Беляк О.А., Дрезин С.В. Динамика материальной точки
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x x
x =
P
p
(k
2
− p
2
)
2
+ 4p
2
h
2
sin(pt − α),
α = arctg
2hp
k
2
− p
2
.
h = 0
x =
P
k
2
− p
2
sin pt, p 6= k.
p k (p = k)
x = −
P t
2k
cos kt.
F
x
0
O |F | = cx
0
F
∗
= fN = fmg
x
0
x
−x
v
x
−x
x
x
x
0
O
N
N
F
F
mg
F
mg
F
v
|F | 6 F
∗
|x
0
| 6
fmg
c
,
F
−x 6 x 6 x x = fmg/c
|x
0
| > x
F
∗
F > F
∗
m¨x = −cx −F
∗
˙x
|˙x|
,
v = ˙x
x
0
= x
01
> 0
v = ˙x < 0
¨x + k
2
x =
F
∗
m
,
x(0) = x
01
, ˙x(0) = 0
x(t) = (x
01
− x ) cos kt + x , ˙x 6 0,
v(t) = ˙x(t) = −k(x
01
− x ) sin kt 6 0.
x
0
= x
02
< 0
v = ˙x > 0
¨x + k
2
x = −
F
∗
m
,
x(0) = x
02
, ˙x(0) = 0
x(t) = (x
02
+ x ) cos kt − x , ˙x > 0,
v(t) = ˙x(t) = −k(x
02
+ x ) sin kt > 0.
(x, ˙x) t
(x − x )
2
(x
01
− x )
2
+
˙x
2
k
2
(x
01
− x )
2
= 1, ˙x 6 0,
(x + x )
2
(x
02
+ x )
2
+
˙x
2
k
2
(x
02
+ x )
2
= 1, ˙x > 0.
M(x, ˙x)
M
1
M
2
i = 3i = 2 i = 1i = 1 i = 2
˙x
M
1
M
2
x
02
x
12
x
11
x
01
7x5x3 x
−5x
x
−x
−3x
O
xx
x
x
11
x
02
t
x sin kt = 0 t = 0 x = x
01
x = x
02
t = π/k
x
11
x
12
x
11
x
12
x
11
= −(x
01
− x ) + x = 2x − x
01
,
x
12
= −(x
02
+ x ) −x = −2x − x
02
,
|x
11
| < | x
01
| |x
12
| < | x
02
|
O
2x
T =
π
k
.
O
i
−x 6 |x
0
| − 2x i 6 x ,
(2i − 1)x 6 |x
0
| 6 (2i + 1)x .
Ox
2x i, i ∈ N
i = 0
x
0
i
T
∗
= iT = iπ/k i = 3
AB A
1 B 19,6
B
C 0,1
x
∆
x
∆
O
B
B
B
A
A
F
mg
C
A
F
mg
C mg F
C Ox
O
mg F
∆
∆
∆ = ∆ + x.
F = c∆ = c(∆ + x)
c ∆
∗
= 1 F
∗
= 19,6
c =
F
∗
∆
∗
= 19,6
C Ox
m¨x = −F + mg
B
˙x(0) = 0, x(0) = −∆ .
∆ C
0 = mg − c∆
∆ =
mg
c
=
0, 1 · 9,81
19,6
≈ 0,05
m¨x = −c∆ − cx + mg.
¨x +
c
m
x = 0
C k =
r
c
m
k
C T =
2π
k
¨x + k
2
x = 0
λ
2
+ k
2
= 0.
λ
1,2
= ±ik C
x = C
1
cos kt + C
2
sin kt
C
1
, C
2
−
C
1
= x(0) = −∆ , C
2
=
˙x(0)
k
= 0
x = −∆ cos kt.
A = ∆ = 0, 05
k =
r
c
m
=
r
19, 6
0, 1
c
−1
= 14 c
−1
,
T =
2π
k
= 2π
r
m
c
≈ 0 , 45 c,
f =
1
T
≈ 2
x = −0, 05 cos 14t .
c = 19,6
m
1
= 0, 5 m
2
= 0, 8
O
m
1
g
(m
1
+ m
2
)g
∆
x
∆
1
Ox
O
∆
1
∆
1
= m
1
g/c
∆ ∆ = ∆
1
+ x
¨x + k
2
x = 0, k =
r
c
m
1
˙x(0) = 0; x(0) = ∆ −∆
1
= ∆
2
=
m
2
g
c
,
∆
∆
2
∆
2
=
m
2
g
c
=
0, 8 · 9, 8
19, 6
= 0, 4
x(t) = A sin(kt + ϕ).
A = 0, 4 , ϕ =
π
2
.
k =
r
c
m
1
=
r
19, 6
0, 5
−1
≈ 6, 261
−1
T =
2π
k
≈ 1 f = 1/T = 1 .
x(t) = 0, 4 cos 6, 26t .
m
1
= 2
c = 98
m
2
= 0, 8
Ox
O
(m
1
+m
2
)g F
∆
∆
∆ = ∆ + x
∆ =
(m
1
+ m
2
)g
c
.
∆
2
m
1
g
m
2
g
m
1
g
m
2
g
∆
F
F
x
∆ = ∆ + x
x
∆
1
O
F
m
1
gm
1
g
(m
1
+ m
2
)¨x = −c(∆ + x) + (m
1
+ m
2
)g,
(m
1
+ m
2
)¨x = −cx.
x(0) = −(∆ − ∆
1
) = −∆
2
˙x(0) = 0 ∆
1
∆
2
∆
1
=
m
1
g
c
=
2 · 9,8
98
= 0,2 ∆
2
=
m
2
g
c
=
0,8 · 9,8
98
= 0,08 .
x = C
1
cos kt + C
2
sin kt,
k =
r
c
m
1
+ m
2
=
r
98
2 + 0, 8
c
−1
≈ 5 , 916 c
−1
T = 2π/k ≈ 1, 062 c C
1
C
2
C
1
= x(0) = −0, 08, C
2
=
˙x(0)
k
= 0.
x(t) = −0, 08 cos 5, 916t
m α
c
v
0
x
N
α
F
mg