Беляк О.А., Дрезин С.В. Динамика материальной точки
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x
F (t)
mg
FF
p = k
x
x = x + x = x(0) cos kt +
˙x(0)
k
+
P
2k
2
sin kt −
P t
2k
cos kt.
x(0) = 0, ˙x(0) = 0
x = (0,2 sin 4t − 0,8t cos 4t) .
Q
c
1
c
2
c
2
l
Q
c
1
c
1
P f =
P l
3
3EJ
,
EJ l
T
l =
3
v
u
u
u
u
u
t
3EJ
c
1
+ c
2
−
4π
2
Q
T
2
g
c
2
4π
2
Q
T
2
g
− c
1
;
x = −Q
c
2
l
3
+ 3EJ
c
1
c
2
l
3
+ (c
1
+ c
2
)3EJ
cos
s
(c
1
c
2
l
3
+ (c
1
+ c
2
)3EJ)g
(c
2
l
3
+ 3EJ)Q
t.
OA l m
O a O
c
m
a
l
O A
cc
OA
k =
a
l
r
c
m
M 0 , 5
c
1
= c
2
= 1
f = 0,2 f
0
= 0,25
O x
0
= 3
M
c
2
c
1
OO
−0,5 < x < 0,5
T = 0,141 x = −0,2
M
A B
T = 0,5
R
R = 0, 42
M
BAA
A
α
S = 180 sin 10t
R = −29, 4v R c = 5
T T
1
A
α
cc
x = (e
−7,35t
(0, 228 cos 49, 46t −0, 72 sin 49, 46t) + 3, 74 sin(10t −3
◦
30
′
))
T = 0, 12 7 T
1
= 0, 628 ε = 3
◦
30
′
α = 3 0
◦
c
1
= 12 c
2
= 4
t = 0
ξ = 1,5 sin 10t
x = (−1,63 cos 1 7 ,32t − 1,3 sin 17,32t + 2,2 sin 10t)
m
F Q
v
Q = mv.
Q
k
= mv
k
, v
k
= ˙x
k
, k = 1, 2, 3 .
dS
d(mv) = dS, dS = F dt.
[t
0
, t
1
]
mv
1
− mv
0
= S, S =
Z
t
1
t
0
F dt.
mv
1
k
− mv
0
k
= S
k
, k = 1, 2, 3 .
F = 0
v = c,
c = (c
1
, c
2
, c
3
)
˙x
k
= c
k
, k = 1, 2, 3.
F
1
= 0 F
1
= F
2
= 0 ˙x
1
= c
1
, c
1
= const ˙x
1
= c
1
˙x
2
= c
2
c
1
, c
2
= const
F
1
= F
2
= 0
x
3
O
G
O
= r ×mv,
r O
O
dG
O
dt
= M
O
(F ),
dG
O
dt
= r ×F .
dG
k
dt
= M
k
(F ), k = 1, 2, 3.
O
O
dG
O
dt
= 0.
G
O
= const,
r × mv = const.
r × v = c,
x
2
˙x
3
− x
3
˙x
2
= c
1
,
x
3
˙x
1
− x
1
˙x
3
= c
2
,
x
1
˙x
2
− x
2
˙x
1
= c
3
.
c
1
x
1
+ c
2
x
2
+ c
3
x
3
= 0
v
0
v 2dσ/dt
2
dσ
dt
= c,
r ϕ O
r
2
˙ϕ = c.
c
c = r
0
v
0
sin γ
0
,
γ
0
r
0
v
0
F (r) = −
mc
2
r
2
d
2
dϕ
2
1
r
+
1
r
.
T =
mv
2
2
.
dT = F · dr.
δA = F · dr
T
1
− T
0
= A,
A M
0
M
1
A =
Z
M
1
M
0
F · dr.
α = 3 0
◦
T l = 39,2
f = 0,2
Ox
x
mg N
F = fN
O
y
mg
N
α
F
xx
Ox
F = mg(sin α −f cos α).
Ox
d
dt
(m ˙x) = F.
m
˙x = g(sin α −f cos α)t + C
1
.
˙x(0) = 0
C
1
= 0
x = g( sin α − f cos α)
t
2
2
+ C
2
x(0) = 0 C
2
= 0
x = g(sin α − f cos α)
t
2
2
.
x(T ) = 39,2 T = 4,948 ≈ 5