Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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v = const
K K
0
r = r
0
+ Vt
0
, t = t
0
,
V K
0
K
r(t)
t v(t) =
˙
r ≡
dr/dt w(t) =
˙
v =
¨
r w(t)
v r v
t
0
w(t)
r(t)
F
i
i m
i
N
m
i
¨
r
i
= F
i
, i = 1, 2, . . . , N.
r
i0
= r
i
(t
0
) v
i0
= v
i
(t
0
)
t = t
0
r
i
= r
i
(t; t
0
, r
10
, . . . , r
N0
, v
10
, . . . , v
N0
), i = 1, 2, . . . , N.
F r v t
F
i
i
F
i
= F
(int)
i
+ F
(e)
i
,
F
(int)
i
=
N
X
j=1,j6=i
F
ij
i
F
(e)
i
i
i j
F
ij
= −F
ji
.
F U(r, t)
F = −grad U(r, t) ≡ −∇U ≡ −
∂U
∂r
.
U ∂U/∂t = 0 F
F
v v
e H
F =
e
c
[v × H].
F
F = −γv, γ > 0.
P =
N
X
i=1
p
i
, p
i
= m
i
v
i
,
L =
N
X
i=1
L
i
, L
i
= [r
i
× p
i
]
E = T + U,
T =
N
X
i=1
m
i
v
2
i
/2 ,
U =
N
X
i=1
U
(e)
i
(r
i
) +
1
2
N
X
i=1
N
X
j=1,i6=j
U
ij
(|r
i
− r
j
|)
F
(e)
i
= −
∂U
(e)
i
(r
i
)
∂r
i
, F
ij
= −
∂U
ij
(|r
i
− r
j
|)
∂r
i
.
dP
dt
=
N
X
i=1
dp
i
dt
=
N
X
i=1
[F
(int)
i
+ F
(e)
i
] =
N
X
i=1
F
(e)
i
= F
(e)
.
F
(e)
i
= 0, P = const = P
0
dL
dt
=
N
X
i=1
dL
i
dt
=
N
X
i=1
[r
i
× F
(e)
i
] =
N
X
i=1
M
(e)
i
= M
(e)
,
M
(e)
F
(e)
i
F
(e)
i
=
0 M
(e)
= 0
dL
dt
= 0 L = const = L
0
.
E
m e
E = E
0
cos ωt E
0
ω t = 0
r(0) = r
0
v(0) = v
0
r(t) v(t)
r(t) = r
0
+ v
0
t +
eE
0
mω
2
(1 − cos ωt) v(t) = v
0
+
eE
0
mω
sin ωt
m
O
F = −kr
t = 0
r
0
v
0
O
r(t) = r
0
cos ωt +
v
0
ω
sin ωt v(t) = v
0
cos ωt −r
0
ω sin ωt
r
0
v
0
x r
0
x
2
sin
2
α − xy sin 2α + y
2
(cos
2
α + r
2
0
ω
2
/v
2
0
) = r
2
0
sin
2
α
α r
0
v
0
, ω =
p
k/m.
m
F
c
= −kv
r(t) v(t) r(0) = r
0
=
{0, 0, r
0
} v(0) = v
0
= {v
0
cos α, 0, v
0
sin α}.
r(t) = r
0
+
m
k
v
0
(1 − e
−kt/m
) +
m
k
g
h
t −
m
k
(1 − e
−kt/m
)
i
v(t) = v
0
e
−kt/m
+
m
k
g(1 −e
−kt/m
);
v
0
g
z = r
0
+ x
µ
tg α +
mg
kv
0
cos α
¶
+
m
2
g
k
2
ln
µ
1 −
kx
mv
0
cos α
¶
.
t
t
r(t)
r(t) − r
0
(t) =
m
k
v
0
(1 − e
−kt/m
),
r
0
(t) = r
0
+
m
k
g[t −
m
k
(1 − e
−kt/m
)]
v
0
r
0
(t)
R(t) =
m
k
|v
0
|
³
1 − e
−kt/m
´
z = z
0
+
v
2
0
2g
−
gy
2
2v
2
0
.
U(x) = −Ax
4
x(t) =
x(0)
h
1 ± tx(0)
q
2A
m
i
m z = 0 x = a ch kt, y =
b sh kt.
F = {mk
2
x, mk
2
y, 0}.
L = {0, 0, mkab}
E =
1
2
mk
2
(b
2
− a
2
)
z p
z
x
2
a
2
−
y
2
b
2
= 1
e H
A = (L·H)+
e
2c
[r×
H]
2
L
e H = q
r
r
3
A = L −
eq
c
r
r
U(r) =
α
r
= [v × L] +
αr
r
a
E
< F >= 2E/a
H =
{0, 0, H
0
cos
y
a
} r(0) = 0, v(0) = {0, ωa, 0} ω =
eH
0
mc
˙x = aω th(ωt), x = a ln(ch ωt)
˙y =
aω
ch ωt
, y = a arcsin(th ωt).
U(x) = U
0
tg
2
(
x
a
)
F
y
= F
z
= 0 ⇒ ˙y = ˙y
0
= const, ˙z = ˙z
0
= const
∂U
∂t
= 0 ⇒
E = E
0
= const.
m( ˙x
2
+ ˙y
2
0
+ ˙z
2
0
)
2
+ U
0
tg
2
³
x
a
´
= E
0
.
E
1
= E
0
−
m( ˙y
2
0
+ ˙z
2
0
)
2
= const.
dx
dt
= ±
r
2
m
h
E
1
− U
0
tg
2
³
x
a
´i
.
x = a arcsin
(
r
E
1
E
1
+ U
0
sin
"
±
t − t
0
a
r
2(E
1
+ U
0
)
m
+
+ arcsin
r
E
1
+ U
0
E
1
sin
x
0
a
#)
.
τ
sin
2π
τ
a
r
2(E
1
+ U
0
)
m
= 2π, τ = 2πa
r
m
2(E
1
+ U
0
)
.
τ
2
=
Z
x
2
x
1
dx
q
2
m
£
E
1
− U
0
tg
2
(
x
a
)
¤
,
x
1
x
2
m l
x = l cos ϕ, y = l sin ϕ;
˙x = − ˙ϕl sin ϕ, ˙y = ˙ϕl cos ϕ.
-
y
?
x
A
A
A
A
A
A
lϕ
t
mg
?
T =
m
2
( ˙x
2
+ ˙y
2
) =
m
2
l
2
˙ϕ
2
.
U = −mgx = −mgl cos ϕ.
Φ
0
E = T + U = U(Φ
0
) ⇒ ml
2
˙ϕ
2
/2 − mgl cos ϕ = −mgl cos Φ
0
.
dϕ
dt
= ±
r
2g
l
(cos ϕ − cos Φ
0
).