Беляк О.А., Дрезин С.В. Динамика материальной точки
Подождите немного. Документ загружается.
C
M ϕ OC
mg + N + J = 0, J = J
n
+ J
τ
.
J
n
= −mw
n
,
N = mgcosϕ +
mv
2
l
.
M
1
N = 0
cos ϕ
1
= −
v
2
1
gl
,
cos ϕ
1
< 0 ϕ
1
>
π
2
mg
T
1
− T
0
= A
01
;
mv
1
2
2
−
mv
0
2
2
= −mgl(cos α − cos ϕ
1
).
ϕ
1
mv
1
2
2
−
mv
0
2
2
= −mgl
cos α +
v
2
1
gl
,
M
1
v
2
1
=
v
2
0
− 2gl cos α
3
.
M
1
v
2
0
> 2gl cos α,
ϕ
1
cos ϕ
1
= −
1
3
v
2
0
gl
− 2 cos α
.
α = 60
◦
g = 9,8 v
0
= 3,5
cos ϕ
1
= −0,5 ϕ
1
= 120
◦
; v
1
= 1,565 M
1
D = l sin 30
◦
= 0,25
2gl cos α − v
2
0
> 0 .
M
1
ϕ
∗
1
v
∗
1
= 0 M
∗
1
−
mv
2
0
2
= −mgl(cos α − cos ϕ
∗
1
),
cos ϕ
∗
1
= cos α −
v
2
0
2gl
.
ϕ
∗
1
6 ϕ
1
cos ϕ
∗
1
> cos ϕ
1
,
cos α −
v
2
0
2gl
>
2
3
cos α −
v
2
0
3gl
,
C M
∗
1
M
∗
0
v
1
β = π − ϕ
1
M
1
t
m¨x = 0, x(0) = 0, ˙x(0) = v
1
cos β = −v
1
cos ϕ
1
,
m¨y = −mg, y(0) = 0 , ˙y(0) = v
1
sin β = v
1
sin ϕ
1
.
x = −(v
1
cos ϕ
1
)t,
y = −
gt
2
2
+ (v
1
sin ϕ
1
)t,
y = x tg ϕ
1
− x
2
g
2v
2
1
cos
2
ϕ
1
.
α = 60
◦
, ϕ
1
= 120
◦
y =
√
3x − 8,0025x
2
,
x y
M
2
xM
1
y
(x − l sin β)
2
+ (y + l cos β)
2
= l
2
,
ϕ
1
(x − l sin ϕ
1
)
2
+ (y − l cos ϕ
1
)
2
= l
2
.
M
2
(x
2
, y
2
) t
2
x
2
, y
2
t
2
(−v
1
cos ϕ
1
t
2
− l sin ϕ
1
)
2
+ (−
gt
2
2
2
+ v
1
sin ϕ
1
t
2
− l cos ϕ
1
)
2
= l
2
.
t
2
t
2
2
(t
2
2
−
4v
1
sin ϕ
1
g
t
2
+
4
g
2
(v
2
1
+ gl cos ϕ
1
)) = 0.
v
2
1
+ gl cos ϕ
1
= 0
t
1
= 0 M
1
t
2
=
4v
1
sin ϕ
1
g
,
M M
1
M
2
α = 60
◦
v
0
= 3,5
t
2
=
4 · 1,565 ·
√
3
2 · 9,8
≈ 0 ,5 53 .
M
2
x
2
= −
2v
2
1
g
sin 2ϕ
1
, y
2
= −
4v
2
1
g
sin
2
ϕ
1
.
ϕ
1
∈ (
π
2
; π) x
2
> 0 y
2
< 0
ϕ
1
v
1
M
2
x
2
= −
2 · 1,565
2
9,81
·
−
√
3
2
≈ 0 ,4 3 , y
2
= −
4 · 1,565
2
9,81
·
−
√
3
2
2
≈ −0,75 ;
C
α = 45
◦
v
0
= 4
α = 30
◦
v
0
= 5
3
m
O ϕ
0
v
0
l
ϕ
v
0
n
τ
ϕ
0
ϕ
v
mg
N
OO
τ n
m
dv
dt
= −mg sin ϕ,
m
v
2
l
= N − mg cos ϕ.
v
mv
2
2
−
mv
2
0
2
= A, A = A(N) + A(mg).
N
A(N) = 0 N · dr = 0 A = A(mg) = −mgh
h = l(cos ϕ
0
− cos ϕ)
mv
2
2
−
mv
2
0
2
= mgl(cos ϕ − cos ϕ
0
),
mv
2
l
=
mv
2
0
l
+ 2mg(cos ϕ − cos ϕ
0
).
n
N = mg cos ϕ +
mv
2
l
= 3mg cos ϕ − 2mg cos ϕ
0
+
mv
2
0
l
.
M A
R v
0
v
0
A
τ
R
N
n
v
ϕ
M
mg
v
0
OO
n, τ
m
dv
dt
= mg sin ϕ,
m
v
2
R
= mg cos ϕ − N.
mv
2
2
−
mv
2
0
2
= mgR(1 − cos ϕ).
v
2
= v
2
0
+ 2gR(1 − cos ϕ).
N = mg cos ϕ −
mv
2
R
= 3mg cos ϕ − 2mg −
mv
2
0
R
.
N
ϕ=ϕ
1
= 0
3mg cos ϕ
1
− 2mg −
mv
2
0
R
= 0,
ϕ
1
3g cos ϕ
1
= 2g +
v
2
0
R
, cos ϕ
1
=
2
3
+
v
2
0
3gR
.
ϕ
1
= arccos
2
3
+
v
2
0
3gR
.
ϕ
1
= ϕ
0
= 0
v
0
v
∗
0
N|
ϕ=ϕ
0
= 0 cos ϕ
0
=
2
3
+
(v
∗
0
)
2
3gR
1 =
2
3
+
(v
∗
0
)
2
3gR
, (v
∗
0
)
2
= gR, v
∗
0
=
p
gR.
A
v
0
>
p
gR.
K A
R
v
0
K
f
n
τ
ϕ
A
F
R
N
ϕ
K
mg
v
0
OO
F = fN
m
dv
dt
= mg sin ϕ − f N,
m
v
2
R
= mg cos ϕ − N.
v = R ˙ϕ
mR ¨ϕ = mg sin ϕ − fN,
mR ˙ϕ
2
= mg cos ϕ − N.
N m
¨ϕ − f ˙ϕ
2
=
g
R
(sin ϕ − f cos ϕ).
ψ = ˙ϕ
2
¨ϕ =
d ˙ϕ
dt
=
d ˙ϕ
dϕ
˙ϕ =
1
2
d
dϕ
( ˙ϕ
2
) =
1
2
dψ
dϕ
.
dψ
dϕ
− 2fψ =
2g
R
(sin ϕ − f cos ϕ).
dψ
dϕ
− 2fψ = 0 ψ = Ce
2fϕ
ψ = c
1
sin ϕ + c
2
cos ϕ
ψ
(c
1
− 2fc
2
) cos ϕ − (c
2
+ 2fc
1
) sin ϕ = −
2gf
R
cos ϕ +
2g
R
sin ϕ,
c
1
c
2
c
1
− 2fc
2
= −
2gf
R
,
c
2
+ 2fc
1
= −
2g
R
,
c
1
= −
6gf
R(1 + 4f
2
)
,
c
2
= −
2g(1 − 2f
2
)
R(1 + 4f
2
)
.
C
v(0) = R ˙ϕ(0) = v
0
, R
2
ψ = v
2
0
.
v
2
0
R
2
= −
2g(1 − 2f
2
)
R(1 + 4 f
2
)
+ C, C =
v
2
0
R
2
+
2g(1 − 2f
2
)
R(1 + 4f
2
)
.
v
2
=
v
2
0
+ 2gR
1 − 2f
2
1 + 4f
2
e
2fϕ
−
2gR
1 + 4f
2
h
3f sin ϕ + (1 − 2f
2
) cos ϕ
i
,
v
2
0
e
2fϕ
= v
2
− 2gR
1 −2f
2
1 + 4f
2
e
2fϕ
+
2gR
1 + 4f
2
h
3f sin ϕ + (1 − 2f
2
) cos ϕ
i
.
ϕ
0
0 6 ϕ
0
6
π
2
˙ϕ = 0 ¨ϕ = 0
0 =
g
R
(sin ϕ
0
− f cos ϕ
0
),
ϕ
0
tg ϕ
0
= f, ϕ
0
= arctg f.
0 6 ϕ
0
6
π
2
sin ϕ
0
=
f
p
1 + f
2
cos ϕ
0
=
1
p
1 + f
2
v
2
0
=
2gR
1 + 4f
2
3f
f
p
1 + f
2
+
1 − 2f
2
p
1 + f
2
e
−2fϕ
0
− 2gR
1 − 2f
2
1 + 4f
2
=
=
2gR
1 + 4f
2
h
p
1 + f
2
e
−2fϕ
0
− (1 − 2f
2
)
i
.
v
0
6
s
2gR
1 + 4f
2
h
p
1 + f
2
e
−2fϕ
0
− (1 − 2f
2
)
i
,
O OM l M m
OM α M
O
1
h = OO
1
β