Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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m R
U(r) = −
α
r
2
U(r) = −
β
r
v
0
= 0
∆t =
r
m
2α
R
2
∆t = πR
s
mR
8β
.
r
0
U(r) =
1
3
κr
3
v
0
±
π
2
v
0
v
0
=
r
κ
m
r
3
0
.
r(ϕ) =
A
ϕ
, A = const.
U(r) = E −
L
2
2m
µ
1
A
2
+
1
r
2
¶
E L
U(r) =
½
0, r > a
∞, r 6 a.
ρ χ
ϕ
0
=
π − χ
2
; ρ = a sin ϕ
0
= a cos
χ
2
.
dσ
dΩ
=
a cos
³
χ
2
´
sin χ
·
a
2
sin
³
χ
2
´
=
a
2
4
.
O
-
χ
ϕ
0
ϕ
0
ρ
J
J
J
J
J
J
J
J
J
J
J
J
a
σ =
Z
a
2
4
dΩ = πa
2
,
m
1
m
2
a ε
ε = ε
0
sin
2
χ
2
ε
0
=
2m
1
m
2
2
(m
1
+ m
2
)
2
v
2
1∞
=
4m
1
m
2
(m
1
+ m
2
)
2
E
1
χ = π
dε = ε
0
sin
χ
2
cos
χ
2
dχ.
dσ = πa
2
dε
ε
0
.
m
U(r) =
α
r
2
.
dσ
dΩ
=
2π
2
|α|(π − χ)
mv
2
∞
(2π − χ)
2
χ
2
sin χ
.
U(r) = −
α
r
2
.
U =
L
2
2mr
2
U(r)
L
2
2m
< α
ρ 6 ρ
0
=
s
2α
mv
2
∞
.
σ = πρ
2
0
=
2πα
mv
2
∞
.
U(r) = −
α
r
n
(n > 2, α > 0).
σ =
πn
n − 2
·
α(n − 2)
mv
2
∞
¸
2
n
.
m
1
m
2
(m
2
À m
1
) R
U(r) = −γ
m
1
m
2
r
.
σ = πR
2
µ
1 +
2γm
2
Rv
2
∞
¶
.
x
1
, x
2
, x
3
X, Y, Z v
P
P
v
P
= V + [Ω × r],
r P
x
1
, x
2
, x
3
, V
Ω
x
1
, x
2
, x
3
T L
T =
1
2
X
i,k=1,2,3
I
ik
Ω
i
Ω
k
,
L
i
=
X
k=1,2,3
I
ik
Ω
k
,
I
ik
=
X
a
m
a
¡
r
2
a
δ
ik
− x
ai
x
ak
¢
V
I
ik
=
Z
V
ρ(r)(r
2
δ
ik
− x
i
x
k
) dV.
I
ik
(I
ik
= I
ki
) x
1
, x
2
, x
3
I
ik
I
ik
= I
i
δ
ik
,
I
1
, I
2
, I
3
T =
1
2
¡
I
1
Ω
2
1
+ I
2
Ω
2
2
+ I
3
Ω
2
3
¢
,
L
i
= I
i
Ω
i
.
T =
mV
2
c
2
+
1
2
X
i,k=1,2,3
I
(c)
ik
Ω
i
Ω
k
,
V
c
I
(c)
I
1
= I
2
=
m
1
m
2
m
1
+ m
2
l
2
, I
3
= 0
I
3
= 0
I
1
= I
2
=
X
a
m
a
z
2
a
− M
P
a
m
a
z
a
M
2
=
=
1
M
X
a
m
a
z
2
a
X
b
m
b
−
Ã
X
a
m
a
z
a
!
2
=
1
M
X
a,b
¡
m
a
m
b
z
2
a
− m
a
m
b
z
a
z
b
¢
,
M =
X
a
m
a
X
a,b
m
a
m
b
z
2
a
=
X
a,b
m
a
m
b
z
2
b
I
1
=
1
2M
X
a,b
¡
m
a
m
b
z
2
a
+ m
a
m
b
z
2
b
− 2m
a
m
b
z
a
z
b
¢
=
1
M
X
a,b (a>b)
m
a
m
b
l
2
ab
,
l
ab
= z
a
− z
b
a b
a h
l =
m
2
h
2m
1
+ m
2
I
1
=
2m
1
m
2
h
2
2m
1
+ m
2
, I
2
=
m
1
2
a
2
, I
3
= I
1
+ I
2
.
t t
t
%
%
%
%
%
%
%
e
e
e
e
e
e
e
a
h
m
1
m
1
m
2
x
2
x
1
l
I
1
= I
2
=
1
12
ml
2
, I
3
= 0.
R
I
1
= I
2
= I
3
=
2
5
mR
2
.
R h
I
1
= I
2
=
m
4
(R
2
+
h
2
3
), I
3
=
m
2
R
2
.
a, b, c
I
1
=
m
12
(b
2
+ c
2
), I
2
=
m
12
(c
2
+ a
2
), I
3
=
m
12
(a
2
+ b
2
).
h R
I
1
= I
2
=
3
20
m(R
2
+
h
2
4
), I
3
=
3
10
mR
2
.
a
x
x, ˙x
U = −mgx sin α;
T =
1
2
m ˙x
2
+
1
2
I
3
˙
θ
2
,
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
&%
'$
s
H
H
H
H
H
H
H
H
H
H
H
Hj
x
?mg
¢
¢
a
α
θ
˙x = a
˙
θ
I
3
L =
3
4
m ˙x
2
+ mgx sin α
¨x =
2
3
g sin α.
l
ϕ
ϕ
U = −mgl cos ϕ ≈ −mgl +
1
2
mglϕ
2
.
T =
1
2
mv
2
+
1
2
X
I
i
Ω
2
i
,
-r
?
@
@
@
@
@
r
ϕ
l
v = l ˙ϕ Ω
i
α, β, γ
α, β, γ
L =
1
2
ml
2
˙ϕ
2
+
1
2
¡
I
1
cos
2
α + I
2
cos
2
β + I
3
cos
2
γ
¢
˙ϕ
2
−
1
2
mglϕ
2
.
ω
2
=
mgl
ml
2
+ I
1
cos
2
α + I
2
cos
2
β + I
3
cos
2
γ
.
3
8
R
m
I =
2
5
mR
2
− m
µ
3
8
R
¶
2
=
83
320
mR
2
.
ϕ Z
c
=
R(1 −3/8 cos ϕ)
L =
1
2
m
˙
Z
2
c
+
1
2
I ˙ϕ
2
− mgZ
c
L =
1
2
I ˙ϕ
2
−
3
16
mgRϕ
2
.
?
r
Z
c
@
@
@
@
@
@
@
@
@
@
@
@
/ / / / / / / / / / / / / /
ϕ
R
ω =
r
120g
83R
;
ω =
r
15g
26R
.
T =
ml
2
3
(1 + 3 sin
2
ϕ) ˙ϕ
2
.
#
#
#
#
#
#
#
c
c
c
c
c
c
c
c
c
/
/
/
/
/
/
/
/
/ / / / / / / / / / / / / / / / / / / / /
ϕ
l l
R
T =
3
4
m(R − a)
2
˙ϕ
2
.
&%
'$
¡
¡
¡
¡
¡
@
@
@
@
@R
B
BM
a
ϕ
R
ϕ
V = a ˙ϕ cos α α a
Ω =
V
a sin α
= ˙ϕ ctg α.
A
O
:
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
a
¢
¢
¢®
x
-
y
6
z
ϕ
Ω
Ω sin α Ω cos α
T =
ma
2
2
cos
2
α
˙
ϕ
2
+
1
2
µ
I
1
cos
2
α + I
3
cos
4
α
sin
2
α
¶
˙
ϕ
2
=
3mh
2
40
¡
1 + 5 cos
2
α
¢
˙
ϕ
2
,
h I
1
, I
3
, a = 3h/4
L
m U(r)
L(r, v) =
mv
2
2
− U(r)
K
0
. K
K
0
V(t) Ω(t)
L(r, v) =
mv
2
2
− m(v · [r × Ω]) +
m
2
[r × Ω]
2
− m(
˙
V · r) − U(r)
m
dv
dt
= −
∂U
∂r
− m
˙
V + 2m[v × Ω] + m[r ×
˙
Ω] + m[Ω × [r × Ω]].
F = 2m[v × Ω],
F
..
= m[r ×
˙
Ω],
F
..
= m[Ω × [r × Ω]],
F = −m
˙
V
˙
V(t)
Ω = const p =
∂L
∂v
L = [r × p]
K
0
p = p
0
, L = L
0
,
E E
0
E = E
0
− L · Ω.
m
dv
0
dt
= −
∂U
∂r
0
+ 2m
£
v
0
× Ω
¤
+ m
£
Ω ×
£
r
0
× Ω
¤¤
,
r
0
Ω
O
r
0
= R + r R O
˙
R = 0
m
dv
dt
= mg + 2m
£
v × Ω
¤
+ m
£
Ω ×
£
r × Ω
¤¤
,
g = −
γM
R
3
R +
£
Ω ×
£
R ×Ω
¤¤
M
g
Ω
m
dv
dt
= mg + 2m
£
v × Ω
¤
.
v = v
(1)
+ v
(2)
+ v
(3)
+ . . . .
Ω
v
(2)
m
d
dt
¡
v
(1)
+ v
(2)
+ . . .
¢
= mg + 2m
£¡
v
(1)
+ . . .
¢
× Ω
¤
Ω
dv
(1)
dt
= g;
dv
(2)
dt
= 2
£
v
(1)
× Ω
¤
;
v
(1)
= gt+v
0
dv
(2)
dt
= 2
£
v
0
× Ω
¤
+ 2
£
g × Ω
¤
t.
v
(2)
= 2[v
0
× Ω]t + [g × Ω]t
2
.
v
(1)
, v
(2)
v = v
0
+ gt + 2[v
0
× Ω]t + [g × Ω]t
2
+ . . .