Беляк О.А., Дрезин С.В. Динамика материальной точки
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1
2
d
dy
(u
2
) = −g(1 + ku).
d(u
2
)
2(1 + ku)
= −gdy,
udu
1 + ku
= −gdy.
u
k
−
1
k
2
ln(1 + ku) = −gy + C.
y(0) = 0 , u(0) = ˙y(0) = v
0
sin α,
C
C =
1
k
v
0
sin α −
1
k
2
ln(1 + kv
0
sin α).
y u = v
y
y =
1
g
1
k
v
0
sin α −
1
k
2
ln(1 + kv
0
sin α) −
v
y
k
+
1
k
2
ln(1 + kv
y
)
.
h h = y
u=0
h =
v
0
sin α
gk
−
1
gk
2
ln(1 + kv
0
sin α),
m e
E = A cos kt A k v
0
F = −eE
O Ox
v
0
Oy
x
y
F
v
0
E
O
m¨x = 0,
m¨y = −eA cos kt,
x(0) = 0, ˙x(0) = v
0
,
y(0) = 0 , ˙y(0) = 0.
x = C
1
+ C
2
t,
y = C
3
+ C
4
t + y ,
y = B cos kt B =
eA
mk
2
C
1
C
2
C
3
C
4
C
1
= 0, C
2
= v
0
,
C
3
+ B = 0 , C
4
= 0.
x = v
0
t,
y = −
eA
mk
2
+
eA
mk
2
cos kt.
t
y = −
eA
mk
2
1 − cos
k
v
0
x
.
M m
O
1
O
2
O
1
O = OO
2
O
1
O
2
c
A
0
v
0
O
1
O
2
M
O
1
O
2
A
0
O 2a
O Ox
O
1
O
2
M (x; y)
MO
1
= (−a − x; −y) MO
2
= (a − x; −y)
F
1
= cMO
1
= −c(a + x; y), F
2
= cMO
2
= −c(x − a; y) .
x
y
M
OO
1
O
2
A
0
aa a
F
1
F
2
v
0
m
¨
r = F
1
+ F
2
,
m¨x = −c(a + x) − c(x − a),
m¨y = −2cy.
m¨x + 2cx = 0,
m¨y + 2cy = 0.
A
0
v
0
x(0) = −2a, ˙x(0) = 0,
y(0) = 0 , ˙y(0) = v
0
.
¨x + k
2
x = 0,
¨y + k
2
y = 0,
k =
p
2c/m
x = C
1
sin kt + C
2
cos kt,
y = C
3
sin kt + C
4
cos kt.
C
2
= −2a, C
1
= 0,
C
4
= 0, C
3
=
v
0
k
.
M
x = −2a cos kt,
y =
v
0
k
sin kt.
t
x
2
(2a)
2
+
y
2
(v
0
/k)
2
= 1,
2a v
0
/k = v
0
p
m/2c
M O
1
O
2
y =
v
0
k
sin kt = 0
sin kt = 0 t
n
=
πn
k
n ∈ Z
M O
1
O
2
t
0
= 0
x
0
= −2a y
0
= 0 A
0
M
O
1
O
2
t
1
= π/k x
1
= 2a y
0
= 0
t
2
= 2π/k A
0
x
2
= −2a y
0
= 0
M
T = 2π/k = 2π
p
m/2c
M
0,5mge
−kt
k m
M
α = 30
◦
x =
g
2
t
2
2
−
t
k
−
1
k
2
e
−kt
+
1
k
2
x
A m v
0
mk
f α
x =
1
2k
ln
g(sin α + f cos α) + kv
2
0
g(sin α + f cos α) + kv
2
m
v
0
= 0
v = k(1 −e
−
gt
k
) S = kt +
k
2
g
(e
−
gt
k
− 1) k
m v
0
α
y = x tg α − x
2
g
2v
2
0
cos
2
α
m e
E v
0
F = eE
E
mv
2
0
/(eE)
P = 1,96
l = 0,35
v
0
= 4,9
f = 0,25
v = 2,45 T = 3,43
x
F = cx c (c > 0)
[c] = =
2
Ox O
x
x
O
F
M
M
M
m¨x = −cx; ¨x + k
2
x = 0,
k =
r
c
m
, k
λ
2
+ k
2
= 0 λ
1,2
= ±ik
x = C
1
cos kt + C
2
sin kt,
C
1
, C
2
x(0) = x
0
= C
1
, ˙x(0) = v
0
= C
2
k.
x(t) = x
0
cos kt +
v
0
k
sin kt.
T =
2π
k
f =
1
T
[k] = c
−1
, [T ] = c, [f] =
x(t) = A sin(kt + ϕ).
A ϕ
A =
p
x
2
0
+ v
2
0
/k
2
,
tg ϕ =
x
0
k
v
0
.
mg
R = −βv,
β β > 0
m¨x = −cx −β ˙x
¨x + 2h ˙x + k
2
x = 0,
k
2
=
c
m
, 2h =
β
m
(h > 0).
λ
2
+ 2hλ + k
2
= 0
λ
1,2
= −h ±
√
h
2
− k
2
.
h
2
< k
2
λ
1,2
= −h ± ik
1
, k
1
=
√
k
2
− h
2
,
x(t) = e
−ht
(C
1
cos k
1
t + C
2
sin k
1
t) .
C
1
, C
2
x(0) = x
0
˙x(0) = v
0
x
0
= C
1
, v
0
= −hx
0
+ C
2
k
1
,
x(t) = e
−ht
x
0
cos
√
k
2
− h
2
t +
v
0
+ x
0
h
√
k
2
− h
2
sin
√
k
2
− h
2
t
,
x(t) = Ae
−ht
sin(
√
k
2
− h
2
t + ϕ),
A =
r
x
2
0
+
(v
0
+ x
0
h)
2
k
2
− h
2
, ϕ = arctg
x
0
√
k
2
− h
2
v
0
+ x
0
h
.
h
2
> k
2
.
λ
1,2
= −h ±
√
h
2
− k
2
x(t) = C
1
e
λ
1
t
+ C
2
e
λ
2
t
, λ
1
< 0, λ
2
< 0.
C
1
, C
2
x
0
= C
1
+ C
2
, v
0
= C
1
λ
1
+ C
2
λ
2
,
C
1
=
x
0
λ
2
− v
0
λ
2
− λ
1
, C
2
=
v
0
− x
0
λ
1
λ
2
− λ
1
.
h = k.
λ
1
= λ
2
= −h,
x(t) = (C
1
+ C
2
t) e
−ht
.
e
−ht
x(t) t → ∞ C
1
C
2
x
0
= C
1
, v
0
+ hx
0
= C
2
.
m¨x = −cx −β ˙x + Q
0
sin pt,
¨x + 2h ˙x + k
2
x = P sin pt,
2h =
β
m
, k
2
=
c
m
, P =
Q
0
m
.
x = x + x ,
x
¨x + 2h ˙x + k
2
x = 0,