Яковенко Г.Н. Краткий курс теоретической механики
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ωωω
ωω
i
{ωωω
ωω
i
}
{F −F}
F −F
M
O
M
O
6= 0
6= 0
6= 0
M
O
M
O
M
O
O
M
O
(R, M
O
) 6= 0
i
1
i
2
i
3
V
A
= v(i
2
+ i
3
) V
B
= v(i
2
+ i
3
) V
C
= −vi
3
A B C
r
A
= ai
1
r
B
= ai
2
r
C
= ai
3
V
B
= V
A
+ [ωωω
ωω
, AB], [ωωω
ωω
, AB] = 0 (V
A
= V
B
),
ωωω
ωω
= kAB = k(r
B
− r
A
) = ka(i
2
− i
1
).
ωωω
ωω
V
C
= V
A
+[ωωω
ωω
, AC] AC = r
C
−r
A
k = −v/a
2
ωωω
ωω
= (i
1
−i
2
)v/a
O V
O
= V
A
+ [ωωω
ωω
, AO] = vi
2
AO = −r
A
= −ai
1
R = ωωω
ωω
= (i
1
− i
2
)v/a
M
O
= V
O
= vi
2
r =
1
R
2
[R, M
O
] + λR = a(λi
1
− λi
2
+ i
3
/2).
x
1
= λa x
2
= −λa x
3
= a/2
λ
x
1
+ x
2
= 0 x
3
= a/2 R
M
R = ωωω
ωω
= (i
1
− i
2
)v/a, M = V = (i
2
− i
1
)v/2.
V = |M| = v/
√
2
ω = |R| = v
√
2/a
i
1
i
2
i
3
i
3
G = −G i
3
F = F i
2
P = P i
1
r
G
= l i
3
r
F
= H i
3
r
P
= h i
3
M
O
O
R = P i
1
+ F i
2
− G i
3
, M
O
= −F H i
1
+ P h i
2
.
R
2
= P
2
+ F
2
+ G
2
, (R, M
O
) = P F (h − H),
[R, M
O
] = P Gh i
1
+ F Gh i
2
+ (P
2
h + F
2
H) i
3
r =
P Gh i
1
+ F Gh i
2
+ (P
2
h + F
2
H) i
3
P
2
+ F
2
+ G
2
+ λ(P i
1
+ F i
2
− G i
3
).
x
3
λ = (P
2
h + F
2
H)/{G(P
2
+ F
2
+ G
2
)}
{x
1
, x
2
}
x
1
=
P (P
2
h + F
2
H + G
2
h)
G(P
2
+ F
2
+ G
2
)
, x
2
=
F (P
2
h + F
2
H + G
2
H)
G(P
2
+ F
2
+ G
2
)
, x
3
= 0,
R = P i
1
+ F i
2
−G i
3
R
M
R =
√
P
2
+ F
2
+ G
2
, M =
P F (h − H)
√
P
2
+ F
2
+ G
2
.
m
V =
˙
r =
Q = mV = m
˙
r
Q = mV = m
˙
r = .
F
t
˙
Q = m
˙
V = mW = m
¨
r = F(t, r, ˙r). (15.1)
t
r V =
˙
r
m
mW = mW = mW + mW + mW = F,
mW = F + J + J . (15.2)
J = −mW , J = −mW = −2m[ωωω
ωω
, V ], (15.3)
e
3
ωωω
ωω
(t) = ω(t)e
3
e
2
r = ye
2
f
O
e
1
e
2
e
3
J
t
ïåð
J
êîð
F
N
G
J
n
ïåð
m
J = −mW = −mW
n
− mW
τ
= J
n
+ J
τ
= mω
2
(t)ye
2
+ m ˙ω(t)ye
1
,
J = −mW = −2m[ωωω
ωω
, V ] = 2mω(t) ˙ye
1
F = −F e
2
sign ˙y G = −mge
3
N
e
2
m¨y = mω
2
(t)y − F F F = fN f
N
N
{e
1
e
3
}
0 = N + G + J
τ
+ J = N − mge
3
+ m ˙ω(t)ye
1
+ 2mω(t) ˙ye
1
.
N
N =
r
m
2
g
2
+
³
m ˙ω(t)y + 2mω(t) ˙y
´
2
N
m
¨y = ω
2
(t)y − f
r
g
2
+
³
˙ω(t)y + 2ω(t) ˙y
´
2
sign ˙y
{F , −F }
F
r
N
m
1
m
2
m
N
r
1
r
C
C
r
2
...
F
âíåøí
-F
âíóòð
F
âíóòð
r
C
=
1
m
N
X
i=1
m
i
r
i
, m =
N
X
i=1
m
i
. (16.1)
r =
N
P
i=1
λ
i
r
i
λ
i
≥ 0
N
P
i=1
λ
i
= 1 λ
i
= m
i
/m
Q =
N
X
i=1
m
i
V
i
=
N
X
i=1
m
i
˙
r
i
. (16.2)
mr
C
=
N
P
i=1
m
i
r
i
t
m
˙
r
C
= mV
C
=
N
X
i=1
m
i
˙
r
i
=
N
X
i=1
m
i
V
i
= Q, (16.3)
Q = mV
C
. (16.4)
t
m
¨
r
C
= m
˙
V
C
=
N
X
i=1
m
i
¨
r
i
=
N
X
i=1
m
i
˙
V
i
=
˙
Q,
m
i
˙
V
i
= m
¨
r
i
= F
i
+ F
i
˙
Q = m
˙
V
C
= mW
C
= m
¨
r
C
=
N
X
i=1
m
i
¨
r
i
=
N
X
i=1
m
i
˙
V
i
=
N
X
i=1
(F
i
+ F
i
)
N
P
i=1
F
i
= 0
˙
Q = R , mW
C
= R , (16.5)
R
R =
N
P
i=1
F
i
R
z
R
z
Q
z
=
V
z
=
R R
R
R = −2
N
X
i=1
m
i
[ωωω
ωω
, V
i
] = −2[ωωω
ωω
,
N
X
i=1
m
i
V
i
] = −2m[ωωω
ωω
, V
C
].
R
R = −mW
C
R = −2m[ωωω
ωω
, V
C
] R = −mW
C
O
K
O
=
N
X
i=1
[r
i
, m
i
V
i
], (17.1)
r
i
O i
K
O
O
A
O
K
O
OA
r
i
m
i
V
i
O A
K
A
= K
O
− [OA, Q], (17.2)
t
˙
K
O
=
N
X
i=1
[r
i
, m
i
˙
V
i
] +
N
X
i=1
[
˙
r
i
, m
i
V
i
],
m
i
˙
V
i
= F
i
+F
i
˙
r
i
= V
i
−V
O
˙
K
O
=
N
X
i=1
[r
i
, F
i
+ F
i
] +
N
X
i=1
[V
i
− V
O
, m
i
V
i
],