Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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r = r
0
+ v
0
t +
gt
2
2
+ [v
0
× Ω]t
2
+
1
3
[g × Ω]t
3
+ . . . .
θ g X
Y
g
x
= g
y
= 0, g
z
= −g;
Ω
x
≈ −Ω cos θ, Ω
y
= 0, Ω
z
≈ Ω sin θ.
t = 0
r
0
= (0, 0, h), v
0
= 0.
@
@
@
@I
R
6
Ω
À
x
H
H
H
Hj
y
@
@
@I
z
θ
z z = h −
gt
2
2
z = 0
t =
p
2h/g x y
x = 0, y =
1
3
[g × Ω]
y
t
3
=
gΩ
3
cos θ
µ
2h
g
¶
3/2
.
gΩ
3
µ
2h
g
¶
3/2
cos θ.
v
0
.
r = v
0
t +
gt
2
2
+ [v
0
× Ω]t
2
+
1
3
[g × Ω]t
3
+ . . .
zx v
0
z = 0
v
0z
t −
gt
2
2
+ v
0x
Ω
y
t
2
= 0.
t =
2V
0z
g − 2Ω
y
V
0x
'
2V
0z
g
.
y
y = (v
0z
Ω
x
− v
0x
Ω
z
)
µ
2V
0z
g
¶
2
−
gΩ
x
3
µ
2V
0z
g
¶
3
=
4V
2
0z
g
2
µ
V
0z
Ω
x
3
− V
0x
Ω
z
¶
.
U = mg(l −
p
l
2
− x
2
− y
2
).
|x| ¿ l, |y| ¿ l.
U
∼
=
mω
2
2
(x
2
+ y
2
),
A
A
A
A
A
A
A
A
A
A
A
l
u
?
g
p
x
2
+y
2
ω
2
=
g
l
. Ω
m
dv
dt
= mg + 2m[v × Ω],
¨x + ω
2
x = 2 ˙yΩ
z
− 2 ˙zΩ
y
;
¨y + ω
2
y = −2 ˙xΩ
z
+ 2 ˙zΩ
x
.
z = l −
p
l
2
− x
2
− y
2
˙z =
x ˙x + y ˙y
p
l
2
− x
2
− y
2
|˙z| ¿ |˙x| + |˙y| ˙zΩ
y
˙zΩ
x
i
¨
ξ + 2iΩ
z
˙
ξ + ω
2
ξ = 0,
ξ = x + iy ξ = e
iαt
α
−α
2
− 2αΩ
z
+ ω
2
= 0.
α
α = −Ω
z
±
p
Ω
2
z
+ ω
2
= −Ω
z
± ω
r
1 +
Ω
2
z
ω
2
∼
=
−Ω
z
± ω.
ξ = e
−iΩ
z
t
¡
A
1
e
iωt
+ A
2
e
−iωt
¢
,
A
1
A
2
ξ
=
x
(
t
) +
iy
(
t
)
x
0
(t) iy
0
(t)
x(t) + iy(t) = e
−iΩ
z
t
[x
0
(t) + iy
0
(t)] ,
x
0
(t) y
0
(t)
x(t) = x
0
(t) cos Ω
z
t + y
0
(t) sin Ω
z
t;
y(t) = −x
0
(t) sin Ω
z
t + y
0
(t) cos Ω
z
t.
Ω
z
m
Ω z
, y, z
E =
mv
2
0
2
+ mgz −[r × p] · Ω =
mv
2
2
+ mgz −
m
2
[r × Ω]
2
,
v
0
= v + [Ω × r], p = mv
0
x
v = 0 [r×Ω]
2
z =
Ω
2
2g
x
2
+
E
mg
.
q = q
0
∂U
∂q
¯
¯
¯
¯
q=q
0
= 0,
∂
2
U
∂q
2
¯
¯
¯
¯
q=q
0
≡ k > 0.
q
0
ω =
r
k
m
,
x ≡ q − q
0
x = a cos(ωt + α),
a α
E =
1
2
mω
2
a
2
¨x + ω
2
x =
1
m
F (t).
F (t) = f cos(γt + β)
x(t) = a cos(ωt + α) +
f
m(ω
2
− γ
2
)
cos(γt + β).
F (t)
x(t) =
1
ω
Im ξ(t), ξ(t) = e
iωt
Z
t
t
0
F (t
0
)
m
e
−iωt
0
dt
0
+ ξ
0
e
iω(t−t
0
)
,
ξ
0
= ˙x(t
0
) + iωx(t
0
)
F (t) → 0 t → ±∞
∆E =
∆p
2
2m
, ∆p =
¯
¯
¯
¯
Z
∞
−∞
F (t)e
−iωt
dt
¯
¯
¯
¯
f = −α ˙x
¨x + 2λ ˙x + ω
2
0
x = 0, λ =
α
2m
.
λ < ω
0
x(t) = ae
−λt
cos(
q
ω
2
0
− λ
2
t + α).
E(t) = E(0)e
−2λt
.
F (t) = f cos γt
x(t) = ae
−λt
cos(
q
ω
2
0
− λ
2
t + α) + b cos(γt + δ),
b =
f
m
p
(ω
2
0
− γ
2
)
2
+ 4λ
2
γ
2
, tg δ =
2λγ
γ
2
− ω
2
0
.
I(γ) =
f
2
4m
·
λ
(γ − ω
0
)
2
+ λ
2
.
a α
x
0
V
0
a =
r
x
2
0
+
V
2
0
ω
2
, tg α = −
V
0
ωx
0
.
ω ω
0
m
1
, m
2
m
0
1
, m
0
2
.
ω
0
ω
=
s
m
1
m
2
(m
0
1
+ m
0
2
)
m
0
1
m
0
2
(m
1
+ m
2
)
.
m,
A l l
F.
U(x) =
k
2
³
p
l
2
+ x
2
− l
0
´
2
,
l
0
U(x) x ¿ l :
-
6
A
l
0
m
x
t?
r
q
q
q
q
q
q
q
q
q
q
q
q
q
q
U(x) =
k
2
"
l
r
1 +
x
2
l
2
− l
0
#
2
∼
=
k
2
·
l
µ
1 +
x
2
2l
2
¶
− l
0
¸
2
=
k
2
µ
l − l
0
+
x
2
2l
¶
2
∼
=
∼
=
k
2
·
(l − l
0
)
2
+
l − l
0
l
x
2
+ ...
¸
= U(0) +
k(l − l
0
)x
2
2l
+ . . . ,
U(0) ≡
k
2
(l − l
0
)
2
x = 0
¯
¯
¯
¯
∂U(0)
∂l
¯
¯
¯
¯
=
k(l − l
0
) k(l − l
0
) = F, U(x) = U(0) + F x
2
/2l.
ω =
p
k/m k F/l
ω =
p
F/lm.
r.
ω =
r
F (r + l)
lmr
.
t
6
6
r
?
l
Á
r
ϕ
q
q
q
q
q
q
q
q
m
2
m
1
˙x(m
1
+m
2
)+m
2
l ˙ϕ cos ϕ = const,
x m
1
l ϕ
˙x = −
m
2
l cos ϕ
m
1
+ m
2
˙ϕ.
˙x ϕ ¿ 1
L =
m
1
m
2
l
2
2(m
1
+ m
2
)
˙ϕ
2
−
m
2
gl
2
ϕ
2
+ const.
ω =
s
g(m
1
+ m
2
)
m
1
l
.
a À |z|
U(z) |z| ¿ a
U(z) =
κ
2
(
p
a
2
+ 4z
2
−a)
2
−mgz =
κ
2
(2a
2
+ 4z
2
−2a
p
a
2
+ 4z
2
) −mgz =
=
κ
2
(
2a
2
+ 4z
2
− 2a
2
"
1 +
1
2
·
4z
2
a
2
−
1
8
µ
4z
2
a
2
¶
2
#)
−mgz =
2κ
a
2
z
4
−mgz.
z
0
U(z)
∂U
∂z
=
8z
3
κ
a
4
− mg = 0; z
0
=
3
r
mga
2
8κ
.
U(z)
∂
2
U
∂z
2
¯
¯
¯
¯
z=z
0
=
24κz
2
a
2
¯
¯
¯
¯
z=z
0
=
24κz
2
0
a
2
U(z) = U(z
0
) +
1
2
·
24κz
2
0
a
2
(z − z
0
)
2
+ . . . ,
ω =
r
24κz
2
0
ma
2
=
s
6
µ
κg
2
ma
2
¶
1
3
.
m
κ l
0
x
∨∨∨∨∨∨∨∨∨∨∨∨∨∨∨∨
t
m
¾ -
2l
U(x) =
κ
2
(l + x − l
0
)
2
+
κ
2
(l − x − l
0
)
2
= κ x
2
+ κ(l − l
0
)
2
.
ω =
p
2κ/m.
κ
1
l
1
κ
2
l
2
ω =
r
κ
1
+ κ
2
m
κ
2
(l − l
2
) − κ
1
(l − l
1
)
κ
1
+ κ
2
.
m
U(x) = V cos(αx) − F x, (|F | < |V α|) .
ω
2
=
|V |α
2
m
s
1 −
µ
F
V α
¶
2
.
a
s(t)
L =
m
2
¡
l
2
˙ϕ
2
+ 2 ˙s ˙ϕl cos ϕ + ˙s
2
¢
+ mgl cos ϕ.
¨ϕ +
³
a
l
´
cos ϕ +
g
l
sin ϕ = 0,
¨s = a ϕ
0
g
l
sin ϕ
0
+
a
l
cos ϕ
0
= 0
ϕ
0
= −arctg
a
g
. ϕ = ϕ
0
+ θ
¨
θ +
³
a
l
cos ϕ
0
+
g
l
sin ϕ
0
´
cos θ +
³
g
l
cos ϕ
0
−
a
l
sin ϕ
0
´
sin θ = 0.
¨
θ +
p
a
2
+ g
2
l
sin θ = 0.
ω =
µ
a
2
+ g
2
l
2
¶
1
4
.
±
ω =
r
a + g
l
ω =
r
|g − a|
l
a > g
π
ω t = 0
x
0
˙x
0
m¨x + kx = F,
F
k
F
mω
2
ω =
r
k
m
. x = A cos(ωt +
α)
x(t) = A cos(ωt + α) +
F
mω
2
.
˙x(t) = −ωA sin(ωt+α).
α = 0 A = −
F
mω
2
x(t) =
µ
F
mω
2
¶
(1 − cos ωt).
ω
F = at
v
0
x
0
¨x + ω
2
x =
at
m
a
mω
2
t
x(t) = A cos ωt + B sin ωt +
at
mω
2
˙x(t) = −Aω sin ωt + Bω cos ωt +
a
mω
2
.
A = x
0
, B =
v
0
ω
−
a
mω
3
x(t) = x
0
cos ωt +
³
v
0
ω
−
a
mω
3
´
sin ωt +
at
mω
2
F = F
0
e
−βt
m¨x + kx = F
0
e
−βt
.
Be
−βt
B
Be
−βt
¡
β
2
+ ω
2
¢
=
F
0
m
e
−βt
,
ω =
r
k
m
x(t) = A cos(ωt + α) +
F
0
m(β
2
+ ω
2
)
e
−βt
.
˙x(t) = −Aω sin(ωt + α) −
βF
0
m(β
2
+ ω
2
)
e
−βt
.
0 = A cos α +
F
m(β
2
+ ω
2
)
; 0 = −Aω sin α −
βF
m(β
2
+ ω
2
)
.
α A
tg α =
β
ω
, A = −
F
m(β
2
+ ω
2
)
1
cos α
.
α, A
x(t) =
F
0
m(β
2
+ ω
2
)
µ
e
−βt
− cos ωt +
β
ω
sin ωt
¶
.
ω
0
F = F
0
cos ωt t = 0 x = 0 ˙x = 0
ω = ω
0
.