Решение задач по физике Баканина
TRANSCRIPT
-
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 )