Чертов А.Г. Решебник
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J
F =
2Jε
D
; a = g −
2Jε
mD
a = εR = εD/2
εD
2
= g −
2Jε
mD
; J =
mDg
2ε
−
mD
2
4
ω = εt
ε =
ω
t
; J =
mDgt
2ω
−
mD
2
4
;
J =
2 · 0, 6 · 9, 8 · 3
2 · 9
−
2 · (0, 6)
2
4
= 1, 78
·
2
.
m
1
= 50 ; m
2
= 60
D = 4 ; ε = 1, 5 / .
J
&%
'$
*
R
m
1
m
2
?m
1
g
?
m
2
g
?
T
1
6
T
1
?
T
2
6
T
2
?
a
T
1
T
2
ε =
M
J
ε M
M = RT
2
− RT
1
= R(T
2
− T
1
) =
D
2
· (T
2
− T
1
)
a
m
1
a = T
1
− m
1
g; m
2
a = m
2
g − T
2
;
T
1
= m
1
(g + a); T
2
= m
2
(g − a).
a = εR = ε ·
D
2
J =
M
ε
=
D(T
2
− T
1
)
2ε
=
D
h
m
2
g − ε
D
2
− m
1
g + ε
D
2
i
2ε
=
=
0, 04 · [0, 06 · (9, 8 − 1, 5 · 0, 02) − 0, 05 · (9, 8 + 1, 5 · 0, 02)]
2 · 1, 5
= 1, 58 · 10
−3
·
2
.
v = 8 / ; s = 18 .
k
-
v
m R
T =
mv
2
2
+
Jω
2
2
,
J =
1
2
mR
2
ω v
v = ωR; ω = v/R.
T =
mv
2
2
+
1
2
mR
2
·
1
2
·
v
R
2
=
mv
2
2
+
mv
2
4
=
3
4
mv
2
.
A = F s F = kmg.
kmgs =
3
4
mv
2
; k =
3mv
2
4mgs
=
3v
2
4gs
;
k =
3 · 8
2
4 · 9, 8 · 18
= 0, 272.
D = 30 ; m = 6 ;
∆t = 8 ; n = 12
−1
.
J
&%
'$
*
R
n
?
M
ω = ω
0
−
M
J
t
J ω
0
= 2πn
J = mR
2
=
mD
2
4
.
ω
0
−
M
J
∆t = 0;
M =
Jω
0
∆t
=
mD
2
2πn
4∆t
=
πmD
2
n
2∆t
;
M =
3, 14 · 6 · (0, 3)
2
· 12
2 · 8
= 1, 27
· .
m = 0, 4 ;
m
1
= 0, 3 ; m
2
= 0, 7
T
1
, T
2
&%
'$
*
R
m
1
m
2
?
m
1
g
?
m
2
g
?
T
1
6
T
1
?
T
2
6
T
2
?
a
a
a =
T
1
− m
1
g
m
1
; a =
m
2
g − T
2
m
2
(1)
T
1
, T
2
M = R(T
2
− T
1
)
ε =
M
J
=
R(T
2
− T
1
)
J
J =
1
2
mR
2
a = εR
ε =
a
R
=
R(T
2
− T
1
)
J
; a =
R
2
(T
2
− T
1
)
J
=
2(T
2
− T
1
)
m
a
2(T
2
− T
1
)
m
=
T
1
m
1
− g;
2(T
2
− T
1
)
m
= g −
T
2
m
2
(2)
T
1
m
1
− g = g −
T
2
m
2
;
T
2
m
2
= 2g −
T
1
m
1
; T
2
= 2gm
2
−
m
2
m
1
T
1
T
2
2(2gm
2
−
m
2
m
1
T
1
− T
1
) =
m
m
1
T
1
− mg
4gm
2
− 2
m
2
m
1
T
1
− 2T
1
=
m
m
1
T
1
− mg
(4m
2
+ m)g = (2
m
2
m
1
+ 2 +
m
m
1
)T
1
T
1
=
m
1
(4m
2
+ m)g
2m
1
+ 2m
2
+ m
T
2
= 2gm
2
−
m
2
m
1
·
m
1
(4m
2
+ m)g
2m
1
+ 2m
2
+ m
= 2gm
2
−
m
2
(4m
2
+ m)g
2m
1
+ 2m
2
+ m
=
m
2
(4m
1
+ m)g
2m
1
+ 2m
2
+ m
T
1
=
0, 3 · (4 · 0, 7 + 0, 4) · 9, 8
2 · 0, 3 + 2 · 0, 7 + 0, 4
= 3, 92
T
2
=
0, 7 · (4 · 0, 3 + 0, 4) · 9, 8
2 · 0, 3 + 2 · 0, 7 + 0, 4
= 4, 57
.
n
1
= 8
−1
; n
2
= 10
−1
;
m
1
= 70
.
m
2
&%
'$
&%
'$
u
-
n
1
u
-
n
2
m
1
m
2
J = J
0
+ J ,
J
0
J = m
1
R
2
R
J = J
0
ω
1
(J
0
+ m
1
R
2
) = ω
2
J
0
ω
1
= 2πn
1
; ω
2
= 2πn
2
.
n
1
(J
0
+ m
1
R
2
) = n
2
J
0
J
0
=
1
2
m
2
R
2
.
n
1
(
1
2
m
2
+ m
1
) = n
2
·
1
2
m
2
;
(n
2
− n
1
) ·
m
2
2
= n
1
m
1
; m
2
=
2n
1
m
1
n
2
− n
1
;
m
2
=
2 · 8 · 70
10 − 8
= 560
.
D = 0, 8 ; m
1
= 6 ;
m
2
= 60
; m = 0, 5 ;
r = 0, 4 ; v = 5 / .
ω
&%
'$
~u
6
r
-
vm
m
2
3
r
O
-
D
m
1
Jω
O
Jω = mvr
J O
J = J + J + J
0
J
= mr
2
J = m
2
r
2
J
0
=
1
2
m
1
r
2
(m + m
2
+
1
2
m
1
)r
2
ω = mvr
ω =
mv
(m + m
2
+
1
2
m
1
)r
=
0, 5 · 5
(0, 5 + 60 +
1
2
· 6)0, 4
= 0, 098 / .
ω
1
= 4 / ; J = 5 ·
2
;
l = 1, 8
; m = 6 .
ω
2
6
?
l
ω
1
ω
2
J
J =
1
12
ml
2
J + J
ω
1
J = (J + J )ω
2
ω
2
=
Jω
1
J + J
=
Jω
1
J +
ml
2
12
; ω
2
=
5 · 4
5 +
6 · (1, 8)
2
12
= 3 / .
D = 3 ; m
1
= 180 ;
m
2
= 70 ; v = 1, 8 / .
ω
1
f
v
ω
1
m
1
m
2
ω
1
ω =
v
R
=
2v
D
ω
2
= ω − ω
1
ω
1
J
1
− ω
2
J
2
= 0
J
1
=
1
2
m
1
R
2
; J
2
= m
2
R
2
.
ω
1
·
1
2
m
1
R
2
− ω
2
· m
2
R
2
= 0; ω
1
·
1
2
m
1
− (ω
− ω
1
) · m
2
= 0
ω
1
·
1
2
m
1
+ m
2
= ω
m
2
; ω
1
=
m
2
1
2
m
1
+ m
2
· ω =
m
2
1
2
m
1
+ m
2
·
2v
D
ω
1
=
70
1
2
· 130 + 70
·
2 · 1, 8
3
= 0, 62
/ .
m
1
= 280 ; m
2
= 80 .
ϕ
&%
'$
u
6
ω
1
m
2
m
1
ω
2
ω
1
J
1
ω
1
= J
2
ω
2
; ω
2
=
J
1
J
2
· ω
1
.
J
1
=
1
2
m
1
R
2
R J
2
= m
2
R
2
ω
2
=
1
2
m
1
R
2
m
2
R
2
· ω
1
=
m
1
2m
2
· ω
1
.
ω = ω
1
+ ω
2
,
t 2π
t =
2π
ω
=
2π
ω
1
+ ω
2
=
2π
ω
1
+
m
1
2m
2
· ω
1
=
4πm
2
ω
1
(2m
2
+ m
1
)
.
ϕ = t · ω
1
=
4πm
2
(2m
2
+ m
1
)
;
ϕ =
4 · 3, 14 · 80
(2 · 80 + 280)
= 2, 28
= 130
◦
m = 60 ; l
1
= 1, 2 ;
n
1
= 2
−1
l
2
= 0, 6
.
n
2
, A
u
?
l
X
Xy
n
J
1
ω
1
= J
2
ω
2
J
1
, J
2
J
1
= ml
2
1
; J
2
= ml
2
2
.
ω
1
= 2πn
1
; ω
2
= 2πn
2
; .
ml
2
1
· 2πn
1
= ml
2
2
· 2πn
2
; l
2
1
n
1
= l
2
2
n
2
;
n
2
=
l
2
1
l
2
2
· n
1
=
(1, 2)
2
(0, 6)
2
· 2 = 8
/ .
A =
J
2
ω
2
2
2
−
J
1
ω
2
1
2
=
1
2
[ml
2
2
(2πn
1
)
2
− ml
2
1
(2πn
1
)
2
] = 2π
2
m[l
2
2
n
2
1
− l
2
1
n
2
1
];
A = 2 · (3, 14)
2
· 0, 06 · [(0, 6)
2
· 8
2
− (1, 2)
2
· 2
2
] = 20, 4
.
D = 75 ; m = 40
F = 1 ; r = 12 ; t = 10 .
ε, n
&%
'$
-
6
r
~
F
-
D
ε =
M
J
M = F r J
J =
1
2
· mR
2
=
1
2
· m
D
2
2
=
mD
2
8
ε =
8F r
mD
2
=
8 · 10
3
· 0, 12
40 · (0, 75)
2
= 42, 7
/
2
.
ω = εt
n =
ω
2π
=
εt
2π
.
t = 10 c
n =
42, 7 · 10
2 · 3, 14
= 67, 9
.
, h = 1000 .
G
&%
'$
u
*
R
6
r
?
6
h
r = R+h R
F = γ
mM
r
2
,
γ M r
G =
F
m
= γ
M
r
2
r = R R G = g
g
g = γ
M
R
2
; γM = gR
2
;
G =
gR
2
r
2
=
gR
2
(R + h)
2
=
g
1 +
h
R
2
;
G =
9, 81
1 +
1000
6370
2
= 7, 33
2
.
m = 2 ;
1) h = 1000 ; 2) h = ∞
A
&%
'$
u
*
R
6
r
?
6
h
r = R+h R
F = G
mM
r
2
,
G M
A =
Z
F dr = GmM
r
Z
R
dr
r
2
= −GmM
1
r
r
R
= GmM
1
R
−
1
r
= GmM
1
R
−
1
R + h
.
h = 1000 = 10
6
A = 6, 67 · 10
−11
· 2 · 5, 98 · 10
24
·
1
6, 37 · 10
6
−
1
6, 37 · 10
6
+ 10
6
!
= 1, 7 · 10
7
.
h = ∞
A =
GmM
R
=
6, 67 · 10
−11
· 2 · 5, 98 · 10
24
6, 37 · 10
6
= 12, 5 · 10
7
.
m = 30 ;
g = 9, 8 /
2
; R = 6, 37·10
6
.
A
&%
'$
u
*
R
6
r
m F = mg
r r
F = mg
R
r
2
,
R
A = −
R
Z
∞
F dr = −mgR
2
R
Z
∞
dr
r
2
= mgR
2
1
r
R
∞
= mgR
2
1
R
= mgR;
A = 30 · 9, 8 · 6, 37 · 10
6
.
v = 5 / = 5000 / .
h
&%
'$
u
*
R
6
r
?
6
h
mv
2
2
=
2
−
1
,