Беляк О.А., Дрезин С.В. Динамика материальной точки
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m
mv
F
d
dt
(mv) = F , F =
n
X
i=1
F
i
.
m
dv
dt
= F , mw = F .
v w
m
¨
r = F ,
r
Ox
1
x
2
x
3
m¨x
k
= F
k
, k = 1, 2, 3.
τ n
b
m¨s = F
τ
, m
v
2
ρ
= F
n
, 0 = F
b
.
s(t)
v = ˙s, w
τ
= ¨s, w
n
=
v
2
ρ
,
ρ
F F
F
b
= 0
x
k
= x
k
(t), k = 1, 2, 3
m
F
k
k = 1, 2, 3
m
t C
j
(j = 1, 2, . . . , 6)
x
k
= x
k
(t, C
1
, . . . , C
6
), k = 1, 2, 3.
C
j
r
0
v
0
t
0
x
k
(t
0
, C
1
, . . . , C
6
) = x
k0
, ˙x
k
(t
0
, C
1
, . . . , C
6
) = v
k0
, k = 1, 2, 3.
C
j
F =
n
P
i=1
F
i
R
m
¨
r = F + R
m¨x
k
= F
k
+ R
k
, k = 1, 2, 3.
k = 1, 2
k = 1
x, y, z
2,04
x = 10 sin
πt
2
x
F = m¨x.
w = ¨x = −10
π
2
4
sin
πt
2
.
F t
F = −
π
2
4
m10 sin
πt
2
= −
5
2
mπ
2
sin
πt
2
,
F x
F = −
π
2
4
mx = −
2,04π
2
4
x = −5,03 3 x.
F t
1
= 3 sin
πt
1
2
= −1
F
max
= 50,33 .
m = 600
α = 60
◦
f
0,2 r = 0,2 ϕ = 0,4t
3
2
mg N
F
S
Oxy
O Ox
O
x
y
α
F
N
S
mg
m
¨
r = S + mg + F + N ,
m¨x = S − mg sin α − F ,
0 = N − mg cos α.
N = mg cos α
F = fN = fmg cos α.
S w = ¨x
x
x = rϕ = 0,2 · 0,4t
3
= 0,08t
3
.
w = ¨x = 0,48t
S = m¨x + mg(sin α + f cos α).
m α f S
S = 600
0,48t + 9,8
√
3
2
+ 0,2 ·
1
2
= (288t + 5680) = (0,288t + 5,68) .
2
S(2) = (0,288 · 2 + 5,68) = 6,256 .
M 10 l = 2
ϕ =
π
6
sin 2πt ϕ
t S
1
S
2
M
1
ϕ = 0 t
1
sin 2πt = 0 t =
k
2
k ∈ Z
t
1
= 0
M
2
M
3
ϕ = ±
π
6
sin 2πt = ±1 t =
1
4
+
k
2
k ∈ Z t
2
=
1
4
t
3
=
3
4