Беляк О.А., Дрезин С.В. Динамика материальной точки
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O
ϕ
M
1
M
2
M
3
S
1
S
2
mg
mg
M mg S
l
S
mw = mg + S
ϕ
mw
n
= S − mg cos ϕ.
w
n
=
v
2
l
S = mg cos ϕ + m
v
2
l
= mg
cos ϕ +
v
2
gl
.
v = l ˙ϕ =
lπ
2
3
cos 2πt,
v
1
= v(t
1
) =
lπ
2
3
v
2
= v(t
2
) = 0 v
3
= v(t
3
) = 0
M
1
S
1
= mg
1 +
lπ
4
9g
= 10
1 +
2π
4
9 · 9,8
≈ 3 2,1 .
M
2
M
3
S
2
= S
3
S
2
= mg cos
π
6
= 5
√
3 ≈ 8,66 .
0,2 x = 3 cos 2πt
y = 4 sin πt t
F
x
= 0,0789x F
y
= −0,0197y
300
R = 600
80
12784
α = 30
◦
v
0
= 15 f = 0 ,1
β
α
mg
v
0
N
F
α
y
x
OO
Oxy O
Ox
mg
N F
Ox
m
¨
r = mg + N + F
m¨x = −mg sin α −F ,
0 = N − mg cos α.
N = mg cos α F = fN = f mg cos α
m¨x = −mg(sin α + f cos α).
Ox
m
v
x
x
v
x
(t) = ˙x(t) = −g(sin α + f cos α)t + C
1
,
x(t) = −g(sin α + f cos α)
t
2
2
+ C
1
t + C
2
.
C
1
C
2
x(0) = 0
˙x(0) = v
0
C
1
= v
0
C
2
= 0
T v
x
(T ) = 0
T =
v
0
g(sin α + f cos α)
=
15
9,8
0,5 + 0,1 ·
√
3
2
≈ 2 ,6 1 .
v
x
= 0 β
tg β = f = 0, 1
α tg α =
1
√
3
≈ 0, 577 α > β
t = T
s = x(T ) = v
0
T −g(sin α + f cos α)
T
2
2
=
=
v
2
0
g(sin α + f cos α)
− g(sin α + f cos α)
v
2
0
2g(sin α + f cos α)
2
=
=
v
2
0
2g(sin α + f cos α)
=
15
2
2 · 9,8
0,5 + 0,1 ·
√
3
2
≈ 19,57 .
M
y
R
mg
M
O
R |R| = kv
2
k > 0 k
M Oy mg
R
Oy
m ˙v = mg − kv
2
, v = ˙y.
˙v =
dv
dt
=
dv
dy
·
dy
dt
=
dv
dy
v =
1
2
d
dy
v
2
.
v
2
= u
1
2
m
du
dy
= mg − ku,
du
2(mg − ku)
=
dy
m
.
ln |mg −ku| = −
2k
m
y + C.
|mg − ku| = mg − ku mg > R
C t = 0 v(0) = 0 y(0) = 0
v
y =0
= 0, u
y =0
= 0 C = ln(mg)
v y
ln
mg − kv
2
mg
= −
2ky
m
, ln
1 −
kv
2
mg
= −
2ky
m
.
m
v
max
0 = mg − kv
2
max
, m =
kv
2
max
g
,
w = ˙v = 0 v = v
max
ln
1 −
v
2
v
2
max
= −
2gy
v
2
max
, 1 −
v
2
v
2
max
= e
−
2gy
v
2
max
, v = v
max
q
1 − e
−
2gy
v
2
max
.
y
y(0) = 0
H T m
v
0
k
2
mgv
2
v
Oy M mg
R Oy
y
R
mg
M
O
Oy
m
dv
dt
= −mg − R, v = ˙y.
R = |R| = k
2
mgv
2
m
dv
dt
= −g − k
2
gv
2
.
dv
1 + k
2
v
2
= −gdt,
1
k
arctg kv = −gt + C
1
.
C
1
v
t=0
= v
0
C
1
=
1
k
arctg kv
0
.
v t t = t(v)
t =
arctg kv
0
− arctg kv
kg
.
T
v
t=T
= 0
T =
arctg kv
0
kg
.
H
1
2
d
dy
v
2
= −g(1 + k
2
v
2
).
v
2
= u
1
2
du
dy
= −g(1 + k
2
u),
v
du
1 + k
2
u
= −2gdy,
1
k
2
ln(1 + k
2
v
2
) = −2gy + C
2
.
Oy
C
2
y
t=0
= 0 v
t=0
= v
0
C
2
=
1
k
2
ln(1 + k
2
v
2
0
).
y v
2gy =
1
k
2
h
ln(1 + k
2
v
2
0
) − ln(1 + k
2
v
2
)
i
=
1
k
2
ln
1 + k
2
v
2
0
1 + k
2
v
2
,
y =
1
2gk
2
ln
1 + k
2
v
2
0
1 + k
2
v
2
.
v
t=T
= 0 H = y
v=0
H =
ln(1 + k
2
v
2
0
)
2gk
2
.
m v
0
α
R
h
s
R = kmgv
v
0
mg Ox Oy
O
x
y
α
R
H
s
v
0
mg
mw = F .
F mg
Oy R
R = kmgv
Ox Oy
mw
x
= −kmgv
x
,
mw
y
= −kmgv
y
− mg.
¨x + kg ˙x = 0, x(0) = 0, ˙x(0) = v
0
cos α,
¨y + kg ˙y = −g, y(0) = 0, ˙y(0) = v
0
sin α.
λ
2
+ kgλ = 0,
λ
1
= 0, λ
2
= −kg
x = C
1
+ C
2
e
−kgt
,
y = C
3
+ C
4
e
−kgt
+ y ,
y = −
t
k
C
1
C
2
C
3
C
4
C
1
+ C
2
= 0, −C
2
kg = v
0
cos α,
C
3
+ C
4
= 0, −C
4
kg −
1
k
= v
0
sin α,
C
1
=
v
0
cos α
kg
C
2
= −
v
0
cos α
kg
C
3
=
1
k
2
g
+
v
0
sin α
kg
C
4
= −
1
k
2
g
−
v
0
sin α
kg
x =
v
0
cos α
kg
1 − e
−kgt
,
y =
1
kg
v
0
sin α +
1
k
1 − e
−kgt
−
t
k
.
h s T
H v
y
(T ) = ˙y(T ) = 0
s = x(T ) h = y(T )
˙y(T ) = (v
0
sin α +
1
k
)e
−kgT
−
1
k
= 0,
e
−kgT
=
1
kv
0
sin α + 1
.
T
−kgT = −ln(kv
0
sin α + 1), T =
1
kg
ln(kv
0
sin α + 1).
h =
v
0
sin α
gk
−
1
gk
2
ln(1 + kv
0
sin α),
s =
v
2
0
sin 2α
2g(kv
0
sin α + 1)
.
h
Oy
¨y + kg ˙y = −g.
v
y
= ˙y = u
˙u = −kgu − g, ˙u = −g(1 + ku),
˙u =
du
dt
=
du
dy
·
dy
dt
=
du
dy
u =
1
2
d
dy
(u
2
),