, , , , , , , . , .
15. 1. , , . . a. ? k, m.
24 , t. .
t? . , . (3.6) (3.8) , . ? (3.6) x (. 24):
S = a t2 / 2. (15.1)
? , S. ? S , , . (13.3) .
(13.3) F (. 24):
F = k S. (15.2)
.
, . . , . . , , 24: m , . . x (. 24)
m a = m g F. (15.3)
, : t, S, F. t, . (15.3) (15.2)
m a = m g k S. (15.4)
S (15.4) (15.1):
m (g a) / k = a t2 / 2 Þ t = .
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, . .
15. 2. , a, m. a . , . m.
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, , m F. ? , , .
, . . : , . ? . , . , . ? , ? (3.5):
= + = 0 + = .
, = 0 , = .
. , (. 25): m , , , . , 25, , .
? ? , a, , .
y ( 25):
m a = N m g + F sin a. (15.5)
, . (15.5) .
? , : F N. , N, ? ? m, (13.7):
F = m N. (15.6)
F. , F. ? , (15.5). : y. , x (. 25):
0 = F cos a F. (15.7)
, . . N F , ( ):
F = . (15.8)
. m m, . a > 0 ( . 25) F , a < 0 ( ) , .
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(3.5).
, .
(a) .
15. 3. m V v , r. , 2v?
? ? ? ? ?
, 26. , , , . m . ( ).
F = r g V. (15.9)
C , (13.6).
x (. 26) . :
0 = m g k v r g V, (15.10)
0 = m g + 2 k v F r g V. (15.11)
F :
F = 3 g (m r V). (15.12)
v , m / V > r, .
( ) .
, , ( , , ).
15. 4. , . 55 . .
, (g) , (g0). ? , . ? , ? , ( .27). . , R (. 27), , , (13.10):
a = w2 R = (2 p / T)2 R. (15.13)
, , , , . ? (. 27). ? , .
, g g0 , (15.3), . 27 27. , .
. x :
m g0 = F, (15.14)
m a = F. (15.15)
F (13.4):
F = G m M / R2, (15.16)
M .
(3.5):
= + = + Þ
a = g + a. (15.17)
? M r:
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M = r 4 p R3 / 3, (15.18)
, g0 / g = 3.
r
r = 9 p / (2 G T2).
: . . .
, , .
. . , .
16. 1*. 10 , 20 ?
28 : 1 , t1 ; t; m ; ; , .
? . , , , , . .
? , . ? , , , .
? . . , .
x (. 28):
F = m a. (16.1)
F , (13.8),
F = m N. (16.2)
N , y (. 28),
N = m g. (16.3)
a (4.7) (. 28):
v1 = a t1. (16.4)
(4.6) 1 (. 28):
S1 = a t12 / 2, (16.5)
S S1 = v1 (t t1). (16.6)
m :
m = 0,36. (16.7)
(16.7) , m , t , .
, , , ( ).
.
16. 2. , r, m? ?
29 , . , . , :
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= + . (16.8)
. . . , . , , .
m ( ) .
? , , a . ? , , . , , . , . , , . ( ), , (-). - . , , , . , . .
, , a = v2 / r. x (. 29)
m v2 / r = F, (16.9)
y
0 = N m g. (16.10)
, , ( ). , , , , m . a, 29,
tg a = F / N. (16.11)
C (13.8). (16.10)
F £ m N = m m g. (16.12)
(16.9) (16.12), :
m v2 / r £ m m g Þ v £ v = . (16.13)
, , v, , m ( r).
(16.11), (16.9) (16.10) , :
tg a = Þ a = arctg (v2 / r g). (16.14)
(16.13) (16.14):
tg a £ m. (16.15)
(16.14) , v a, ( r). m (. (16.15)).
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, , .
, ( ) , , (-).
, . .
16. 3*. S. (S1) (S2). 0,80. , 1 / 4 .
30 : m ; 1 2 ; 1 2, ; ; ; l; h .
? - ? , , -, . S1 S2? . , (4.6) (4.8).
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(4.8) x (. 30),
S = v2 / (2 a). (16.16)
S1 S2 :
S1 = v2 / (2 a1) S2 = v2 / (2 a2). (16.17)
S1 S2 , ? a, a1 a2 x (.30).
m a = F1 F2. (16.18)
, (13.7)
F1 = m N1 F2 = m N2,
,
N1 + N2 = m g.
(16.18) a = m m g, (16.16)
S = v2 / (2 m g). (16.19)
F2 = 0, (16.18)
m a1 = F1 = m N1. (16.20)
N1 N1 + N2 = m g , :
l (N1 N2) / 2 m N1 h = 0. (16.21)
.
N1 = (16.22)
(16.20)
a1 = . (16.23)
, (16.20) (16.21):
m a2 = F2 = m N2 ,
l (N1 N2) / 2 m N2 h = 0.
(16.23):
a2 = . (16.24)
(16.23) (16.24), (16.19) (16.17) :
S1 = 2 S (1 m h / l) = 1,6 S; S2 = 2 S (1 + m h / l) = 2,4 S.
, , , .
, , . .
, , , .