Беляк О.А., Дрезин С.В. Динамика материальной точки
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f
v = 20 6
Ox
T = 6
v
0
= 20 v
1
= 0
Ox mg
N F = fN, N = mg
Ox
mv
1
− mv
0
= −
Z
T
0
fmgdt,
v
1
= 0
mv
0
=
Z
T
0
fmgdt.
f =
v
0
gT
.
f =
20
9,81 · 6
≈ 0 ,3 4 .
M
v
2
v
1
= 30 r
2
r
1
O
F O
M
0
(F ) = 0
dG
0
dt
= 0,
G
0
= const
O mv
1
r
1
= mv
2
r
2
v
2
=
v
1
r
1
r
2
.
r
2
= 5r
1
v
2
= 6
v
2
v
1
r
2
M
M
1
r
1
M
2
F
OO
m
a
T
R
v
0
O M
0
M
0
O = R
t = 0 v(0) = v
0
+ a |v(0)| =
p
a
2
+ v
2
0
α v(0)
t = 0 v(0 ) γ
0
= 180
◦
− α
M
r
O
M
0
a
ϕ
γ
0
v
0
v
α
TT
r ϕ
r(t) = OM(t) = R − at,
a ϕ(t)
r
2
˙ϕ = c, c = r
0
v(0) sin γ
0
,
(R − at)
2
˙ϕ = c,
c
c = R
q
a
2
+ v
2
0
sin α = R
q
a
2
+ v
2
0
v
0
v(0)
= R
q
a
2
+ v
2
0
v
0
p
a
2
+ v
2
0
= Rv
0
.
˙ϕ =
Rv
0
(R − at)
2
.
ϕ(0) = 0 ϕ(t)
r = R − at, ϕ =
v
0
t
R − at
.
T
T = F (r) = −
mc
2
r
2
d
2
dϕ
2
1
r
+
1
r
.
t =
Rϕ
v
0
+ aϕ
, u =
1
r
ϕ
1
r
=
aϕ + v
0
v
0
R
,
d
dϕ
1
r
=
a
v
0
R
= const,
d
2
dϕ
2
1
r
= 0
T = F (r) = −
mR
2
v
2
0
r
3
= −
mR
2
v
2
0
(R − at)
3
.
K
α v
0
s
f
v(0) = v
0
v
1
= 0
T
0
= mv
2
0
/2 T
1
= 0
mg
A(mg) = smg sin α N
N = mg cos α
A(N) = 0 F = fN = f mg cos α
A(F ) = −fmgs cos α
A = mgs(sin α − f cos α).
T
1
T
0
A
−
mv
2
0
2
= mgs(sin α −f cos α),
s =
v
2
0
2g(f cos α − sin α)
.
m F
r
2
= a cos(2ϕ) a r
r = r
0
v
0
α
F
r
F =
3ma
2
r
7
r
2
0
v
2
0
sin
2
α
3
5
30 55
F = 6 A = 4, 5
70 100
4,3
x
v
x
(0) = 0
100
O
A
x
N
mg
mg
BB
OA |OA| = 100
A
O A
T
0
= T
O
=
mv
2
O
2
= 0, T
1
= T
A
=
mv
2
A
2
.
OA
A = mg|OA|
mv
2
A
2
= mg|OA|,
v
2
A
= 2g|OA|
v
2
A
= 2 · 9,81 · 100
2
/
2
= 1962
2
/
2
, v
A
≈ 4 4,3 .
AB
T = 5 v
B
= 4,3
N
x
mv
B
− mv
A
=
Z
T
0
(mg − N)dt,
N
N = m
g +
v
A
− v
B
T
= 70
9,81 +
44,3 − 4,3
5
= 1246 .
AB
α = 30
◦
BC
v
y
= 1
h = 9 f = 0,08
CD β = 45
◦
l
v
C
v
Cx
v
Cy
y
CB
D
h
A
x
k
β
K
mg
N
α
F
xx
AB A v
A
= 0
T
0
= T
A
= 0 B T
1
= T
B
=
mv
B
2
2
s = AB = h/sin α
mg
A(mg) = mgh;
N N = mg cos α
A(N) = 0
F = fN
A(F ) = −sfmg cos α = −
h
sin α
fmg cos α.
mv
B
2
2
= mg
h − f cos α
h
sin α
,
v
2
B
= 2gh(1 − f ctg α) = 2 · 9,81 · 9 · (1 − 0, 08 ·
√
3)
2
/c
2
≈ 1 52,11
2
/c
2
, v
B
≈ 12,3 .
BC
C
v
Cx
= v
B
≈ 1 2, 3 .
CK
C K C
v
Cx
v
Cy
= 1
xCy
C
m¨x = 0, x(0) = 0, ˙x(0) = v
Cx
= 12,3,
m¨y = −mg, y(0) = 0, ˙y(0) = v
Cy
= 1.
x(t) = 1 2 ,3 t, y(t) = −
gt
2
2
+ t.
l = CK
CD y = −x
−
gt
2
2
+ t = −12,3t, −
g
2
t
2
− 13,3t = 0;
t
1
= 0 C t
2
= 2,7 K
l = x(2, 7) ·
√
2 ≈ 47,4 .
U = x
2
+ xy + y
2
x = 3 y = 4 10
x
x = (3, 5 cos
√
3t +
5
√
3
3
sin
√
3t − 0, 5 cos t + 5 sin t)
y = (3, 5 cos
√
3t +
5
√
3
3
sin
√
3t + 0, 5 cos t − 5 sin t)
l
a
T =
2π(l
2
− a
2
)
1/4
√
g
.
l = 0,5 O
M
0
α = 60
◦
v
0
3,5
M
1
v
1
M
∗
0
M
∗
1
M
β
D
C
C
C
y
y
x
M
1
M
1
M
1
O
O
v
1
v
1
v
1
M
0
M
0
J
n
J
τ
α
α
ϕ
1
ϕ
1
v
0
M
2
x
ϕ
v
0
mg
N
OO