,

= 2 ** 147

,

1 2 1, 11.

39. : ^ = ig ,

m 2a = m2g - F Tp,

(, - m 2) g - F 7p) + 2

1= -g g 1,6 / 2,

= ,2m2g + FTp

ft!) + /2 4 1 7 .

40. (. 37). Af, 2

,, =,Mig-T, 2 2 = 27' M 2g .

, , = 22 (?). , 2 Mi-

2 + ~ 2g-

27) > 2, \ .41. Af ,

Af. () ().

------------- g ,

m. s t , , t = Y %1/ = 4 .

42. (. 168). , , . F

F = ,

, . F , ,

F max = k P cos .

F ~ sin .

148

, (Af )

n k k g Mmax~ Y T T W 7 T + F

k g M g

Y T + W m ~ 2* ,

.

43. , , , ,

= = k g - 0,98 / 2.

, .

44. , ,

- --------- ,m

m . , , v = Y 2 g h , I,

, v 2 2 g h m P k r/== 2 ~ 2 ( - ) ~ - ~ 15 . 169. 45

45. . 169. ,

= kF. ,

F Ala,

,

( 1 )

(2)

. , = v / t . , , v /?,

=/?

t (3)

(1), (2) (3) ,

k F = /?

//? k F '

( 4)

(5)

149

, S = a t I 2/ 2, 5 = 2R n ,

2nRn ~ a t 2 2 * ( 6)

t , a t 2 _ R t 2 Mu>2R

4 n R 4 n R t 4 n k F

46. ( ), . , , (/?)2/2,

(>/?)2 2 = k F 2 n r n ,

, . ,

2/?2 4 n k F r

47. : m g , N F Tp,

. 170. ,

m g sin = m g k cos ,

k . k = tg .

> /:

{ = F F Tp m g sin p - F Tp,

F T p = k m g c o s p. / = ,

a = f ( s i n p - t g a c o s ) = g sin ( a) cos a

7 = //sin , t

I 2

t = [ = / " g si

2h cos asin p sin (p - a) *

150

48. . 171, ,

FT = k m g cos ,F = m g sin ,

, F Tp , F () m g ^N .

/ F,

= g ( k cos sin )

, ,

^ 0 ,66.

49. , ,h

S--

+ g sin g cos

sin . 171.

, , . 170. , F Tp

F T = k m g cos ,

.

= m g sin k m g cos ,

= g (sin k cos ).

v 0 =* V 2 a s = V 2 g h ( l k d g a ) .

. a t (. 172)

F + m g sin , = --------------------= g (sin + k cos ).

'

Sj , ,

hi Si

0 .sm = sm =2

N = m g cos , F Tp = k N k m g cos a.

2m g m g sin a m g cos a,

a = g [2 (sin a + cos a)].

, sin a + cos a . ,

_ . / \ s m + cos a = 2 sm cos l a j .

, = /4. ,

Omln 8 (2

J r ,

m

F

. 174,

51. (. 174)

MaZ FT ~ l ~ F'? (t t l Q , - j p , . , F Tp = t n g k , (1)

F = ( + ) + 2m g k = 24,5 .

52. F , , F = m g sin a m g k i cos a = m g (sin a k i cos a)

(. 47). F r F t = m g cos a (sin a k \ cos a).

, F r , . ,

k N ,

N . , N = m g F sin a,

152

F sin a F . F ,

N = m g ( l sins a + k x sin a cos a) = mg cos a (cos a + ft, sin a). , F rp ^ F t , F T F r,

1 ^ ft tg a + 1

53. ,

M g ,

= ( F r + F ) ;

, FTp , F = Mg cos a. , F T P = * k M g cos . ,

2 = Mg + f T + F = Mg (1 + ft cos a + sin a).

, /2 . F :

- ) -54.

, = , (1)

. m : m g , . m , , . ,

t n g T = m ( a b ) . (2)

(1) (2) m (g + b)

M + m '

. t m g - M ba b ~,------ M + m

, , , M b m g , 6 = 0. m . > m g , b < 0 .

153

. , (2 ) :

(g + b) = .

, - , , , g .

5 5 . , b .

, ( g ~ ) +

+ b =mg + Mb +

, g .

5 6 . , 5 4 5 5 . 2 . .

2mg = 2 m ( b); (1 )

. m m g = m ( + b). (2 )

(1 ) (2 ) :

2 T ~ j m ( g + b ) ,

. . , 5 4 , , 6 .

5 7 . 5 6 . :

2 < = ^ m ( g - b ) . 5 8 . , , - 1 /0 ,

. 175. N , F . ( . . ) , , , , . :

N c o s + F s in mg = , F c o s N s in = .

F = kN. ,

_ ft c o s a s in ft t g c o s + ft s in 1 + ft t g

fc = t g a , .

154

59. 58, , (. 176). , , ( ), :

= N sin + F cos , = N cos F sin m g .

F = k N . ft + tg = ---ui 1 ft tg

ft t g a = 1, .

69. , . , ,

F , , . , = S i p , , S , / .

N

TTlff. 176.

= v / t .

, 2 18 /

0,7 103 -^- 0,3 60 63 -4-.61. F

1 - F = m v n , = 1 0 \ / . ,

F = 10-2 8 102 / 10 1 / = 80 .62. ,

, : ( m v ) = F /, (1)

, () , A t . , , , (1)

Ant , At. (. . ), , ,

= v S p At , (3)

(2) (3)

F v 2S p = 98 108 ~ 10 .

63. . , ,

1 . .

= m v S a 300 .64.

.

F p t = m v , m , /,

Fp ~ v = 0,2 ! 20 ! = 4 .

N , , . 177, F :

' = = F-v + 2 20,4 .

, , 65. m g h

m g h = Ei l + 2 S , (!)

m , E] . 2 , I , ,

hE, = mgfecosa, / = , F z m g k .1 sin a 5

(1)

ctg a ) 49,05 .

156

66. :

IIWq2 = k m g l cos + m g l sin .

k

,nvo , , , 4~ 2 ~ = k m g l o , k = ,

, ,_____ v}h _____Uq cos a + 2g l 0 sin a

67. ( , ). F,

a i q r \ = F I .

. . :

a 2qr\ F I + M g h l ,

2 . ,

2 1 =M g h l

q 2,2 .

, -

68. . 178 , . ,

F = F -jp + t u g sin a, F rp = k i n g cos = m g cos

( , fe=l ) .

p = F v = m g v (sin + cos ) = 2 m g v sin cos ^-5- a j .

a . - -,

= m g v V 2 .

157

69. .

mv2 2 + P s = F s .

m = P f g ,

F = l) 2,5 10s 250 .

, < F , P s . , . , F , ( ) . F .

70. F . W , . W = F S , S . ,

W Qqr\.

S ,

W Q q 4 S S

3- 4 .

71.

( F A ~ m o ) h ~ W ,

W , a . h u

(^ ~ m &) h W m g h it

W - F Ah - n t g (A + AJ.,

f a = J n R 3pg,

p , ,

W = 219 520 2,2 10 2 .

, .

72. , //4

158

. //4 , , m g l f 4. 2/2.

2 m g l . f g l ,~ Y ~ = ~ 4 > v ~ ~2 ~ ^03 ! .

. .

73. . , , ,

( + ) -g - = M lg l + M 2g - 2~,

v _ y f M^gL + VAjgLMi +

74. . , , F (. 179).

= 1 / 2 = 9,8 4 / 2, 10 (?); , , . 10 , 5 , . , , . F 0 = P S = 9,8 102 ,

~ 10 980 + 5 980 = 9,8 103 .

73. , 74, . . , , (, 180).

159

.

mv2 1 .- - = m S k L + m g k x ,

= 0 ,84, , ,

S = L + = 2 + 0,84 = 2,84 .

76. . 181 , (. 74 75)

= F a + F a = F a ,

77. , ,

W - = S L ( > g ( H 0 + H ) , (1)

S . W + , , 7476, .

: , F a , a F Q= S L g p 0 , , W + . 182

W + = 0S L g ( 0 - L ) + P--~ g - . (2)

(1) (2), :

=

- 0.

. , , . . L < 20 (1 /0). , .

160

78. , , , AF = k M g A x / L . , , s , . . 183 s ( ). , , ,

k M g ,

. . k M g / L , . 183.

, k M g / L '

= k M g a L .

79. S 2 = 2S,, L / 4 L / 4 . , , 1/4, , , L / 2 . h . 184. , 7477, :

^ " ( /?A - ^ T T (p- - p)gZ's iT T -

8 .

, ?(,?! + 2/2) g = 2 ~ + jj ,

, 2 , 2. ,

/, 1 . . . 161

.2 2g(m1?i + m2y =

, + : 2 /?

81. , , ,?, > 2/2. , (. 185).

, ? m 0v l 2------1------ 2 '

/77?0 0777,

|/77 , i

. 185.

, t>2 , 2. , : / , = 2 : /2,

. .2 2 g ( O T , / , - O T 2/2)

1----------/| *

82.

/,:

m i g h t . (i)

, , . , , ,, E Ki, , , E Xti

, ? A m v i/ifirAo = 2----- ------1 " (2)

, , 2 . , , , , . :

4 m v F _______ 2

, ( ), . ,, ,. , ( = E Kt) , , ,, 0. , ,, 2

162

. , , . i ( ) >2

( , 2) g h =4 m v

+ m, v;2V2 (0

v , V i v 2 m, m, ra2 . ,

V - V , . (2)uj 2

(2) (I), 2 2 (,-'1

/) gAI2

. 1 + 2 + 4 ~ ^

v \ = 2 , :

12 *, + 2 + 4

. . 187. rri\

t ,

t r i ia = m t g T v

2

2 = 2 m 2g .

( 3 )

(4)

, T j 2 . a t :

^ . (5)

. 187. 2 ( T i ~ T 2) r .

F m , . .

{ T i ~ T 2) r = = F l , (6 )

F 4 m a i . (7)

, , , (6 )

161

. (3 ) - (7 ) .

84. 1 h ( ). N \, ,

N \ = p Q g h .

, v = Q / S . N l t ,

N N1 +

85. W , = m v 2/ 2 , , 1 , . , .

W2 2 (2)2 2

8W7,.

8 6 . :

, = 3 = = V % g R

, ( ). :

(. 188)

= R - ~ R , co sa = - | - ,

a = arccos 8

165

87. P i , , :

m v o = m v + M v t ,

, ,

W = - ~ M | [

( ">-_ 2 ( - )1

2

Q =

IF, = > (pq ~ 2)

J L , : 1 -

mv "2 Vq Vq + v

2 ( 0 - )2 2

= 0,998,

;

1F IF, Af v 0 + = 0,002.

8 8 . , , . .

V( + ) V m v cos + 3

89. v t =* = m v / M 2.

Wl ~ (1) ,

2 = - . I + 2

w m V 2 2 ( M t + M 2) '

(1) (2) Q , :

(2)

Q = nrv Af.2 M 2 (Af, + Af3) *

90. = M V ,

V , . = (L h ) / t , V = h / t , t , . v V ,

, ! -- 1 ,

+

, .

. h , v . m v = M V . , , . . , .

91. , , , (. 189):

[(otj + m2) v ] 2 = (OTjCj) 2 + ( m 2v 2)2;

v=^ r h ^ r(miVi)2+(m^ }2' , tg = m 2v 2/ m l v i .

m2V2

WW

7 )' / + )

. 189. . 190.

92. , . , , 0 (. 190)

167

, , , . , .

. , , . ( . 190) . , . , , .

93. V :

= (- ) V , V m v / ( M + ).

h , (. 1ST), :

(Af + ) V 2 2

=

= (- ) g h ,

I I2 g '

:

cos : h (4 + m ) 2 2 g l '94.

4 [, = M 2v z,

. 191.Afjtij 42|

,

, . v \ v 2 M i 3.

, V2E l'l = -- - - -__ f V2 =-- - " -- {\ + 2\ {\ +

95. RaC M V 2

---- 2 , 4= 0 = 2140, 0 , 1,67 -10 ^ . V :

M V = ,

168

, : = 9,1 I0 - 2 8 . ( . )

mev1 2

W.

, meW

1 ,3 - 1 3 = 1,3- 20 .

96. , , a mCd ,

mnv = {mn + mcd) .

, = 1,67 10~ 24 , mc d = 112,4 ,

: W,

V = J - 2mnlV- 104 /. + ,Cd97. , 0.

m v 0 4 [ m v t ,

mv ,2 4 m v \ m v 'h 2

4 m , , , 1 . ,

30 |" 0-

m v { 4 m v 2 mo2.

m v \. / 2 2 4mo* t t i v i

2 4-

2 1 2

3 9v 2 -

5 Vi 25 0-

, 625/81 7,7 .

169

98. (. 192):

m D V D = V He cos ,

mnVn ~ mH e - F H e s in *

m nvl , " ' ^ m DV I 2 + ---- 2 --------- -------2 ~ + Q-

,

= m nV l " 2 +

Q + ~ 1 2 .

99. , ,

, .

| / ~ 2-|- g l cosct = -^y-, ______

V z g iv a => v cos a = -------

(. 193). h , ,

. ^ 32 g = 8 g 8 '* 100

100. . , v t t>2 ,

m v i + m v t = 0 , v i = 2.

170

, ,

22 g

\ (2 0) 2

+ m g H = ---- , ^ . , 2 =

= - v 0 V 3 , , . t t 2 :

( . , +

101. , , , , , .

I = 3 + + 3,

f \ \~~2~ ~ 2 2~ *

0 2 + 3 , a + . 2 = 3 ( \ + |= ( 2 + )2~ 2 2 3; 2 3 2 + 3 , 3 = 3).

= + 2 2/

o ? = ~ T - 2 o l 1 m 2

, Oj 1500 / .

102. V m i

1 / 2 = 2 g h = 2 g ( l , - l i cos a) = g t t ( 1 )

( a = 60). , :

m i V m \ O i + m3 s 3. ( 2 )

171

,

,2 I(3)

(2) (3) n i l trij, + 2 V,

2, -f- m2

< 2, V . , h i = v xj 2 g . h 2 v \ ! 2 g . 2 (. 194),

, :

//,

COS 1 = 1hx h C O S 0 2 = 1 - (4)

" 0 0 -fl)f. 194.

.Sf1 , , (5 (8) (9) >|S. [Q0 mi = 2OT2 = 6 ms.

3 > 2 > 1 .

107. v (. 196):

m V i 1 - m V 2,

v = J ^ ( V i - V 2).

W , ,

W--m V ? m V \ M v 2

. 197.108. . 197.

:m v = * M V + m v lt ( 1 )

V , vt . V2 = 2g h ,

=

v j = 2 g x ,

{ m v - V 2 j h ) 2 2 g m 2

. ( 1 ) , ,

174

(), . (1), :

FTp - M g = . (2)

FTp , t .

v v }FTp - m g = } (3)

( Oi) . , (2) (3) (1), M g m g Fxp.

109. , , . , . Vi , 2 , V ' ,

m v \

m v i = M V , 2 =* M V ' ,

m v l M V 2 ME ' 4 + - - +

( 1)(2)

(3)

. v V (1) (2) (3), !

M 2V 2 t n v \ M V 2 m2 t |" 2m ' = 2~ + ~ + 2M '

M 2 ( M ) 2 2 ( + ) 1

2 2 ( - )^ = 2 g ~ 2 g m 2 ( + )

. , , , , , . , , ,

175

110. :

mgh -

/ 2 i , mv \ =* MV\ (mv2:= MV, (2)9 M V 2

,2MV'

2 + 2 + 2 (3)

] / 2 gh ( ) 2 ( + ) '

111 . , . vt . c o s a Moi = m(v c o s a t>i),

m o c o s a M + m

112 . , + , ,

, ( ):

,

m g h ~ ^ +

2

2

1 + ! )1 1 3 . ,

;

0 = ( + ) j , (1 )

, 0 , v 1 , .

:

mvg ( + ) u f~2~ ~mgH + ----- ----- ( 2 )

0 = 2gH (M + m ) ~ 6,9 /.1 14 . , 2 =

*= V2.gH; I, ,

1/6

, . 3 t 0. , ,

t + t = ^ k x = to=tV2gH. ___ L + t V ^ g H .

115. , . v . , ( ). i = ( ) + = = 0 + 2. ,

(2 - )2 2~~m , m A u v , 22 2 +~~

2~2~ 2 m u v .

< , u ! < ^ u v , . , ,

| [ _ 2 m u v _ 2

2

116.

.

\ = 2 JR, 0)

2

m , ' . : ____

117. (. 198), ( m g N ) . N = :

2m g = , R

, -2 ,

= - , ( 1 )

v Y R g = 2 / .

118. ( 2, , F H . () 2, 22, ,

12 . . . 177

, nti(a2ri FH, 222 =

= 2 2. 1 + I,

_ __ 1f l + 2 1 + 2 '

2 .

- -(/.\ + 2

= 4- 2 9 2OTj * 2 /*2

2 + 21 |22 (-f- 2 2/2.

119. , , ,

. 199.

( N ) (. 199). ,

f rtuo2R ,

to , R = , . :

I 2 _ . ./ = m g tg , , R = L sin ,

, 4%2L sin 4 2L cos i~ tg a ~ = g

120. R = (. 200)R = L sin a + d .

, ,

/ V L sig* g

sin a + d 8,3-

121 . , . , . 2 0 1 , , m a2L s in a = m g t g a , c o s a = g/ g / L 3 ! . , , >! = 2 / , , 7 = 7 ^ = 100 . , , ( ? ) . = =* mg / c o s a = mco2L = 160 .

122. , , . m g mv2/R (v , R ) .

v = V R g ** 8 !,

1 2 3 . m i 2 , \ 2 , : m2 = r2 : r t. , m 1 + m 2 = 2 M , , R,

24, ...2 = (1)

, .

( 1 ) (2 )

4 2f2 i = V

\R 2

42 2 2 R * ~ '

(2)

(3)

:

4 2 = (/2)2 R* (4)

R 0 , m3 + . (3) (4) R 2R a 300 10 .

12* 179

124. m g N (, , ). m g N--

mtr -jr- .

V

m g, , N = , :

. 202.

2 2 8

127 .

125. , () N F Tp (. 202).

,

() kN mamaxi

N

p - N = f ~ J2 '

= ft (g - = 2 ,9 4 /?.

126. > . m g N , , .

m g + N =

127. N , (. 203). , .

- - 1

o 2 = 2 g h , a h = L (1 cos ), = m g (1 + 2 2 cos ) = m g (3 2 cos ).

/0,

3 fo 1cos =2 m g 2 . . = 60.

= 90, cos = 0. / 0

/o = 3mg = 2,94 .

128. , , . ,

N m g cos = .

2 = 2 g h , h , . .

h = L (1 cos ) L (1 cos ) = L (cos cos ).

N = m g cos + 2m g (cos cos ) = m g (3 cos P 2 cos a).

, = 1,6 .129. , , ,

m g = m w ? R , ( 1 )

to , R , . g

m g = ~ (2)

, .

= 4 '3 t f

(3)

181

(1) (3), ,

, :

mvo ...

R , , 0 . ;

- . 204. N . , , . .

ffir2m g cos + N = (4)

v :

m g H = + m g R (1 + cos ). (5)

(3), (4) (5)

N = 3 m g (1 cos ), (6 )

= 0, N = 0 .133. ,

, m g f . , , . / m g :

} + m g = 2

( 1)

. :

i>2 = 2 g ( t f -2 r ) = | - g t f . (2)

(1) (2) / + m g = 2 m g . (3)

183

, / = + m g , , , ,

/=11 ,76 .

134. m g (. 205).

(6 ) 132 = 90,

N = 3 m g .

F , , N m g :

F = V N 2 + ( m g ) 2 = m g V T O .

, F = g .

h (6 ) 132:

m g = 3m g ( 1 cos ).

co sa = - g , h R (1 + cos a) = R.

135. ; , m g . (. 206).

136. (. 207); N ( ), F , , m g ( ).

F ~ k N , (1)

k . .

184

( ' . 207) to2R cos , -. .

N cos F sin = m a F R cos . (2)

, :

m g F cos N sin = 0. (3), m g c o s a F = 0 . , to2/? cos a sin . m g cos a F , , : m g cos F = m

138. ) .

L , v , . 2 = 4g L .

) , . m g , (. 209). -

I . (. 54).

140. m g , N F (. 210). ,

( 1)

v J j L , Vi , = 0

/

2niVym g = - j . (2 )(2)

/ ] , . 209 . , ( 1 )

( 3 )

. 209.

(2) (3)

v = V 5 g L .

139.

{ ). , , ;

2R (2)

F d sin , N N d cos ( d ). ,

F d sin N d cos .

(1) (2) (3), 2tga

(3 )

R -g

8,5 .

. , , . , , (. . 210) . , , -, , , -, .

141. , (. . 210). , :

. 210.

,

N = m g , ( 1 )F = k N , (2 ) m v 2 F ~ R * (3)

N F ( ) :

F sin = N cos . (4) (1) (4)

ctg = k = 0,4, v = V R g k 18,8 / .

90 22 (. 140).

142. v , 140 141:

v = V R g k , (187

- . , , . 211. JV

, m g , F

F = k N . (2)

F N

. 211.

F cos + N sin = -mv

R(3)

1 .

, :

N cos F s i n m g * = 0. (4)

(2) (4) fe + tg g ! klga

, ( 1 ),

Qi _ - \ f fe + tg v V / ( 1 f e t ga ) '

, Vj , (2 ) (. 140).

143. , , . 2 1 2 . : F Tp = m g . ,

Ftp - k N( N ). N :

R '

N = -

. 212.(. 140).

144. 141:

v

Rg'

k , -

( 1)

188

, , . 213. ,

F Tp = k N . (2)

,

F Tp - m g . (3)

N ,

JV = -m v

Ri ' (l) (4)

2 Rit _ RiRe2

(4)

vv

, - N F T :

F Tp d sin = N d cos (5) . 213.

( d ). (2) (5)

, , v2ctg a k = f r

(. 140).

145.

, R + R 2 . , : ( T / T 0) 2 = ( a / R ) 3.

=2 nR

v2_ - f R_ R3

(. 131). ,

= 2 , / R 3 ( R + R* \ /2 3 g \ 2R ) 189

(. . 51).

_ ( R + R 3 \ / 22 /7 1 2 ) *

146. , , , N , ( ). M m / R 2 ( , , ). M m / R 2 N a 2R == -y j- R , , ,

N

( R + h ) 2

~R N = 4 2 _R 2 Y {R + h ) 2 ~ ~ W R - (

4 = (1) :

L - . / YP *{ 2 - h = R h , + .

147. 1 . ,

( 1)

g a , . ,

= 5 , (2)

ga = -J- ~ 0,008 / 2. (3)

. , :

m ga.hu = m g h , (4)

190

tn , fta , . (3) (4)

h a = S ft 64 .

, , g a , ( 1 ).

148. g c 4 4

g 1

= ncPcY>3 " " 3 ( 1)

R c , . ( 1 )

3 = -|-, (2) R 3 , 3 . (1) (2)

8%g - ^ ( 3 )

R c Pq -jj = 108, =0,23, (3)

R g c 265 / 2.

149. , . , , :

4;2

: Af3, , Afc , , R 3 i?jj , ' . , .

.Afo

4*14 '

351 103.

,

(0

_

1

= 390.

151. , 147 148:

= J ^ ..PnV .

:4g = yJttf3p3Y.

R 3 R n , 3 . p3 = pn, a R 3 2 R n,

gn = g R n _2

:2

/ = 42 2 = 49,6 .

152. F , , Aim 4/

(d /2)2~ 4 . , / 8 ( 1/8 ).

F = Y-Wm rf2

153. :

m g H =2m v j

~~2 +M v \

2 192

v \ , a v 2 .

t>i _ R

m g H =m v \

~2~M I ? v \

2r2 2 m g H

m + M

154. , . W = F v \ = F , , it] ,

. , . , , . 2 /?, 2, . . 2/?. , o a

, k , v a :

= cosT ^ + gL(Sin ~ COS *g ^ (5) :

2 Vl + 8L Va '

158. k , 3, ,

6 = t g a 3 ( 1 )

(. 47). t

S

, , . -

S] S2 \ 2

(3)

Sj , , 0 |, S2 2. h ,

1 I , 6 2 (4) .cos a cos a 2

47 ,

ai = g ( s i n a , - - k cos a t), (5) = g (sin a 2 6 cos a 2).'

(4) (5) (3),

^ _ sin ot] cos ot| sin a 2 cos a 2 ,g.cos2 a j cos2 a 2 * '

(6 ) (1), :

6 = 2 3 = t g a 3.159. , .

, , , , M g = m g k ,

13*

= k m . (195

, . , ,

M g M b . (2)

. m ,

m g k m (b ) . (3) (1) (3)

b = k a

k -1- 1 (4) (4) , > , . . m .

160. g , . , .

2g, . , , , . 139, 2 = 5L g , L , a v . , m g :

- , }2 = m g j = 6 m g .161. () ,

, , . 0 , , , 139,

0 = 2 / " > = 2 1 ^ = 2 g V g + a V 1 lg-

^ - = V n . ( i )

t , 50 , :

tn 32 . (2), , t 0 , . /, ,

1 == 6 . (3)

196

, g . , ( ). , (3). , ; , 1 0 + , t 0 (2 ), a t \ , .

v ~ Y 2 a h 3130 ! ,

, ,

/, = 320 ,

10 + 352 .

162. , (. 215). pSgh, SgH

. Sha = = pShg pSHg, h = H g / ( g - a ) .

163. 75.

, , F h , . 216. , h h 0 = 8 0 = 0,08 , , , F 0 .

F a, , . , , . , W , , W = Q m v 2/ 2 , ,

197

. . 216, W = F ah 0/2 ; ,

150 .0 , ,

. , , , , (. 72). ( 3%). .

164* , , (. 15):

/ ~ / = 6 , ( , / 0 , / , , 6 . , = m g , , / = 2 /0, ( 1 )

6 = /m g ' (2)

R , . F , ( 1 ),

- /

(3)

,

= 1 (4)

(2) (4)

R I / 12= V + R , + .

165. , , . 217: F = k (I /0)

, M g N . , , , . ,

198

:

N sin M g - k ( l - l 0) cos = ,

= _4ls_sin + k (l l0)

cos sin '

:

Ji = k U l 0) sin + N cos = M g :------ 1------ .sm sin

, I /1- / 2:

_ M g cos ct k >2 sin2 a

, . f t ( / 2 (0 (. 218). , (/2) , tgy=M co 2 sin . (/,)

^ = " ) = (Mg cos a -

/ 2 = 1 sin .

, . 218 ( ). , I , /, , , . , I , / , f 2, , . . . , .

. 218, :

cos . M g -------- < 0,sin a sin a

k l 0 > M g cos a, ( 0

k:----- > Mco2 sin a,sin a

^ 1 - , / Tto < ----- 1 / - 7 7 .sm a (2 )

199

, (1) (2). , , , . : ; , .

166. , , / 0 , I , ,

. tw s F R = 26R - R , . . :

R . , .

167. 164, 166, :

2l l = l = bmg. (1)

/ , , 6 . (1)

. , , L sin Leo2 sin . F ,

F sin = mco2L sin , F = m w ? L ;

(3) ( 1 )

(4)

(2) (4) :

200

.168. ( 165 167). , , . 219, :

k ( / /) N cos a = 12

M g N sin = 0. N , I 211 tg ,

= M g + k l a tg ( 2 k 1 2) tg 2

, 165.

169. , v ] v 2 , . .

, = 2= (1) -

v 0 = R .

F , , , .

F = M Xv 0 - v

~ h ~ ' (2)

F=M22 (3) , t F , (2 ) (3)

M 2v = 1 (t>o ~ v )f

_ MjVpv , + 2

(4)

, , - xv y 2 , ^---- 1-----^ . . (4),

M xv q I 1 | 0 2 M 2v 2 ^ M xM 2m 2R 22 \ 2 1 2 / 2 (Mi + 2) *

201

170. , . v

v = V 2 . (1)

v

tg a =

2 n R (2)

. 220. N , (. 2 2 0 ),

N = m g cos a. (3)

, (,

):

/ . F =

R(4)

. 220.

, cos2 a( 1 ) (6 )

___ 2 n m R g

:

= v cos a. (5) F N . N F ,

P i = Y N 2 + F 2

A n 2R 21 + tg 2 a A n 2R 2 + H 2

V A n 2 R 2 + 2 2 2 + 2

(6)

A n 2R 2 + 2

171. , , (3), (4) (5) 170:

N = m g cos a,

F = - cos2 a;(0(2)

N , , F ( ), , tg a = H j 2 n R ,

v 2 = 2 g n H . (3)

, , . 221.

202

, . . , a D , D ( ); N , (1), . F . , (-- ) . N - - = = F h , ,

N_2 F . (4)

(1) (3),

. 1 4n 2R 2c o s = j + t g 2 = 4 n 2 R 2 + 2 ,

(4)

R V A n 2R 2 + 2 1 cos 8 sin * mff , ZZI. ,

D . N . F N , N , F . , D , .

172. , , ,

= F cos ((, , . 222). 6 ,

m b m g 2 F sin . (2)

h tg , :

= 6 tg . (3)

( 1 ) (3) :m g tg q

+ 2 tg 2 (4)

. h , .

203

V , .

m g h = 2 M V 2 m v 2 2 + _ 2~", V v tg , = - V 21 , - (

2 h tg V , /z tga), (4).

173. m g h :

m g h + - (

, ' . ,

v n = v B = v (2 )

(. 156). (1) (2) ,

v 2 = g h . (3)

, , , , V 2 .

, , . ,

m v \~2~=&> (4)

204

t>j v . ,

Pi = sin ; (5)

h . ~~2 sir*2 -

(4) (5)

174. 156 173, :

= = , ( 1 )

, , - . v , (3) 173,

gfi, (2)h = ( R ) cos a ^ R cos . (3)

R . , , . 223.

N m g cos = R

(4)

N . ( (4) , R . )

N = m g , (5)

(2) (5) : R

h 2

'

175. . 224, N i N 2 ,

. , , . .

= + (1)

,

F 1, (2)

205

(3) , ,

N 1 + N 2 m g = 0 . (1) (3) N i f TD,

. mg(rf + 2fer) 2 ( d + k r ) * (4)

' 2 7820 .

= 0, (4), , 2 => m g j 2. k , (4) d . N 2 = m g . , ,

n = N 2 - 2 m g k r 2 d + k r 500 .

176, , m i f . F , ,

= mig + /. ( 1) . 225.

( i ), 2 , F, ,

, , . 2 , m 2g ( ). (., , 164), ( ) . F, ( 1 ),

F = * k x 2, (2) k . . 226.

206

(3)

, F ,

F x 2 k x \2 2

, , t r i i g ( 2 + h ) , = 0.8, . ,

kx% k h 2 = ig (* 2 + A) + - y - . (4)

, k h = m 2g.

(4) (5) ,

/ ?2 2m-itn,r,a2+ ----W ^ -| = 0, -

2 m {g

2

* 2 =W|g + (!+OT2)g

k ~ k

(5)

(6)

( 1 ) (2) , / = F - m 1g = Ax2 - m 1 g.

2 (6 ),

f = ( m l + m 2) g .

177. m g . , , V g , k v ( V , , v ). Vo, ,

k v 0 + p > < < V g = p V g , 0 )

== ^ g ( p - )

0 , , :

V g - T ,

T + p * V g - p V g + V g ( ~ * ~ V,v 0

207

178. : , N , , , , ,

(. 227). :

= N m 4g , \ = N + m g , | (1) m ag . J

, , + = , (2 )

: + = 0 ( !)

+ m n v n m 4 v 4 = const, (3)

v . , (3) , . ,

. , , , . , , , , , , .

, , , , (3)

rn.iV + t n nv n t n 4v 4 = 0. (4)

, S, , ( ), 2 = 2 v = v . i , (4)

Sq 3sn 3s.i . (5)

s4, s.i , (1), , , . () , , 5 + $.

S 4 + s.i = nl. ( 6 )

(5) (6 )

5 nly (7) , ,

= h + s 4 = n l = 3,8 .

208

179. . 0,01 . , , (. 228). 2 0 ( , ) , , , . , m g

m g = 2 ,

m g 2 ,

10~ 5 9,8 / 26,28 2 1 0 4

0,078 / . . 228.

180. (. 229).

F = .

, , . 229, 2 f fa sin . , 4. , , 2,

14 . . . 209

:

2 = 4 2 0 sin ,

8 ffsin

sin ,

_ 8 _ 4 2

, = 2 , . ,

, . ( ) 2 / .

181. , , , 0:

= 0 + g h .

(. 180)

. .

D I 2 ( 7 + g h = ---- ,

= __ 2__ + g h

0 , , ( ) . , : 8 .

182. V R d , n P } d m / p ,

d = - 4 r = 9,36 10- 4 = 9,36 16 . j210

d / R , d V ( , V : V n d : 47?) d .

d / 2

, , 181, . , , , . , , . ( .)

, , , :

2 2-0,465-3,14-25 10~d 9 ,3 6 -10_6

780 .

183. (. 230), , (. 182)

2 d = -

. 230. , ( ) d/2. , :

2 2( 2 0,073 10 5

-

= P S = d d d 2 10*. 10-12

1,46 10 .

184. .

2 0 = n r 2p g - i - ,

0 , .

, TV

2 n r = 2 { 0).

14* 211

P - P 0 = p g //2 .

P V = P aV 0:

2 = 0 2 , ,

P {lih - 1 ) = P ^ Y

:

/h ------------- = 9,9 .

2(1 + 2)185. , ,

, (. . , ). , . , , , . . , , //3,

= _ _

/ ,

n r 2h p g = 2

186. t \

2, :

( 2 ]) t

_ m q~~W

q ( - ,)

2,26. 1 0 4,2 103 90 1 0 = 60

(q = 2,26- I06 / , = 4 , 2 1 0 3 / ).

187.

W

q 2,26 10 / . (. 209)

v = m_ R T _ W R T q\x '

= 1 100 , R = 0,082 --------------

= 373 . ,

v S.

W R T

P S

103 103-0,082-373 2,26- 1 0 6 . 18 - 1 1 0 ~ 2

= 75 / = 7,5 / .

188. , . , , , .

W = m e ( 2 ,),

m = p v S . ,W ^ p v S c ( ,).

189. Q , , . :

Q =m(E2- F 0

2 m c ( T 0 ) + ,

213

, mi .

, 1

v\-v% ( ,)] =0,015.

190. , 273 , Q, :

Q =* tn c ( 0) = { .

- ( + | ) . = - ^ 3, i =J m /(

,

= ^ (~ )-

'+ ' -

, . , . . , . , , , . ,

Q] 1,41 105 , Q2= 1,29 106 , Q = Qi + Q 2 = 1,43 106 .

. , , , , , ,

Q = Qi -jt-5

Q = 1,53 105 .193.

, 100 , 100 , . . 1 . (373 ), (1 ) () (. 209)

P V = .

:M2m ( \ ) = ^ ( \ ) 4-

= 273 , ~ , 4,2 103 / . :

= , +4] [ + (] ) +

M2cM

= 1053 .

194. / - , m , .

Q i = ~ c v ( T t - ~ T \ ) .1215

. , , Q , \ 2, Q] , . , A V ,

A = P A V = - , ) .- A V

2 \ . ,

Q = Q x+ A = ^ { cv + R ) { T 2- T i ). Q=110

197. t W * = w [ t .

Q=sm - JL w ^ m ~1,

1 1 1 1

, = 1 0 ,

, , ____ Q ____ 100 1 ) w f t 1 ) '

100 - 60 6 - 500 - 3000.103V = 60 4180.103- 10 ~ -

198. Q] Q 2, , :

Q] = ( ,)Q 2 = ( 2)

200. ,

(. 231) , . :

( - ) = / + / / 1 = / 1 7,

^ + y - j = / + /2, / 2 = / 2 ,

( 1 + , ) ( + ) = ( 1 + 2 ) ( ~ )

,

[2 + (ct] + 2) ]

. 231. ( 2 () = 2 5 '

201. .

1 = 1 [ \ - . ( 2 - { ) ] , - = (2 - .h , , I , / 0 I .

1 ~ 1 / '

. ( / 0 /)//> :

= ( 2 - {) = 4 ,02 108 / 2 = 40 ,2 .

, .

202. :

/ = 0 [ 1 + ( , - 2)], 1 ^ _ = ( 1 - 2),(

I 10 ] = 573 2 9 , a . , , I . , , (/ l o ) l h

218

:

/-/ I F h = S '

F - , S , . F

F = ( 1 2) = 3360 .

, , , . , , .

203. , , . ,

/, = 1 ( 1+) . ( 1 )

, , D d .

D . - D d + ),

, , , 1\ / 2 2 //2 \ . , , , / /1 , , ( / /).

(1) (2)

F = T T l t E'E2S( AL ,

, (, ) :

1 CtiPi + 0t2 g 22 E t + E 2 j 2 T l ) '

205.

Plgtfl = P2g#2, \ 2 , ] 2 .

= Pi (1

,^ , = Pi (1 ) / / 2,

112 - , = 2 ,.

2 = 1,56 .

206. - , . , , , , .

, :

X t n + ( 0) = (] ) ,

m , X , , 7'j . :

X j + m T 0 m

_____________ +

220

V 0 ( + ) 1 ,

AK-KoPtr-r,),X

, + m

m + 0,95 .

, 0,95 .207.

: _ OTic i^ 1 + 2 2 _ [C jT j + 2 2 2

m Ic I + m 2c 2 11 + 22

, . -

, = - ,) ,

2 = 03 2 ( - 2).

, + 2 2 Vo ~ [ , ( - , ) + ( - )] =

32 (, + 2)

,|, 4- 2^ 2 Pic i + 2

= - 8,9- 1 4-

.208. t = 0 V 0,

0. , .

1 P ig^i 2 P 2 g V 2 '

/ 2 \ (1+

pfMif2 , , ].

p lt

t ~ t *1 *2*2 . ,

, . , . , 4 .

209.

( )

P V = R T , , , (. . ) . R , - 22,4 . , m/ = 1 ( -), = = 1 , 0 = = 273 , V 0 *=22,4 .

R = l f = ^ = 0 - - UTp~ -8,31-----*2* 273 -

m / V = ,

V R T

.

. , , , = 28 ! = 28 1 0 _ 3 /

R T V

1 0,082 30028 3 103

0 , 8 8 .

222

210. ,

m |R T *

, 18, (190 . . = = 0,25 ),

m = 17-0,25-20 0,082 300 = 34,6 .

211. , 10 20 , , , , ,

/ = P V ] iR (_L , = 2 , 2 .

212. , .

Pi P V . . . . m R T ! = =, , = -R T

I ' j m g ^ I 3 , 5 m g .

213.

W T ~

]R T

V , , R . , ,

M g < g ( m B - mHe) = g (pB - pHe),

M . V 0 R T J P c , 22,4 , V , :

f e r 2 p < - p r - ( p B- p He) - f -

Train ~0

^ 223

,

1 3 -22,4- 1 0 ~ 3 1 ( 2 9 - 4 ) - 10- 3

2,71 .

214. . 0 m

0 = p.PoVR T 0

V 0

] i P 0V _ V 0

R T V0 T

, , , V0 = R T 0/ P 0 = 22,4 - .

, ,

\ :

I ___ _ 0 145 22,4- 10~3 11 V 29 10~ 3 224 2

= 270 = 546 = 273 .

215. P i > P o + 2,

0 , , = V ^ , 2 Vp2g .

,

R T 1: V u P g

r t 2

tn _ pP~v"~~Wr

T " V V T \p 8V p P g - p o R T

= 305 .

, , 5 .

216. F = V g ( ), - j j f r ,

F = ( - ) = 7870 = 803 .

224

217. = 0, = 12 3 4- + 6 + 16 = 58. , , ,

til R T 0,716-0,082-473 ~ P V = 58-0,987-0,2426 ~ 2,

218. .^, = 1 2 + . :

m R T 0,65-0,082-300 = = ------- ------ = 16-

= \ , = 4. , 4.

219. . = 1 2 + 1 6 (/. :

m R T

= 44.

= 1 , (/ = 2. 0 2.220. -

m

R T

- JV0 = 6,023 23. . , ,

N = 2 N 0Z = 2 N 0Z = 4,3 P i

221. U 7 + p' = 2 4.

,

8m He = 7 fflLi = M 4 .

m R T 1,14 - 0,082 - 303 V = J ~ F ^ -----4^92-----= 77

222. , , . j, 2

15 . . . 225

3, ', 2 3 ,, 2 3 (. 232). , ( . .) , ( ) ,

,= \ + ,, 2='2 + 2, 3 = '3 + 3.

, , . .

/, + 1 2, + [ 3.

, :

p [ h S = P 2 ( h + H i ~ H 2) S = P'3 { h + H { ~ H 3) S .

h (1- 1)( ,- 2)

= 16 ,

3:

H \ - ( H ^ h + P 3) 3 ++ ( P 3 - P l + H l ) h + P 3H l = 0 ,

3 = 751,2 . .223. 0 0

_ m _ ~ ~ 0 '

0. N 0f g , ( 0)/ N a) / g . ,

P V = -

R T , iF = , - R T ,

( P - P i ) V -

, __

R

-

- 1ug

- 1 :

R T .

( P ~ P i ) g V

0,

{ P - P i ) g V T o 224.

= - = 0,58 .

, /*= 1 ,58 .

226

, , ,

'

Pv = 7 9 .

. j T i

Pi 1 ' Sj

S ,= P i - P a

2.

225. , , N = 3,7 1010 0,1 60 60 24 365 1,17 1017.

, . , N t = 2 , 2 1 0 20, , , 0 , 1 %. -

JLNo1,17 10176,02 23

= 1,94- ~ 7,

N o . R T _ q

= - = 4,6 1 0 3 .

226. - - m ' - R T u P V = ^ - R T 2, Am = - P Fp 1,6 . " ' " R ( T i - T 2)227. ,

,= 1 m R T

q t = t

q = v S ,

v 3_s J __ m_ R TS t 0,87 / .228. ,

, . . , ,

15* 227

( ). : 2

,

P l v = H h . R T t p 2v = r t .Hi P2

{Pl + P2) VJ .! lL + J ^ \ RT,\ P i p 2 /

p = = ( + .") (P. + ) = o,48 / \v ( nii | 1

l Pi p 2 /I r t

229. , pg7;S< = P 0S P S , P = p g ( H h ) . ,

Pol

P V = P aV 0, P { l - h ) S = P 0 . S ,

p g H l2 ( l - h )

, p g ( H - h ) = HI2 ( 1 h ) * -( + 0 + ^ - 0 ,

H + l , \ _ - , n 21,2 = ~ - ? '^ 2+ /2.

( . . .) ( ) -. 0 < h < l , a Ai = ^---- -g- Y 2 + > 1, , .

h = J i + l - ^ Y W ^ p .

, .

230. , ,

,

P V *= R T ,

PV = Hh-RTl,

1.

m = m =

m =

228

m R T M R T

V = - iiP 985 3.

231. , w m g < > - .

ItlmRT ^ R T V u

> 4/3 124 .

232. , . , ,

p P V 4 3"! = 1 = =12'3 -

233.

P V ^ P ^ ' u = + 0,10 = 1,1.

V = -P~ L = 360 3.

i

c iKsaJIk)

h. 233.

V , V = 500 360 = 140 3 .

V 234. , = + , = F (1 + ),

, 1/335, 1

1 + +

= - = 335 .

235. = 0,0107 1 3.236. . 233 ,

P iS = P 0S - m g ,P 2S = PoS + m g .

, P \ l jS j = P 2l 2S 2.

,

( - g h ) li = ( 0 + pgh) 1,

h + hP0 = pgh /1 U229

237. ,

iVi _ 2 2| 2 *

i = 2, , I IT t

l i7 V

, , = 0 + , 2 = - , , = 0 + , 2 = 0 - , + - . . . 0 7, , = - _ i = - 7 \ / = - =

sr0 29 . , (1 )

( 1 2 3); 0 .

238. = 1250 . , . , , 20 , .

240. 1 = 2 = , = 2 = .

, , mi = = . ,

I _ 1*2 2 Pi

, ]+ 2 = ,

| -1*2

1*1 + 2 =1*1

P i + 2,

/] = 5 (), / 2 = 80 ().241. 2 1 / 8 - .

1/8 2 (0,25 ) . , 1,75 .

R T = 2 .

242. P i - P t ~ P , Ti = r 2 =

= "1 R T> 1 * 1 2 = ^

1* 2

230

,Vi ... Win3 F 2 2 !

, Fi + F 2 = F, Vi miV-2 V 12 + 2 1 0,71.

P T243. =Pi 11 r 2 = 7i + Ar,

_, = 1 ] '

Pi ,- />, 1 ,0 0 2 ,- , = 500 .

244. , 2 = 2 F 2 = F i/2;

F , + F 2 F i.

, :

P [ v \ = p i V b P [ = * y p i -

, ( 2 = 2 1),

* 2 22 1

1 Fi,

2 = -- = 400 = 700 .4 4

245. P i 2 :

2 m R T i m R T 2 1 4 F 2 2 F

P 2 P i ~ M g f S , H>2-/ > , ) S mR(T2- T j )

g 2 g V / S

231

246. 1 , . S i/S 2 *=2 , 2 , 2

P = 2 P 0 + - f PgH.

2H S 2 ( 2 + h P g H ) H S 2P 0I

Ti-4T0 3 \4 )

247. , : 760 ; . . ( )

P i = P < s - p i g h ,

pi .

Pi = ~ tgh, 2 .

, , , ] 2 /. , (. . p ig h p2 g7i) 0. pi 2

Pi 2 = R T 0 , R (. 209).

P i-P 2 ) 0 = 273 ( ), R T 0/ P 0 , - . 1/0,= 22,4. , - :

r = S ( P 1 - P 2) = Sg/i - b - ( l - ^ - ) .

, i > 2. , -

232

F. F

3 ,5-9 ,8 -224-29- 10~3

22,4- ~ 3

273 \ 287 )

- ~ 500 ~ 50 .

p t 2 , .

248. 247. 0 ,

J \ _ = F V = _ 8,7-9,8-22,4 10~ 3

S g h p 1 .9 ,8 22,4 29 10~ 3

, 0 7 4

= W = 6T = 390K = 117- , , , , .

249. ;

= ,+ = R T ! / .V U a

m \ /

'R T ( m

^ \ \

2 )

2 V . , />=1,5 , /> '= 1 ,

Hh. = = = to -m B 2 2 2

259.

= P i - 2 + -(I 1 2

(. 249)

R T

V

\ 0,082-293 /2 0 , 2 \ . .7 ----- 1-----10+ 12) = 24

233

281.

M g = p i V g , (A f-A m )g = p2 Vg,

pi 2 , .

= (, 2) V.

pj 2 \ 2 j 2 (. 209),

A m _R

V = 4,2 .

252.

-? | ~~ I; Pi V

, , 2 = 1(. :

__ \ 2 1 R T 2 5 ( 1 1 % 1 0 3 - 2 - - 6 2 3 = 304 5-28-293 .

253. , ,

1 ,

Ti - 2 = 2

-

254. ( ):

Pi = ~ -

: 2 = . = + .

, 2 ~ \ - , , , :

1 = Pi' R T

= R T *

,

= =

=

(._- ^ .4&( P 2 ~ P l ) P r

234

255. :

= 0 + Mg S

AV 2 ^

= 2 - 1 = - ^ - | - ( 2 - 1))

m R ( m R P S ( 2 P 0S + Mg (2 - , ) = 41 .

256.

=* +M g

A = F ( h 2 - h l ) = P S (A2 - A ,) = P (V 2 - V t).

V = V2 Vi ( ), :

= ( 2- 0 = 1 7 1 5 ,] 9 V = V 2 - V, = -- = 0,0167 3.

257. 256,

R ( T 2 - T , ) *

,

= 0,23 .

258. , , , ( ), A P ( V 2 Vi) ( . 256).

Q = V ev (2 ~ .) + R - .) = ^ {cv + R ) ( - ,) = 109 . > >235

259. (. 234), 0 = 2 > 2 . , 2 = 2 .

'2 = 2 0 .

P ' V ' ^ P . V ,

' = ---- V ' = -

TJ_P , .

2 '2, , ' \

, 2-,2 0 + ( i )

260. (. 259)

P 2 = 2 P 0 - P l. (1)

,

1. = EjL, '

( 1 ) (2) 20

(2)

. = - 2 = 21 +I + -

]

261. :

m R T = - 2V 1

, , , :

plV = H}RTb

V = - ^ - R T -

{ 2.

236

,

, m , T J R

1 |1(, + )

nil + /2 = ,

2 +1

2 1SPTJR { _,iiViT + T^RT 7 + ,

2 ,T + Ti

262. n i p m 2 / , i = 4 , 2 = 2 0 7 3 = 80 . , , ,

m i P T i m /? 2 m 3 /?7~3p p V p V '

P , , V .

, ,,

m R T ]

/n = mi + m2 + m3.

3/, 3 0 ~ m ~ j 2_ + _

mi mi , , , .

= 2,4.

3 , , , ,

1 + ----1 2 3

263. 0. , j S h ,

S h V . V T i ft = -g--3p .

264.

niiti + 2m 1 + m 2 17 = 290 .

237

: , 2 R T >0,85 , m 2 2 R T V 1,27 .

265., 1 , P 0 = R T 0/ V . 1 - 20 0,5 - . (, ). P t , 2 , 2 V P i = R T , 2 V P 2 = 0 , 5 R T . , >

2 V P 1,5 R T , ,, 1,5/? R T 02 P2 0 l , 5 R F

'6 ,

266. . , : \ = ' = />' = . , : vf2 : V'z = : P 2 F 2 : P aF 3. , V [ + V'2 + F 3 = Fj + V o + F 3.

P ( V i + V 2 + V a) = P t V i + P 2 F 2 + P 3 V 3.

,1 + 22 + 33 , ,, p lVi(vi + v2 + v3)F ,+ F 2 + F 3 * 1 P i K ,+ P 2 F 2 + P 3 F3 *

/ 2 2 ( 1 + 2 + 3) , P 3 F 3 (F , + F 2 + F 3)^ 2 = P l V 1 + P 2V 2 + P 3 V 3 v ' ^ P iF i + P ,F , + P ,F ,

267. , ,

p,is p- i us 0 *

,

pi - o + y P8L

> =

po + Y

238

2 6 8 . ,

Pi - .. , - * *

T pgL = + j P g L , , - -

2 0 - p g L 2 Po + p g L '

, .

269. P i = + p g h , V 1 = S h .

P 2 = P o + 1 L , V 2 = S - ~ h

( /4 /4).

1 2

= | 2 0 + p g h 2 8 1 P o + p g h '

270. , " . , ; Pi = + 0 2 = .

_ RT _ 2 RT_

^ 2V

\ 2 1 V L^H 2pHJ = 0,37 , 2

R T 2= 0 , 1 2 .

271. , , = ( 1 V 2) (. 256). ,

v2 = v l - f ,

- .Tt Vif

239

,

,

,1 P V , '

PVi

, = 9,5.272. ,

P S = P 0S - M g , = 0 - = 0,5 . 1 ,

P V \ = R T U

P V 2 = R T \

_2 Q, = = 1,64-10 . 164 .

m273. A = P ( V 2 - V t ). { V 2 V \ ) = R ( 2 - T i ) .

*

Q = m c p ( T 2 - ~ T i ) i

m ( T i )

, = . = 1,37- 1 0 3 .

274.

P V = R T

, V :

2 _ mi 2

284296 = 0,96, . . 2 = 96% ,

4?6 . ( = const)

(. 256);

A * = P ( V 2 - V , ) = P V 2 ( \ - ~ ^ .

240

V 2 V, a Vi , , . ,

v2 V ,

= P V ^1 -yi-j = 1,033 104 50 0,04 ~ 2,07 104 = 2 105 .

__ e P V2/. = - ^ == 41,7 - . 0,5 (0,5 ) 500/18 = 27,7 -. , 14 , 28 2.

276. \ = P V= jLi -jr=- = 3430 ; 2 I P V= -==- = 2650 ; , , 780

2Q 26,1 10 . = 4,18 / , 1 0 0

Q =

= 62 .

277.

= = 5,8 . /

Q = c m A t -f = 3,84 106 .

1 q 41,8 - 103 '

2 = 5 8 0 0 -9 2 = 5708 .

= 2 1 ,

T i = ~ =254 .2

278. , , F:

P l S = M g + P 0S + F .

16 . . . 241

----

( )

,

Ms.

279. . , , 2 ' , .

P V 2 P i V i 2 , '

2 = 2 = 2 P i -/- = 8 .V2 1 1280. J 2 ,

1 0,082 1273m R T _

V '254 0,5 = 0,82 625 . .

J 2 J , n m - . m

2 J (1 ) I1 J2.

J J 2, . .

m R T . m R T . m R TP = 2 - + t t - a ) - = (I + a ) - T = ( I + a ) P 0.

, . , _ P _ 700 _ , I 0

+ a P 0 625 2 ,

a = 0 , 1 2 , . e. 1 2 %.281. 280.

0 2 , 0

=m R T 0,2 0,082 - 2600 V 44 1

0 2, 44,

242

0,97 = 735 . .

_ COs .

2 0 2 ^ 2 + 0 2,

0 2 0 2. , (I ) 0 2, ---- ,. 2 J

, :

,, , D T R T R T = { 1 - ) R T + -------- + 2 V

,

1 +-- = " _ _810 735 0 ,2 , . . 2 0 %.

U,

282. = 22,4 0,6 = 13,4 . .

9 . . , ,

p V - ^ R T ,

pPVR T

6 , 2 .

283. 150 0,0164 .HI I 22 = = 0,0075 . ,

R! 2 0,0089 .

284. : = + , = 44,6 0,8 = 35,7 . ., = 724,3 . .

R T33,4 ,

R T1090 .

, ( + ) g = 1123 .

285. 50% 20 8,75 . .

16*

P V

243

2

(-) V R T -

R F

R T 1

760

2 + ~ 1 760 0,4 8,75 = 1,005.

286, = [ ( + ) - ] g ,

, . ' = + 0,2. ,

F ' = [V(pB+ + ctp + 0,2) ] g ,

. ,F = F ' F = V ( + 0,2) g .

, , 0 ,2 , ,

= R T

,

- 0,2R T - 0,2 -

, . , ,

F = - 0 , 2 V \ ^ - - 1 ) p g = - 0 , 1 2 2 V p g . \ - /

, .

287. , ,

R T

200,6 - 0,0011 1030,082 760 293 = 0 , 0 1 2 = 1 2 ,.

288. N = N 0n i / n , ' .

_ R T

244

,

N J N = No 7 ^ ^ 1 6> 023 10 = ! 7 8 - < H .

289. -

VRT

RT _ kT NoP ~

N0 6 ,0 2 3 23 , a k ; , ,

kT

100 1 = *= 1 , 0 3 3 - ^ = 6

- .1 ,4 - 10 -16 3 7 3

! 06 = 3 ,7 1 0 _ / .

290. , , Q = 1. 100 :

Q = cm + qnii.

"h RT11

= 373 , 1 . ,

h = = RT U2rt~ cm S Sppn

,

= . + 2 = (0,2 + 2 0,3) = 0,27.

= 27 % .292. 1003 \ = 760 0,4 = 304 . .

,

\\ _ 22 , 2 ' 2 = = 1400 3.I \

, 1300 3.293. ,

, . , , , 760 . ., 60 . , = - ^ - = 1 2 ,7 . ., 72%.

294. :

P = P i + p , P t = P - - = 1 , 3 7 .1 , , : = m R T I p , V 0,51 . 100 1 , , , . , 0 = 1 , 8 8 .

295. , :

( ) R T 11 ,

a p n V = RT~

0 . , + = ,

V = _______ m R T _______( ) 4"

0 , 8 6 .

296. 76 P \ P o p i = 450 . . 80 2 = P i T 2/ T 1 = = 103 . .

297. V0/2

m =yRVR T

= 1,26 .

, 0,64 .298. , .

m R T / p V = 0,27 .

246

; , . 1,27 .

299. :

m , = = 0,909 , , = .

1 1,900 m 2R T

= 2 0 . .

ai = = 6396.300. .\ = 0,009 (.

299). 2=1,009 . , , = m 2R T / p , V = 1,37 7\=1041 . . , . , , , . 10096.

301. , 100 , 1 . . V a = m 2R T / p aP = 2 , 18 . . , ( , ).

302. , . ,

V a

,

V B + V a0,24.

, :

m BR T

^ -0,1-0,082 373

18 0,24 0,7 .

< = 1 ( 1=100 ), , , .

303. , (. 302), 1 . 1 .

V a =m 2R T

= 0,55 .

247

, 0,45 .

R T0 ,2 6 .

304. 7096 = = 1 2 , 6 . .

, 87,4 . . 1 0 874 . . .

P i = 874 + 18 = 892 . .

, 1 ,

V =

~up R T = 0,11 5

305. , ,

( + ) L = ( + ) 2-

, :

, . ,

(Pu)i 1, () 2>

2 = 1 = 3 6 96.

306. \ = 37 . . , 2, , 2.

= 2 R T '

0 = R T

243

,

aV = 0 - = ( 0 - 2 ) = 17,5 .

307. , 120 . 20 140 . , :

UlRT , = = 0 . 0 0 1 . V

308. 5 20 ,

m R T _ 20 0,82 278 760 ~ 18-6,54

3 - 1 0 3 .

t 1 .309. - ()

:/ \ ( 1 = - 0,0050 -,V / ,

= 0,00275 -.(_\ [ , L ~ R T

, , 0,00550 - . 17,5 . ., , :

2 V; = 0,00096 -.

, 0,0045 , 82 .310. ,

, : ( ) V ,

------------ R T + R T "

, 1 I m R T \

( - ) \ V ]*

= 18 . .

311. ( ); , ,

249

2, , P J J T t ( T i ) , ,

1 = + 1.

2 - j f + -

= 2 ~ 7 7 = 2 ~ ( 1 - .) Y [ =

= 815,5 - (626,7 - 26,7)- - = 149,5 . .

, . , .

312. . ( = const):

~ h = 7 6 0 1360 ~ 8 3 8 - '

= 0 + -2- ( - h ) = 760 + - ^ - 1360 = 860 . . 13,0

3 . ,

, . .; ( ) , , . , ( ) ( ) .

= = 22 . .

313. 1 3 0/ = 8,5 22.

= 1 / = 1,18 10 2 3 3. Z = V = 2,3 10 8 .

314. 1 2 1 = m N = p d , d , 1 2

1 . P = m N v , a v = V 2 W / m .

250

,

m N = P ~ \ f - = ~ \ f __ 2 W N 0 '

, a N 0 ,

d = j / " -2 ^- = 9 - 1 0 3 / .

315. 100 1 .. , . Q = q m . q , (, ):

= "2 ^" 5,9 , Q = 13251 cb/e.

, ( ), . (. 290).

316. , 0 = 1 . 100 . - h , h S ,

= \ i P 0h SR T

= 5,9 .

100 . q (. 290). ,

Q = + q m .

, ,Q + q m 41,6 104 + 1,3 104 W W 500

= 858 = 14 18 .

, , , 1/40 , . . 20 . 5 / , .

317. , , 1 , , 100 .

P V = ^ . Q = q m . , 100 .

Q

qiiPVR T c A T

= 3,17 .

251

318. , (100 ), , , . :

M t f i ( ^ i ) + Qtu = 22 ( 2 ),

= [ 22 (, - ) - 4,, ( - ,)] = 38 .

V =

i n R T 64 . ,

Vh = ~ g 64 .

319. , 100 . 100 :

m e ( 0) = m q .

. { - 0) m = -----------------= 3,34 .

= ^ 5,3 . ,

= 53 .

320. , : PS = P0S + M g . ,

= 0 + g - 2 ..

0 2 , . 200 2 , , , 2 . : V=mRT/] iP ; , ,

, _ mRT _h S pP S 1,9 '

321.

P S = P 0S M g , - - = 0,5 .

100 ( 1 ) , , , 0,5 :

m R T 1 V - - = 61 .

252

0 0,5 , , . , :

h = -- = 6 , 1 .

322 . tMe { i)

/V

p P Vrt2

26 .

323. 100 P i 0 + p g f f 11 . :

(V+v)P I v = 286 .

824. 1 0 0 W \

A = P V = ( P 0 + p g H ) V,M v

2

0 = 1f ! S t ; M p V . ,

= 2 S 4 * A + W PV p V 3

2 S 2f = 113 .

325. :

Mig + P0S + kxi = PtS, i S - P o S k x 2 2 *S,

M i , S , Xi 2 , I 2 , 0 .

k ( x 2 - X i ) = S ( P 2 - P i ) .

2 X i = ; 2 P i M R ( 2 ' - p S 1 h )

, - T i - H, + p k H ( - /) 1 h M R

326. -. , . h. - , h.

253

. F kx, k , .

h , F P S , , fc PS/A. , ,

'_S

.

1,5 , //2 :

1 T J h_2

,

P V - R T i;

4- = 2,25 . 4

327. , - h S , S , P 0 + p g h . , :

P o H S = { 0 + p g h ) j5,

,

+ pgh h i

h - , + / = / - - \ + p g h )

* h 1 + pgg \ + Pgh )328. , ,

, ( ) . h

0 - 5 .

) 0P a + p g h

, P o + p g h

,

30 .

254

329. .

= V i

( ),

,

2 > V i x S .

+

X

_13L3

L S = P 0x S ,

2 1 )3

2L

,

2~

21- 4 0 +

339. i -------

( + L ) L S

( + L ) ( L + ) S

= L , S , - *). -

l l - 1 + -

1 1L 2

*) , , . , . .

255

. , , . ,

331. ( ). , . , , . .

P i = P + p g h 2 = + p g H .

m g = ( - P i ) s = { P 0 - P - p g h ) s ,

M g = ( - P t ) S = (P0 - P - pg H ) S .

P 0 P,

m , T + pA +ptf.

, .332.

, F = k h .

kh g~,

, P t - k h \

V = AS,

V I = A l S .

k h h S k h i h j S

S T SP,

At = A ] / I f - .

/ V2 V+ A2 -jj-g-. 332.

334. P - p g ( H - x ) (. 235). V = x S . ,

/ ~ 2 ~ 4 [xSpg *

, , m R T 2 . t f 2nSpg

j:---- < -, , < . .p S p g 4 4m/?

. 235. . 236.

335. 100 . , L/2; 100 1 , . . 0,

Pa + P i> PgY + \>& ^-

, , . . 236 ,

,

S n ^2 ^ n 2f P i ^ P o - ^ f P S Y ^ P 2 r T '

-jr- = 13,6 . pg Pi

336. , P a + p g ( H ) .

[ P a + p g ( H - x ) ] x S ~ P 0L S ,

- ( + -^ - )\ pg 1

+ ^ = 0, Pg

*1,2' ) * / ( + P o L

pg17 . . . 257

, . . : = const ( 1 . 237). ,

: =* = + p g { ) ( I I ) .

, ; I I I a . , b . , , , , .

, , , . , . ,

X

; ' ( + ) _ ^ '2 \ p g l V 4 \ g ) g

, , . .

, > 2 1 / ^V

_ g'

. 237 , I I I I I .

337. g H 0 = 0. , , , . . : = P S m g , in - O f i H 0S , a g .

, P S = ( + g ) = 1,2g H 0S 1,20. 0 : 0 = : , , / 0 = 1,2.

338. I 2 3 4 (. 238). ,

A i 2 P i i V g V i ) , 34 = 2 ( V t V 2).

P i V 2 = R T 2, P t V i - R T u 22 = 3.

258

,

1 2

=12 + 34---- ( ,- 2)

, 3 4 , 1 2.

339. , 2 3 .

= 2 ( ~ V) R { ).

.

. 2 3 9 ) . , , F i V2 * abed, abed :A ^ P 1{V2~ V l) + j ( P i~ P l)(Vi~ V l)-

- ,( ,- ,)+ jiVz-Vtf-, ,

A = - a ( V l - V * ) .

, , .

3 4 3 . PV = RT,

PV-- RaV: R a V .

, , 3 4 2 . , . 2 4 0 :

l f R a { v \ - V \ ) .

.3 4 4 . ,

, : a PS AP", ; 1

p l n a S

260

2 1

\ \ 02 /

- | j - = 2 - ^ - = 2 3 = 8. 3 4 5 . 3 4 4 .

2 , - - = 4. 1

4 = 2

11 . . 1 = 2 , = 2 '

III.

.

3 4 6 . , , , . :

Q ne. (1 )

. - :

,

Q =

, = - 0.

= -j nR3 .

eN 4 /? 3

= 5,6 107 .

(2)

(3)

3 4 7 . , , ( )

f , ^ .' 4 0/?2 (

R . q ( . 3 4 6 ):

(2 ) (1),

/TvV 4622

4 m 0R 2 QeaA 2R 2= 4,38 1018 .

348. , / /1:

2 . 2! 4e aR 2 ' R2'

= 1,6 10' -19 ,

10 = 36 1 0 ~ / ,

, R = 6 ,7 - 1 0 1 1 - 1 3 - 2 / / ,

2

164 z 0R 2

:

R 2 9

9264302

= 7,6 1

2 ,

tga = _ L

m g (2)

F , ,

(. 242, ) . , , i. ;

1 cos a + F = m g , |

, sin a =

t g a = fe (m g - F ) '

(3)

(4)

(2) (4),

= ! . (5)m g ( m g - F )

m F: = 4/ 3 3, , ; F = 4/ 3jrr3pKg, . (5):

= ( ).

264

=

1 1,6 3352. ,

:

mePR402

q_ _ 40)2R 3 _ g _ J 0 - 6 _ _ q *

. 353.

(?):

Q 1 92 = 9 = 9. (

V = ^, ,

II!>sT>

(2)

F, + F2+ y 3 ?. (3) ( 1 ) (3)

______ %C2 C3______1 + + ,

______ t c f i , _____2 ^ + + ,'

_________iCt_____3 ,* + , , + 8,

, , ( 1 ) (3) :

_1_

_% '

354. . 0 :

, 2 + 3*

_ _ Ci (2 + )0 Cj + +

265

/V'; , .

359. Ci 2;

_'~, + 2'

, ,

( ). :

C C t fQc + , q c > + , -

,

Q = qc ~ < ic = T T c 7 iC~ c,)- V

v . Q = 9 ( - ,) , + 2 ( + C j ) 2

362. / = 0, F 0 . , q ! , q q t . | F, , .

q = V , ( + ,).

, -

V ,= + ] + 1

0

+ , *

, , q 2. q 2 :

y=V,i + Vr2 + r3 + , . + , + Cl + ( + , ) 2 + "

, :

V = = 8000 .1363. q

, ,

2 = , ^

366. . 244. 0 0 ' ,

. , . . , , , ,

. C t 2, . .

1 1 , 1 , 2 , + 2 _ , + 2 *

Cj 2ee0S 2 = 8p

Q , a R , 0 (). Q , . .

Q - N q - 4 n e 0N V r , (2)

. (2) ( 1 ),

Vi- N V rR ,

.. 4 , 4N -7 3 3

8 1

R 3 N

V, =. V N m .

369. - . . . , , Q ' , . .

_1_.+ _ _ = 0402 402

Q = Q.

V u Q 2, Q ' = Q .

V 2 , Q. , , . , . ,

= ____2 40#2 *

Q = 4ne0 Vitfi, ___

v ~ v l + vi = v l - f *'-.

271

370. . ( .) (. 369).

V Q2 4802

Q 4ne0Ki/?i. ( )

RiVi .*371. , ,

-V

q , .

V. V lt q,

, . d a b d ; , , , , ,

da + db 0 .

~ db da>

'db ' 'Vc,~9, Vda=VC,

9, I

((2)

(3)

, ,

CCi = 7 ' (4)

[ ] ( ):

9, = * ,

+ , (5),

9,

+ 2 (6)

(3), (5) (6 ), (2)

= % ( , - , )( + ,) ( + 2) * Cj 2.

:vn-4-v,,> -V,,> V - V h ab ' V Ci V Ci

, .

373. , . d a b d ( );

, + Vab = % 1>

V,

9|_ ,

9 , = ( * , - * 2) , + 2 -

18 . . . 273

(. 372). srA + srA v" " c ,+ 'cs

374. , Cj . . ( 2 , . , , 2% ( + 23). , , 2 2%. .

373. , . .

v * + Ci1 + 2 -f 3 4 - 4

V 2 = t

Ci + , + 2 + 3 + 4

4:_, 3 ( 2 + 4) 4 ( C j + 3)

3 I + 2 + 3 4 4 Ci + 2 + 3 + 4

3 , 4

9 = 94-

4 \ , R

4ne0r |R - r 2

, q i + q 2 Q, :

, , ,

/ = 0, = 1Q

e0S

s .,

/

' q J / = 0

383. , , , , , , (. 381),

2ee0S 2se0S

, V , S . , ,

F ' - H S Pg . (2)

F F ' 22

2fee0S = j H S p g .

C = e e 0S / d , , ( d = 3 H / 4 ) eeS \ 2

V 2

2se0S J-ffSpg,

9 H 3p g

~32 '384. ,

, F q 2/ 2 e aS (. 381). , -

' " dj d2:

. 248.

A F ( d 2 d i ) 22 e0S ( d 2 - d i) = 2C\ C2

C f i 2 385. , ,

= Q t , Q , ? - . Q :

Q (, ), 0 , ! .

' " , (. 248). Ci

J _ = J _ J _ d - d i C i ~ + C " " e0S *

278

d , a d \ . ,

,= 8 0Sd - d x '

, 0 = e cS / d ,

0 Z2Sdt d ( d d = 8,85-10 6 .

386.

mg = 2e, (1) , , . , ,

Vd (2)

-- , (3)

(1 ) (3),

4 n r 3p g d ~ 3 2 V 1,6- 10~19 .

387. , , (. 386)

V _ 4m^ydig_ _ 43gQ s_9

) , , . . V const, ,

V = E t d i = E 2d 2, = % - Ct 2 d 2 > d t. , , F = ; , .

) , , : q const. , (. 377). .

27

388. , , V ,

, 1 = ,(/

1 = 4,8 1 0 - [ 2! 2} = 1,6 10- 1 2 .

I ~ 1,6 3 .

389. , :

m v -,

q = Z e ( ), =

( 0 , 1,67 2 7 ) .

v

-

Vo2 e Q

40$ "392.

2 e V ,

/22 , V . , V . (. 369)

,

f - i l ; )40 \ 2 ) '

- _ 2 (

, .

395. . , . , . , , , , :

22 ~eV; ( 1)

V , .

(. 377)V = E d , (2)

(3)

(3) Q . Q , :

Q e n t S . (4)

(1) (4), . __ 2

2 e 2n d = 1,56- 10- 7 .

396. ( ), eQ/4ne0r. , , , - , . :

e Q _ m v 2 40 2

, 4eoV2r4 7 2

397. -. v t, , ,

Vf*V2ctid, (1) 1 , a d .

= \ . (2)

282

>,= 2 e E l dm (3)

, , . , , ,

(4)

, (3):

v * - y/ 2e E 2d

($)

(3) (5) (4) v 2 t'iVs '398. D

V AD = 0 , , D , . ,

,

mv\~2~ '

~ 8,4 10 / ,

mvc2 - eVAC,

= 6 1 0 6 / .

399. a = e V / m L 0 + , 2 0. ,

, v = v0 + - .

~ L 2 m L 2 ~ v 2 m v ClL + e n T

, , , , . 27", . = ( 2 + ) , < 2 , .

283

400. L ,

2 2

( , , )- , ,

2, . . L n , , (2= ]/2 )

]^ = - (3/1+1).

L n = L , = 8 , t 1,77.

401. :

St2 S = - S7r~, a t 2 d = ,

V = 4 - ,d

.

1 --------------^

1 * ?-;

Vf

a t .

V X '5 = -

. 249.

402. . 249. , ,

, - ( ). ,

vx = , v y = at ,

(1)(2)

:

= m g . (3)

, , -

284

. , m g ( 1 ,6 * 1 0 ~ 1 9 , *9,1 1 0 ~ 2 8 ) . ,

, .

( 4 )