Беляк О.А., Дрезин С.В. Динамика материальной точки
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sin ϕ
0
= 0, ϕ
0
= 0 cos ϕ
0
= g/ω
2
R
A
ϕ
0
6= 0 g/ω
2
R ≤ 1
ω >
p
g/R
N
n b B
N
n
= mg cos ϕ + mω
2
R sin
2
ϕ, N
b
= −J
c
= −2mωv
r
cos ϕ, N =
q
N
2
n
+ N
2
b
.
N
v
r
M
0
cos ϕ
0
= g/ω
2
R v
r
= 0
N
b
= 0
N
M
0
= N
n
= mg cos ϕ
0
+ mω
2
R sin
2
ϕ
0
= mω
2
R.
ϕ = ϕ
0
= arccos(g/ω
2
R)
v
r0
= v
0
ϕ ϕ = ϕ
0
+ θ θ
m
θ M
0
R
¨
θ = ω
2
R sin(ϕ
0
+ θ) cos(ϕ
0
+ θ) − mg sin(ϕ
0
+ θ).
A
A
1
M
M
0
ϕ
0
θ
v
0
θ sin θ ≈ θ cos θ ≈ 1
R
¨
θ = ω
2
R(sin ϕ
0
+ θ cos ϕ
0
)(cos ϕ
0
− θ sin ϕ
0
) − g(sin ϕ
0
+ θ cos ϕ
0
).
θ
˙
θ
¨
θ
R
¨
θ =
ω
2
R(cos
2
ϕ
0
− sin
2
ϕ
0
) − g cos ϕ
0
θ + (ω
2
R sin ϕ
0
cos ϕ
0
− g sin ϕ
0
).
cos ϕ
0
= g/ω
2
R, sin ϕ
0
=
p
ω
4
R
2
− g
2
/ω
2
R,
ω
2
R sin ϕ
0
cos ϕ
0
− g sin ϕ
0
= 0
¨
θ +
ω
4
R
2
− g
2
R
2
ω
2
θ = 0.
(ω
4
R
2
− g
2
)/R
2
ω
2
= k
2
,
¨
θ + k
2
θ = 0
θ = C
1
cos kt + C
2
sin kt.
C
1
C
2
t
M
0
v
0
θ(0) = 0
˙
θ(0) = v
0
/R C
1
= 0 C
2
= v
0
/Rk
M M
0
θ =
v
0
Rk
sin kt, ϕ = ϕ
0
+
v
0
Rk
sin kt.
ω
2
< g/R
ϕ
0
= 0 A
v
0
ϕ ˙ϕ ¨ϕ
R ¨ϕ + (g − ω
2
R)ϕ = 0.
k
2
1
= (g− ω
2
R)/R ϕ(0) = 0 ˙ϕ(0) = v
0
/R
A
ϕ =
v
0
Rk
1
sin k
1
t.
m
c a
ω
J
c
mg
ω
e
v
r
J
e
N
y
F
a
y
x
M
O
′
O
1
O
M
O
′
O
1
ω
e
= ω
Ox O
mg F
O x F = cx
N N = N
y
+N
z
J
e
= −mw
e
= −mw
n
e
Ox w
n
e
= ω
2
(a + x)
J
c
= −2mω
e
× v
r
Oz
mw
r
= mg + F + N + J
e
+ J
c
.
Ox
m¨x = −cx + mω
2
(a + x).
¨x + (k
2
− ω
2
)x = ω
2
a,
k
2
= c/m
k
2
− ω
2
> 0 x = x + x
x(t) = C
1
cos k
1
t + C
2
sin k
1
t +
ω
2
a
k
2
1
, k
1
=
√
k
2
− ω
2
.
x(0) = 0 ˙x(0) = 0
C
1
= 0, C
2
= −
ω
2
a
k
2
1
.
x(t) =
ω
2
a
k
2
− ω
2
1 − cos
√
k
2
− ω
2
t
=
2ω
2
a
k
2
− ω
2
sin
2
√
k
2
− ω
2
2
t.
ω
2
− k
2
> 0
x(t) = C
1
e
k
2
t
+ C
2
e
−k
2
t
−
ω
2
a
k
2
2
, k
2
=
√
ω
2
− k
2
.
0 = C
1
+ C
2
−
ω
2
a
k
2
2
, 0 = C
1
k
2
− C
2
k
2
,
C
1
= C
2
=
ω
2
a
2k
2
2
.
x(t) =
ω
2
a
ω
2
− k
2
1
2
e
√
ω
2
−k
2
t
+ e
−
√
ω
2
−k
2
t
− 1
=
2ω
2
a
ω
2
− k
2
ch
√
ω
2
− k
2
t − 1
.
k
2
= ω
2
¨x = ω
2
a,
x(t) =
ω
2
a
2
t
2
+ C
1
t + C
2
.
C
1
= C
2
= 0 x(t) = ω
2
at
2
/2
O
1
l
O OO
1
= a sin pt
l
ϕ
J
e
w
e
O
1
N
x
O
mg
AA
O
1
O
OA ϕ
mg
N J
e
w
e
= −ap
2
sin pt J
e
= −mw
e
O
1
A O
1
ϕ(t)
I ¨ϕ = M + M
e
,
I = ml
2
O
1
M
mg N N
O
1
M = −mgl sin ϕ.
M
e
O
1
M
e
= −J
e
l cos ϕ = −(−map
2
sin pt)l cos ϕ.
ml
2
¨ϕ = −mgl sin ϕ + map
2
l sin pt.
sin ϕ ≈ ϕ cos ϕ ≈ 1
ml
2
¨ϕ + mglϕ = map
2
l sin pt,
ml
2
k =
p
g/l
¨ϕ + k
2
ϕ =
a
l
p
2
sin pt.
ϕ = ϕ + ϕ = A sin kt + B cos kt + C sin pt.
C
C =
ap
2
l(k
2
− p
2
)
,
k 6= p. p = k
ϕ(0) = 0, ˙ϕ(0) = 0,
B = 0, Ak + Cp = 0,
A = −
p
k
C = −
p
3
a
l(k
2
− p
2
)k
.
ϕ =
ap
2
l(k
2
− p
2
)
sin pt −
p
k
sin kt
.
6
◦
w
T T
1
w = 1, 03 /
2
T −T
1
= 0, 0028 T
500
60
◦
12
y
2
= 2px
Ox ω
x
y
O
M
0
x
0
v
0
ω
M
0
x
0
O
v
0
v
r
=
p
2(ω
2
p − g)(x − x
0
) y
max
= v
0
p
p/(g − ω
2
p) ω
2
< g/p
M m
v
0
α
mg