Чертов А.Г. Решебник
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m
= −G
mM
r
,
G r r =
R + h R
mv
2
2
= −G
mM
r
2
+ G
mM
r
1
= GmM
1
r
1
−
1
r
1
r
1
= R; r
2
= R + h
v
2
2
= GM
1
R
−
1
R + h
;
v
2
2GM
=
1
R
−
1
R + h
;
1
R + h
=
1
R
−
v
2
2GM
; h =
1
1
R
−
v
2
2GM
− R;
h =
1
1
6, 37 · 10
6
−
(5000)
2
2 · 6, 67 · 10
−11
· 5, 98 · 10
24
− 6, 37 · 10
6
= 7, 959 · 10
6
− 6, 37 · 10
6
=
= 1, 59 · 10
6
= 1590 .
; T = 65 ;
g = 9, 8 /
2
; R = 6, 37·10
6
.
h
&%
'$
?
6
h
u
*
R
6
r
m F = mg
r r
F = mg
R
r
2
R
a =
F
m
= g
R
r
2
= g
R
h + R
2
.
a = v
2
/r
v
2
r
= g
R
r
2
; v =
r
g
r
· R;
T =
2πr
v
=
2πr
R
·
s
r
g
; T
2
=
(2π)
2
r
3
R
2
g
; r =
3
s
g
T R
2π
2
;
h = r − R =
3
s
g
T R
2π
2
− R
T = 65 · 60 = 3900 c;
r =
3
v
u
u
t
9, 81 ·
3900 · 6, 37 · 10
6
2 · 3, 14
!
2
− 6, 37 · 10
6
= 5, 35 · 10
6
− 6, 37 · 10
6
= −1, 02 · 10
6
.
T
M = 81 ; r = 60R ;
r
1
r
A
r
-
r
1
r
1
A
γ ·
M
r
2
1
= γ ·
M
(r − r
1
)
2
M (r − r
1
)
2
= M r
2
1
; 81(r − r
1
)
2
= r
2
1
; 80r
2
1
− 162rr
1
+ 81r
2
= 0.
r = 60R R
r
2
1
− 121, 5r
1
+ 3645 = 0
r
1
= 60, 75 ±
q
(60, 75)
2
− 3645 = 60, 75 ± 6, 75.
r
1
< 60R
r
1
= 60, 75 − 6, 75 = 54R = 54 · 6370 = 343980 .
; h = 520 ;
g = 9, 8
/
2
; R = 6, 37·10
6
.
T
&%
'$
u
*
R
6
r
m F = mg
r r
F = mg
R
r
2
R
a =
F
m
= g
R
r
2
= g
R
h + R
2
.
a = v
2
/r
v
2
r
= g
R
r
2
; v =
r
g
r
· R;
T =
2πr
v
=
2πr
R
·
s
r
g
=
2π(R + h)
R
·
s
R + h
g
;
T =
2 · 3, 14(6370 + 520) · 10
3
6370 · 10
3
·
s
(6370 + 520) · 10
3
9, 8
= 5696
= 95 .
h = 1000 ; g
0
= 9, 81 /
2
.
ω, v
&%
'$
u
*
R
6
r
?
6
h
r = R + h R
g = g
0
·
R
2
r
2
= g
0
·
R
2
(R + h)
2
g
0
a
n
= ω
2
r = ω
2
(R + h) = g
0
·
R
2
(R + h)
2
ω
2
=
g
0
R
2
(R + h)
2
; ω =
R
(R + h)
·
s
g
0
(R + h)
v = ωr = ω(R + h) = R ·
s
g
0
(R + h)
ω =
6370
(6370 + 1000)
·
s
9, 81
(6370 + 1000) · 10
3
= 0, 001
/
v = 6370 · 10
3
·
s
9, 81
(6370 + 1000) · 10
3
= 7345
/ .
A = 15 = 0, 15 ; ω = 4π
−1
;
m = 20
= 0, 02 ; t = 0, 2 .
F
x = A sin(ωt)
v = ˙x = [A sin(ωt)]
0
= ωA cos(ωt)
a = ˙v = [ωA cos(ωt)]
0
= −ω
2
A sin(ωt)
F = ma = −mω
2
A sin(ωt)
t = 0, 2
F = −0, 02 · (4 · 3, 14)
2
· 0, 15 sin(4π · 0, 2) = −0, 29 .
E =
1
2
mω
2
A
2
=
1
2
· 0, 02 · (4 · 3, 14)
2
· (0, 15)
2
= 0, 0355
.
; l = 30 .
T
L
L
L
L
L
L
L
L
L
L
L
L
L
L
O
l
L =
J
Ma
J M a
a = l/2
J =
1
3
ml
2
L =
1
3
ml
2
m ·
1
2
l
=
2
3
l
T = 2π
s
L
g
= 2π
s
2
3
·
l
g
= 2 · 3, 14 ·
s
2
3
·
0, 3
9, 8
= 0, 9 c.
A = 15 = 0, 15 ;
v
max
= 30 / = 0, 3 / .
a
max
x = A sin(ωt)
v = ˙x = [A sin(ωt)]
0
= ωA cos(ωt)
v
max
= Aω; ω =
v
max
A
=
0, 2
0, 15
= 2
−1
.
a = ˙v = [ωA cos(ωt)]
0
= −ω
2
A sin(ωt)
a
max
= Aω
2
= 0, 15 · 2
2
= 0, 6 / .
x = 0, 15 sin(2t)
x = A sin ωt;
A = 5
= 0, 05 ; ω = 2
−1
;
= 0, 1 ; F = 5 .
t
v =
dx
dt
= [A sin ωt]
0
= Aω cos ωt; a =
dv
dt
= [Aω cos ωt]
0
= −Aω
2
sin ωt.
= E
0
− T E
0
T
T =
mv
2
2
=
mA
2
ω
2
2
cos
2
ωt.
E
0
=
mA
2
ω
2
2
;
=
mA
2
ω
2
2
1 − cos
2
ωt
=
mA
2
ω
2
2
sin
2
ωt
m =
F
a
= −
F
Aω
2
sin ωt
;
= −
F
Aω
2
sin ωt
·
mA
2
ω
2
2
sin
2
ωt = −
F A
2
sin ωt;
sin ωt = −
2
F A
= −
2 · 0, 1 · 10
−3
5 · 10
−3
· 0, 05
= −0, 8;
ωt = 2π − 0, 927 = 5, 356
; t =
5, 356
2
= 2, 68
.
; d = 20 .
ν
&%
'$
a
L =
J
ma
J m a
O C a = R/2 = d/4
J
c
=
1
2
mR
2
O
J = J
c
+ ma
2
=
1
2
mR
2
+ m ·
R
2
2
=
3
4
mR
2
=
3
16
md
2
.
L =
3
16
md
2
m ·
1
4
d
=
3
4
d
ν =
1
2π
r
g
L
=
1
2π
s
4
3
·
g
d
=
1
2 · 3, 14
s
4
3
·
9, 8
0, 2
= 1, 29
.
; R = 40
T
&%
'$
T = 2π
s
L
g
L = J/ma J
m a O
C a = R
J
c
=
1
2
mR
2
O
J = J
c
+ ma
2
=
1
2
mR
2
+ m · R
2
=
3
2
mR
2
.
L =
3
2
mR
2
mR
=
3R
2
; T = 2π
s
3R
2g
;
T = 2 · 3, 14
s
3 · 0, 4
2 · 9, 81
= 1, 55
.
m
1
= m
2
= m; l = 30 .
L, T
e
C
C
C
C
C
C
C
C
u
u
O
m
m
C
C
CO
C
C
CW
a
l
L =
J
Ma
J M = 2m
a
a =
3
4
l
J = m
l
2
!
2
+ ml
2
=
5
4
ml
2
;
L =
5
4
ml
2
2m ·
3
4
l
=
5
6
l =
5
6
· 0, 3 = 0, 25
;
T = 2π
s
L
g
= 2 · 3, 14 ·
s
0, 25
9, 8
= 1
.
m = 2 = 0, 002 ;
A = 4 = 0, 04 ; ν = 5 .
T
max
x = A sin(2πνt)
v = ˙x = [A sin(2πνt)]
0
= 2πνA cos(2πνt)
v
max
= 2πνA
T
max
=
mv
2
max
2
=
m(2πνA)
2
2
T
max
=
0, 002 · (2 · 3, 14 · 5 · 0, 04)
2
2
= 0, 00157
.