S ′ R = (1, 1) S ′ r′ = (2, 1)
S
S S ′
S
P
cos 30 sen 30
R= −
sen 3 0 cos 30
√ √ 1 2
=
−
3 1
1 3
R = (1, 1) r′ =
R(r − R) S ′
S
r′
r
RT r′ = RT R(r − R) = r − R √ 3 −1 1 2 1 √ + r = RT r′ + R = 1 1 2
1
3
√ √ 3 +
=
3 2
1 2
+2
R ω
S
XY Z XY x
A O
θ
OA
OG x
θ Rdθ = dx Rθ˙ = x˙
ω
S = (0, 0,
−w) = (0, 0, −θ˙)
Y
G R x
.
x
O θ
S
A
X
ω
O x˙ = Rθ˙
r
′ = vP
× OP
ω
v p′ = ωr
OP
Y B
x
O
S
A
X O
v0 = 0 vB = Rθ˙i ′
B
A
vA = ′
O
′
−
Rθ˙ i
ω
S v p = v0 + ω
× OP
A A
S vA = v0 + ω
× OA = x˙ i +
i
j
0 0
0
k θ˙
−
0
−R
= (x˙
− Rθ˙)i ≡ 0
x˙ = Rθ˙ B
S vB = v0 + ω
× OB = x˙ i +
i
j
0 0 0 R
k θ˙
−
0
= (x˙ + Rθ˙)i
≡ 2x˙ i
Y
.
B
G
2x
R
x
.
.
x
x O
S
A
X S
x˙ S
vA = 0
A
B
2x˙
S Y Y
g v = (0,
S
x(t) = x0, y (t) = y0
(x0 , y0)
− 12 gt
2
S ′
S V = (V, 0, 0)
S S ′
v′ = v
− V = (−V, −gt) S ′
x′ (t) = x(t)
− V t = x − V t, 1 y (t) = y − gt 2 2
y ′ (t) =
0
t=
y ′ = y0 y′ S ′
x′
− 12 gt
0
2
x0
= y0
− x′
V
g − 2V (x − x′ )
2
2
0
−gt)
Y
40 20
1
2
3
4
5
6
X
7
x0
20
40
60
80
′
S
S g = 10 V = 1 x0 = 3 y0 = 40
y
x
5
0
, y
0
y 4
5
3
4
0
,
0
3
2
2 1
1 x
3
2
1
0
1
4
3
2
1
1
2
3
4
x
1
1
2
2
3
3
4
4
5
5
0 < t < 10 vg = 1 x0 = 0 y0 = 5 z0 = 0
S
S ′
S Z
′
S
w = 2π
S
S ′ R = 0, V = 0, A = 0 S
S S ′
R(t) = ω
−
cos(ωt ) sen(ωt ) 0
sen (ωt ) 0 cos(ωt ) 0 0 1
= (0, 0, θ˙ )
z
θ α
= ω˙ = 0 x0 = x′0 y0 = y0′ z0 = z0′ S
(x0, y0 , z0 )
x(t) y (t) z (t) S
x˙ (t) y˙ (t) z˙ (t)
− −
x0 y0 vg t z0
=
0
=
vg
0
S ′ r′ (t) =
=
R(t)[r(t) − R(t)] = R(t)r(t)
−
cos(ωt ) sen(ωt ) 0
sen (ωt ) 0 cos(ωt ) 0 0 1
− x′ (t) y ′ (t) z ′ (t)
x0 y0 vg t z0
−
S ′
x0 cos(ωt ) + ( y0 vg t)sen(ωt ) x0 sen(ωt ) + ( y0 vg t)cos(ωt ) z0
=
− −
S ′
− − −
( ωx 0
v′ (t) =
( ωx 0
− vg )sen(ωt) + ω(y − vg t)cos(ωt) − vg )cos(ωt) − ω(y − vg t)sen(ωt)
ω ( ωx 0
a′ (t) =
0
0
0 2
2
0
− ω (y − vg t)cos(ωt) + ωvg sen(ωt) 0
0
x
l v0 h
v0
− vg )cos(ωt) − ω (y − vg t)sen(ωt) − ωvg cos(ωt)
ω (ωx0 + vg )sen(ωt )
h0
M m µe
µd
F
F F F c
ma = F − F r M a = F r , F F r , a F M + m M F r = F M + m a=
M + M + m
F r = µe mg
F c µe mg =
M F c M + m
→
F c = µe mg
M + m M
M
F c =
µemg F r = µd mg ma1 = F − µd mg M a2 = µd mg
F − µd mg m µd mg a2 = M a1 =
M
α m
X Z
Y m
(0, −mg) mg ) p = (0, (0, N ) N ) N = (0,
n = n(− sen α, cos α)
(0, −M g) P = (0,
N = −n = n(sen α, − cos α)
max = −n sen α may = −mg + n cos α M Ax = n sen α M Ay = N − M g − n cos α a = (ax , ay )
A = (Ax , Ay )
ax , ay , Ax , Ay , n , N Ay = 0
x, y
y = tan α x − X X
→ y = tan α(x − X )
ay = tan α(ax − Ax ) max = −n sen α may = −mg + n cos α M Ax = n sen α 0 = N − M g − n cos α ay = tan α(ax − Ax ) mg cos α m/M m/M )) + cos 2 α g sen α cos α ax = − sen 2 α(1 + m/M m/M )) + cos2 α g cos2 α ay = − g + sen 2 α(1 + m/M ) + cos2 α mg sen α cos α Ax = sen 2 α(M + m) + M cos M cos2 α mg cos2 α N = M g + sen 2 α(1 + m/M m/M )) + cos2 α n=
sen 2 α(1 +
M → ∞ α h v0
x (0, 0, h) r(0) = (0,
z (v0 , 0, 0) v(0) = (v m¨r = F
r
F F = Fg + Fr
Fg
= (0, (0, 0, −mg) mg )
Fg
= −α(vx , vy , vz ) z α
mx ¨ = −αvx my¨ = 0 mz¨ = −mg − αvz
mv˙ x = −αvx v˙ = −γv
γ = α/m
v(t) = t = 0 v(t) = v(0)
exp{−γt }v (0)
vx (t) =
α exp − t v
0
m
x x(t) x˙ (t) = t=0
m
0
t=t
t
α exp − t v t
′
′
dt x˙ (t ) =
0
x(t) − x(0) =
α dt exp − t v m α α − v exp − t − 1
′
0
0
m
0
m
α α x(t) = x(0) + v0 1 − exp − t m m
x x(0) = 0 y y(t) = y (0) (0) + vy (0)t (0)t
my¨ = 0
y (0) (0) = 0
y(0) y (t) = 0
vy (0) = 0
vy (0)
y=0 z mv˙ z = −mg − αvz ′
vz = vz + mg/α ′
′
mv˙ z = −αvz ′
′
vz (t) = exp {−α/mt}vz (0) gm α α exp{− t} − 1 + exp{− t}vz (0) α m m
vz (t) =
vz (0) = 0 z˙ (t) =
z
gm α exp{− t} − 1 α m
0, t t
0
d z(t )dt dt ′
′
t
=
gm α exp{− t} − 1 dt α m gm α gm exp{− t} − 1 − t − ′
0
2
z (t) − z (0) =
α2
m
z (0) = h
α
z (t) = 0 hα2 α α + 1 = exp t + t {− } gm 2 m m x
−
e
−x = y
tcaida α α hα2 x(tcaida ) = tcaida v0 tcaida − m m gm 2
′
S ′
S
z m mg
′
mω × (ω × r)
S
P
r
′
S
y
ω m
x x, z z(x) ′
z (x) tan α = z (x)
x α
′
mg + mω × (ω × mg = (0, (0, 0, −mg) mg ) mω × (ω × r) = (ω 2 x, 0, 0)
r) + P = 0 P = (−P sen α, 0, P cos α)
P sen α = mω 2 x P cos α = mg
′
tan α = z (x) =
z (x) =
ω2 x g
ω2 2 x + x0 2g x0 = 0
′
′
′
ma = RFng + mgefectiva − m2ω × v a′ ′ gefectiva Fng
′
RFng
′
′
′
′
′
′
′
′
′
−2ω × v = 2ωT (vy sen λ − vz cos λ)ex − vx sen λey + vx cos λez
a′z = 0, vz′ = 0 Fng
x, y
a′ = 2ωT sen λ vy′ e′x − vx′ e′y ′
′
v˙ x = 2ωT sen λvy ′
′
v˙ y = −2ωT sen λvx
′
S ′
Z
Y
′
′
X ′
Z R = 6370 ◦
54,,8 −54
g = 9, 81 48,,86 48
2
◦
◦
3,75
gefectiva = g − ω × (ω × R)
2 2 N = −ωT R cos λ sen λe′y + (ω (ωT R cos2 λ − g)e′z gefectiva
2 2 S = ωT R cos λ sen λe′y + (ω (ωT R cos2 λ − g)e′z gefectiva ′
Z ′
y gefectiva tan α = z gefectiva ′
′
Z
tan αN =
2 R cos λ sen λ −ωT 2 ωT R cos2 λ − g
2 ωT R cos λ sen λ tan αS = 2 ωT R cos2 λ − g
◦
αN = 0,0981099 αN = 0,0129482 αS = −0,0932381 ◦
◦
5
−
ωT = 7,29 × 10
1
−
m1 m1
µ v0
N’ Fr’
m2
N
T
Fr’ P1=m1g
Vo
F
N’ T
m1
Fr P1=m1g
M F
θ µ
θ
F >> M g F F/Mg
θ
µ
F >> Mg
F θ
M
b sen kt
F
m
x(t) = a cos kt,y( kt,y (t) =
m
h
v dx 1+x
m
= ln(1 + x).
β
e ¨r + ω02 r = −
e r˙ × B mc
Z
B
ωL = iωt
−
r(t) = r0 e
ω
100 cm/ cm/s2
20
150 cm
eB 2mc
≪ ω0
Y X
Z
r1 , r2
S
Rcm Rcm
=
m1 r1 + m2r2 m1 + m2
=0
r1
r1 = −
= (0, (0, r1 , 0)
r2
= (0, (0, r2 , 0)
m2 r2 m1
r
r = |r1 | + |r2 | m2 r m1 + m2 m1 |r2 | = r m1 + m2 |r1 | =
I xx xx X X I xx xx =
2
i mi (yi
2
+ zi ) 2 2 I xx xx = m1 r1 + m2 r2 =
m1 m2 2 r = µr 2 m1 + m2
µ
a
S S ′
x, y
x, y
S r3
1 = a 0, − √ ,0 2
1 = a − √ , 0, 0 2
r4
=
+ zi )
−
N i=1 mi yi xi
−
N i=1 mi zi xi
zi = 0
2
=
N 2 i=1 mi (xi
−
xi yi = 0
N 2 i=1 mi yi
I
−
0
1 2
1 2
a − ,− ,0
r4
1 2
1 2
+ zi 2 )
−
N i=1 mi zi yi
0
N 2 i=1 mi xi
0
−
0
N 2 i=1 mi (xi
+ yi 2 ) r1
I
S
= ma2
N i=1 mi yi zi
N 2 i=1 mi (xi
1
0
0
0
1
0
0
0
2
S ′
+ yi 2 )
= a
= a − , ,0
r2
N i=1 mi xi zi
= ma2
0
0
S ′
N i=1 mi xi yi
1 = a 0, √ ,0 2
N 2 i=1 mi (yi
I
r1
1 2
, 12 , 0
r2
1
0
0
0
1
0
0
0
2
= a
1 = a √ , 0, 0 2
1 2
, − 12 , 0
r3
=
M
x2 a2
x,y,z
+
y2 b2
+
z2 c2
=1
a>b>c
x2 y2 z 2 + 2 + 2 =1 a2 b c x1 , y1, z1
x = ax1 y = by1 z = cz1
(x1 , y1, z1 ) x21 + y12 + z12 = 1
V =
V
V
dxdydz = abc
dx1 dy1dz1
V
1
V 1
4/3π V = 4/3πabc Z
I zz zz = ρ
M (x + y )dxdydz = abc V
2
2
2 1
x dx1 dy1 dz1 =
V
z dx1 dy1 dz1 =
1
drr
2
drr
dθ sen θ
2π
dθ sen θ
drr
2
2π
0
(a2 x21 + b2y12 )dx1 dy1 dz1 =
M I zz (x2 + y 2)dxdydz = abc zz = ρ V V
dφ( dφ(r cos θ )2
dθ sen θ cos θ
0
dφ( dφ(r cos θ )2
0
π
4
0
0
1
0
z12 dx1 dy1 dz1
2π
π
2
0
=
0
1
=
1
π
0
V
V
1
1
2 1
(a2 x21 + b2 y12 )dx1 dy1dz1
V
1
21 4πa 2 dφ = 2π = 35 15
4π 2 (a + b2) 15
1 (a2 x21 + b2 x22 )dx1 dy1dz1 = M ( M (a2 + b2 ) 5 V
1
M 1 I 1 = ρ (z + y )dv = abc (c2 x23 + b2 x22 )dx1 dy1 dz1 = M ( M (b2 + c2 ) V 5 M 1 I 2 = ρ (z 2 + x2 )dv = abc (c2 x23 + a2 x21 )dx1 dy1 dz1 = M ( M (a2 + c2 ) V 5
2
2
I xy xy = −ρ I xy xy = −ρ
V dxdydzxy
xydxdydz = −ρabcab
V
x1
x1 y1dx1 dy1 dz1 = 0
V
1
x1
M a
b
Z I zz zz X, Y
Y b/2 dm
y
X x
O
−a/2
a/2
−b/2
I xx xx =
2
y dm
I xy xy =
I yx yx = I xy xy
I yy yy =
I zx zx = I xz xz
(−xy) xy )dm
I xz xz = 0
x2 dm
I yz yz = 0
I zy zy = I yz yz
I zz zz =
(x2 + y2 )dm σ=
dm
dx
dy
M ab
(x, y )
dm = σdxdy
I zz zz
=
ρ2 dm =
S
a I z = M a2 /6
z
b/2
a/2
−b/2
−a/2
σρ2 dx dy =
b/2
a/2
−b/2
−a/2
σ (x2 + y 2)dx dy =
M 2 (a + b2 ). 12
M
x
y I z = I x + I y = 2I x I x = I y
x I x = I y m
R1
R2
R1 > R 2
y
I = mR2 /5 2 2 R15 − R25 2 2 I = (m1 R1 − m2 R2 ) = m 3 5 5 R1 − R23 m1 , R1
m 2 , R2
2 R15 − R25 ′ 2 I = I + mR1 = m 3 + mR12 3 5
r1
= (b, 0, b)
r2
R1 − R2
= (b,b, −b)
r3
= (−b,b, 0). 0).
X 1, X 2 , X 3 I 11 = 11
mi (xi,2 2 + xi,2 3) = mb2 + m(2b (2b2 ) + mb2 = 4mb2 ,
i
I 22 = 22
mi (xi,2 1 + xi,2 3) = 5mb2 ,
i
I 33 = 33
mi (xi,2 1 + xi,2 2) = 5mb2 ,
i
I 12 = I 21 12 21 = −
mi xi,1 xi,2 = −m(b2) − m(−b2) = 0, 0,
i
I 13 = I 31 13 31 = −
mi xi,1 xi,3 = 0,
i
I 23 = I 32 23 32 = −
mi xi,2 xi,3 = mb2,
i
I = mb2
mb2
λ λ λ
(4 − λ) (5 − λ)2 − 1 = 0
λ1 = 4 λ2 = 4 λ3 = 6
I 1 = 4mb2 I 2 = 4mb2 I 3 = 6mb2 I 3
I 1 = I 2
1m 1 kg
0, 00 m 0, 25 m 0, 50 m 0, 75 m
1, 00 m
l
m
a
m
a M z m I = 1/2 mr 2
r
y
r x m
l
θ
X X,Y.
X ′ , Y ′ .
Y Y’ m
X’
θ
X m
m1
m2
α m2
µ
m1
m2
m2
m1
α
m2 N = N (sen N (sen α, cos α, 0) µN ( cos α, sen α, 0) Ff = µN ( x T (1 T (1,, 0, 0) N = N (0 N (0,, 1, 0) P = m2 g(sen α, m2
±
T′ =
−
T = T ( T (
− cos α, sen α, 0)
P = mg(0 mg (0,,
y
−
′
′
− cos α, 0)
′ = Ff
−1, 0)
µN (1,, 0, 0) ±µN (1 m2
′
′
′
′
′
F = T + N + P + Ff = (T + m2 g sen α
± µN,N − m g cos α, 0)
N = m2 cos α m1 T = m1 g m1 g + m2 g sen α m2
−
m2
2
N
− m g cos α = 0 2
F′
± µm g cos α 2
−m
1
+ m2 sen α
− µm
−m
1
+ m2 sen α + µm2 cos α < 0
2
cos α > 0
m2
m1 < sen α µ cos α (baja) m2 m1 > sen α + µ cos α (sube) m2
−
sen α
− µ cos α < m /m 1
2
< sen α + µ cos α.
m α
v0
OX
L(t)
m, v0 , α x y
z L(t) = mr(t)
v(t)
× v(t)
r(t)
(0, a = (0,
−g, 0)
v(t) = (v0 cos α, v0 sen α
− gt, 0) 1 r(t) = (v t cos α, v t sen α − gt ) 2 0
2
0
x, y
L(t) = m(xy˙
− yx˙ )ez = − 12 mgv
0
cos αt2 ez
m F1
F2 F1
λ>0
r
= f ( f (r ) , r
F2 =
−λv
v
r L0 L L = mr
dL =r dt
×F
F = F1 + F2
dL = dt
−λr × v = − mλ L
L = e−λt/m L0
×v
L = 2mR0 V 0
R
V 0
L1 = 2mR1 V 1 R1 = R0 /2 L1 = L0 V 1 = 2V 0 R V 1 = R V 0
V 1
0 1
V 1
→∞
R1
→0 F =
λv
λ = 4 N s/m ·
v 600 W
F v = λv 2
600 v2 =
600 4
→
v = 12, 12 ,2m/s
500 106 eV
×
1000 2
(1/ (1/4πǫ 0 )q /r
6
× 10
eV = 16
q rmin = 1, 44 10
−11
× 10
−18
J r
m m
L µ h
h/2 h/2
h
L
mgh mgh = mv 2/2
√ v = 2gh F f f = µN
N
mg F f f d
d
d = L/2 L/2 mgh = F f f µ =
L L = µmg 2 2
2h L
h/2 h/2
mgh = F f f L + mg µ =
h 2
h 2L
m 2
V ( V (x, y) = 6x + 24y 24y
2
r = 4 j
j
Y F
F(r) =
=
− −
r = (0, (0, 4, 0)
∂V ∂V ∂V (r), (r), (r) ∂x ∂y ∂z (12x, (12x, 48 48y, y, 0) F = (0, (0, 192 192,, 0)
a = F/m =
(0, (0, 192 192/m, /m, 0) M
α m
h M >> m
V v u
α
t m
vdt
t + dt v
Vdt u+v =V
udt + vdt = Vdt
v
= ( vx , vy )
V
= (V, 0)
−u
− −
u =
(V + vx , vy )
vx , vy , V
u
tan(α tan(α) =
uy vy = ux V + vx
−mvx + M V = 0 H mgH m h m 2 M 2 (vx + vy2 ) + V + mgh = mgH 2 2 vx , vy , V
h
vx =
M V m
vy = (V + vx )tan α
V 2 2
M 1+ m
V =
M
≫m
m/M m/M
M + tan2 α(m + M ) M ) = mg( mg(H
M m
1+
2mg( mg (H h) [M + (m (m + M )tan M )tan2 α]
−
− h)
1/2
vx , vy , V
≈0
V
h
≈0
M V M
V V
2g(H h) V = x (1 + x) [1 + tan tan2 α(1 + x)]
−
1/2
≈
x = m/M m/M
2g(H h) x 1 + tan2 α
−
1/2
m 2g(H h) = M 1 + tan2 α
−
1/2
x vx =
vy = α = π/2 π/2 vy =
2g(H
2g(H
2g(H
− h)/(1 + tan
− h)/(1 + tan
2
2
α),
α)tan α.
− h) m R
M
2
1 3
v1 , v2 , v3 ′
′
′
v1 , v2 , v3
x
r
y
z v1
= (v0 , 0, 0)
v2
= (0, (0, 0, 0)
v3
= (0, (0, 0, 0)
′
v1
= (v , 0, 0)
v2
= (V cos α, V sen α, 0)
v3
= (V cos α, V sen α, 0)
′
′
′
′
−
α R α = arctan r+ R
′
′
′
mv1 = mv1 + M v2 + M v3 m 2 m 2 M 2 M 2 v1 = v + v + v 2 2 1 2 2 2 3 ′
′
′
x ′
′
mv0 = mv + 2M 2M V cos α m 2 m 2 v0 = v + M V 2 2 2 ′
′
′
v , V V
′
′
v0 cos α M 1 + 2m cos2 α v0 M 1 + 2m cos2 α
=
v0 =
m M V M T T M T T
≫ M
M T T
≫m
mv = M V = V = mv/M 0 = mvf (M + M T V f T )V f f = f =
−
⇒
1 1 mv 2 = mvf 2 + 2 2 1 = mvf 2 + 2
⇒
mvf M +M T T
1 M V 2 + 2 1 M V 2 + 2
1 2 (M + M T T )V f 2 1 m mvf 2 2 M + M T T
≃ 12 mvf + 12 M V 2
2
8
L
m
M m 2 v = vf + ( v) = m M 2
2
2
⇒ vf = v
h=
v 2 (1
2
m M
− 1
m/M )) − m/M 2g
V m k
l m 4
v0
− m4 v
0
v
′
5m v 4 v0 5
= =
′
−
x 1 5m 2 v = 2 4 f
1 k(x 2
x = l
2
− l)
±
m V 0 20k 20 k
2000 kg 6000 kg
80 Km/h
m1 v1 x cos 45 y ˆ + m2 v2 y ˆ = (m1 + m2 ) v (sen45 x ˆ + cos ˆ)
m1 v1 = (m1 + m2)v m2 v2 = (m1 + m2)v
√
2 2
√ 2 2
v2 = 80 Km/h
√
m1 + m2 2 v1 = v = 240 Km/h m1 2 E ci 67E E cf ci = 2, 67 cf
m
M
L α v0
M α s m v
ω
k ω
∆L
L
M
m F
t 30o
vi vf α P βv (t)
M L = 0,2m k1 = 0,2 k2 = 0,7 l = 4/5L h u α β r v2f m1
− v f = r(v i − v i) 1
1
2
v1
m2 Q = (1
− r ) 12 µv 2
1
2
µ = m1 m2 /(m1 + m2 )
r
m M
l
v = vxˆ
V
i
X Y
mV = mv + mv1 + mv2 = mv + 2m 2mu V
= V i
v
1, 2
u
= vi
v1,2
= ui x
i
mV = mv + 2mu 2mu
mr × V = mr′ × v + L r′
r L
L
= L p + Lcm
L p
Lcm Lcm
= I· ω
ω
I
mRV = mRv + I ω I 2/5mR2 2/5mR2 + mR2 = 7/5mR2 ω = u/R
I = 14 14//5mR2
1 1 1 mV 2 = mv2 + mu2 + I ω2 , 2 2 2 1 v = − 13 V u =
′ v1 ′
v2
= ω × r2
7 13
V
ω =
5 13
V R
= ω × r1 = (0, (0, 0, ω ) × (0, (0, R, 0) = −ωR i = (0, (0, 0, ω) × (0, (0, −R, 0) = ωR i 2 V 13 12 = −ωR + u = V 13
v1
= v1′ + Vcm = −ωR + u =
v2
= v2′ + Vcm
2l ω0
µ
X Y
Z
dm
r
ρ = m/2 m/2l
dmg = ρdrg F r = µdmg dm r |dM| = |Fr × r| = µdmgr = µgρrdr,
l
M = µgρ
0
rdr −
0
1 rdr = µgρl = µmgl 2 −l
M = I α
α
I
Z l
I =
r 2 dm =
l
−
α=
2 2 ml , 3
M 3 µg = . I 4 l ω (t) = ω0 − αt
t=
ω0 4 ω0 l = α 3 µg
m M
R=
ω0 r
z
1 I 0 = M R2 + mR2 2 r 1 I (r ) = M R2 + mr 2 2
m 1 + 2M M R2 + mR2 I 0 ω0 = I (r )ω (r ) =⇒ ω(r ) = ω0 = ω0 M R2 + mr 2 1 + 2Mmr R 1 2 1 2
ω(r )
2
2
r
1 1 I 0 ω ∆E c = I (r )ω(r )2 − I 0 ω02 = 2 2 2
r
W R→r =
r
F(r)
· dr =
R
=
I 0 ω02 2
1− 1 2
I 0 I 0 ω −1 = I (r ) 2
mω( mω (r ′ )2 dr ′ =
R
2 0
r
dr′ mω02
R
r2 R2
M R2 + mr 2
2 0
1− 1 2
r2 R2
M R2 + mr 2
1+ 1+
2m
M 2 2mr M R2 ′
r ′ dr′
∆E c
V ( V (r) = −W R→r
m1
m2 m0
m0
m2 m1
I ˙ I ω˙ = M
I
ω
M
L
= m1 r1 × v1 + m2 r2 × v2 + I0 · ω m1
r1
I0
L L
|m1 r1 × v1 | = m1 Rv1 |I0 · ω| = I 0 ω = I 0 v1 /R L = (m1 − m2 )vR + I 0 v/R v
I 0
I α = T 1 R − T 2 R T 1 , T 2
m1 g − T 1 = m1 a T 2 − m2 g = m2 a α = a/R
T 1 , T 2 , a a=
(m1 − m2 )gR 2 I 0 + (m (m1 + m2 )R2
a 1 1 1 E = m1 v2 + m2 v 2 + I 0 ω 2 + m1 g(h1 + x) + m2 g(h2 − x) 2 2 2 h1 , h2
x v
v
ω
x
E
˙ = m1 va + m2 va + I 0 v a + m1 gv − m2 gv 0 = E r2 a I 0 a = (m1 − m2 )/(m1 + m2 )g T 1 = T 2 M
R1
R2 ∆t
v0
I α = M
I = r
2
r dm =
ρr 2 dV
ρ h
2πr
dV = 2πrhdr
dV R2
I = 2πρh
R1
1 r 3 dr = πρh( πρh(R24 − R14 ) 2
dr
1 I = πρh( πρh(R22 − R12 )(R )(R22 + R12 ) 2 V = πh( πh(R22 − R12 ) M = ρV = πρh( πρh(R22 − R12 ) 1 I = M ( M (R22 + R12 ) 2 F r a = F r /M t
v0
v (t) = v0 − at = v0 −
F r t. M
α N = R2 F r = I α α = R2 F r /I ω (t) = αt v = R2 ω tR R2 αtR = v0 −
tR =
F r tR . M
M v0 R12 + R22 F r R12 + 3R 3R22
v (tR ) = v0 −
F r 2R2 tR = v0 2 2 2 M R1 + 3R 3 R2
R1 = 0
v (tR ) = 2v0 /3
R1 = R2
v (tR ) = v0 /2 1 2
1 2
I ω(tR )2 ∆E = E fin ini = fin − E ini
R
m
1 1 1 M v(tR )2 + I ω (tR )2 − M v02 2 2 2
M v(tR )2
1 2
M v02
I α = M S
x
z = (mg, 0, 0) ma = mg − T P
T
= (−T , 0, 0)
S ′
z′
M = T R T R = Iα
1 2
I = mR
2
a = αR
a,α,T ma = mg − T 1 TR = mR2 α 2 a = αR
2 g 3 1 mg 3 2g 3R
a = T = α =
2πr t=
6πr/g
V = at =
8πgr/3 πgr/3 a
σ
ω θ
S ′
′
S
X , Y
Y
′
Z ′
S ′ S ′ a/2
′
I xx =
M a2 y σady = 12 −a/2
a/2
′ I yy =
2
x2 σadx =
a/2
−
′
I zz
M a2 12
M a2 = I xx + I yy = 6 ′
′
M = σa2
′ I zz
′
=
I
′
′
′
I xx I xy I xz ′ ′ ′ I xy I yy I yz ′ ′ ′ I xz I yz I zz
M a2 12
=
ω
0 M a2
0 0
12
0 0 M a2
0
6
= (0, (0, ω, 0)
S
′
S S
S ′
θ ϕ = ωt
Z
Y S ′
ω
′
Z Z ′ = ω (sen θ, cos θ, 0)
′ ′ ′ I xx ω˙ x′ − (I yy − I zz )ωy′ ωz′ = N x′ext ′ ′ ′ I yy ω˙ y′ − (I zz − I xx )ωz′ ωx′ = N y′ext ′ ′ ′ I zz ω˙ z′ − (I xx − I yy )ωx′ ωy′ = N z′ext
˙′ =0
ω
0 = N x′ext 0 = N y′ext ′ ′ −(I xx − I yy )ωx′ ωy′ = N z′ext ′ ′ I xx = I yy
S ′ ′ ′ I xx = I yy
S ′ ′
L
′
=
I
ω
′
=
M a2
0
12
M a2
0 0
12
0 ω
0 0 M a2 6
′
M
ω sen θ ω cos θ 0
R
=
M 2 a ω 12
cos θ sen θ 0
m
r v m, R ω0
m
R
m
v
M a ω M
z
M
M
a
M
x M
y
v0 v = 5v0 /7
I = 2M R2 /5
M
R m
R
R
h
M
ω d
20 200
1
90
m R
l 2R ≥ l
m r = a exp (bθ) bθ) a, b (¨ r
−
−
θ r θ˙ 2 ) rˆ +
a =
1 d 2˙ ˆ r dt (r θ ) θ.
F = f ( f (r )ˆr
r¨
− rθ˙2 = m1 f ( f (r ) d (mr 2 θ˙) = 0 dt
J =
r
mr 2 θ˙ θ˙ =
r˙ =
−abθe˙
J m
bθ
−
=
Jb 1 r¨ = r˙ = m r2
1 f ( f (r ) = m
− Jb m
2
1 r3
−brθ˙ = − Jmb 1r
−
− J m
r
′
−V (r) = f ( f (r ) = − V ( V (r ) =
−
1 r2
Jb m
2
2
1 = r4
1 r3
−
J 2 (b2 + 1) m2
1 r3
J 2 (b2 + 1) 1 m r3
J 2 (b2 + 1) 2m r 2 O
r
F ( F (r ) O
O
− a r
F ( F (r ) = k 4
2
3
a r
3
a
√ ak
O O O O (a, 0, 0)
J = mr
×v
J = m(a, 0, 0)
V ( V (r ) =
−km
√
√
2 m m E = v2 + V ( V (a) = ( ak + 2 2
2
ak )
vr = r˙
2
4 ar
−
3
− 3 2ar
2
a2 km 4 a
× (√ ak, √ ak, 0) = ma√ akk
−
a3 3 2 2a
=
− 32 mka. r
vθ = r θ˙
1 1 E c = mr˙ 2 + mr 2 θ˙2 . 2 2 J = mr 2 θ˙ = ma ka
√
E =
−
3 1 1 mka = E c + V ( V (r ) = mr˙ 2 + mr 2θ˙2 2 2 2
−
a2 km 4 r
−
a3 3 2 2r
1 = mr˙ 2 + V ef ef (r ) 2 r˙
1 a3 k V ef V (r ) = m ef (r ) = m 2 + V ( 2 r
−
a2 a3 4k + 2k 2k 2 r r
r˙ = 0 V ef ef (r ) = E =
−
3r 2 rmin = 23 a
3 a2 a3 mka = m 4k + 2k 2k 2 2 r r
− 8ar + 4a 4 a2 = 0
rmax = 2a
rmax rmin r = a rmin < r < rmax m
θ
λ h h
h
Y Z
X dm
”x”
|dg| = Gdm r2 r mdl/L = mRdθ/(2 mRdθ/(2πR πR)) = m/(2 m/(2π π )dθ
dm θ
Y Z √ r = x2 + R 2 Gmdθ 2π 2 (x + R2 )
dg =
x cosφ =
x r
=
gx = gcosφ
x (x2 +R2 )1/2
dgx = dgcosφ = G
2π
gx =
m x dθ 2π (x2 + R2 )3/2
0
dm =
m x dθ G 2π (x2 + R2 )3/2
gx = Gm
x (x2 + R2 )3/2 x
g=
U = U ( U (x) = x R
≪
≫R
g.dr =
− −
Gm
x dx = (x2 + R2 )3/2
− (x2 +Gm R2 )1/2
− (x +Gm R )
U ( U (x) =
x
−
−Gm (x2 +xR2)3/2 i
2
−
2 1/2
Gm = (x2 + R2 )1/2
−
Gm R
1
≈= − 1/2
1+
U ( U (x) =
x2 R2
− Gm x
x2 2R2
−
Gm 1 R
GM/ 2R3
M x E p (x) = M U ( U (x) =
h
E c = 12 mv p2
− (x2GmM + R2 )1/2
v p
E T U (x = h) = 0 T (0) = E c (0) + U (
− (h2GmM + R2 )1/2
v
x=0
1 − GmM + mv 2 R 2 GmM 1 − (h2GmM =− + mv 2 2 1 / 2 R 2 +R )
v= 2
GM R
−
GM (h2 + R2 )1/2
1 2
xm
− (h2 +GM =− 2 R2 )1/2 (x
GM 2 1/2 m+R )
xm = h M Ro
r g(r ) =
ˆr r
M ( M (r ) =
− GM r2(r) ˆr
M ( M (r ) M 3 = R r
ρ 43π r 3
ρ = M 0 /(4πR (4πR 03 /3)
3 0
g(r ) =
− GM rˆr R3 0
m
F = mg(r )
F = ma a a = r¨ˆ r
r¨ =
− GM r R3 0
r¨ + ω 2 r = 0 ω=
GM R30
T = 2π/ω
T /4 r (t) =
−R0 cos(ωt cos(ωt))
r˙ (t) = ωR 0 sen(ωt sen(ωt))
r (0) = t = T /4
vmax = r˙ (T /4) = ωR 0 = U = 1 2 2 mvmax
U =
− R0
2GM/R0
r˙ (0) = 0
GM R0
GMm GM m R0
vmax =
GMm GM m R0
U = m m R
W =
F ( F (r )dr =
0
1 2 2 ω mr 2
2 /2 W = mvmax
vmax = m l
GM/R0
ω ω
(0, (0, l) r
FG
=
(l, l) m
(l, 0) M
G Mr3m r FG = Gm2
1 1 1 (0, (0 , l ) + ( l, 0) + (l, l ) l3 l3 ( 2l)3
√
Gm2 1 = 2 (0, (0, 1) + (1, (1, 0) + (1, (1, 1) l ( 2)3
√
Gm2 1 = 2 1+ (1, (1, 1) l ( 2)3
√
F c
√
ω
r = l/ 2
√
Gm2 1 2 1 + l2 ( 2)3
√
−
m
lω 2 =0 2
√
ω
ω=
2Gm 1 1 + . l3 ( 2)3
√
GM m V ( V (r ) = GMm M r r= r = RT E (r = ) = E (r = RT )
m V (r =
∞
∞) = 0 1 0 = mv 2 2
v=
m − GM RT
2GM RT
∞
−
( 1, 1) = mrω 2
− −
L1
L2 L
L 1 > L2
L = L = mRv ′
′
L = L = v ′
E =
| |
m/2 m/2 2 2 v
m/2 m/2
m 2 Rv
R E =
GMm/2 − GMm/2 = E/ E/22 R
M ′
||
m 2 2v
GM m − GMm R
L′
E
m E =
m 2 L2 r˙ + 2 2mr2
− GMr m
L
r
V ef ef (r ) =
L2 2mr 2
− GMr m m r m
r mr¨ =
′
−V ef (r)
rmin rmin
L2 = , Gm2 M
V ef ef (rmin ) =
−
G2 m3 M 2 2L2 E r
rmin
≡R
m m/2 m/2 m 2 L2 E = r˙ + 4 mr ′
′
m − GM 2r
E m 2 L2 = r˙ + 2 4 4mr2
m − GM 2r
m 2 L2 E = r˙ + 2 2mr2
− GMr m
m m/2 m/2
rmin m
m
E =
m 2 2v
− GMR m
′
E =
m/2 m/2 2 2 v
GM m m − GMm E + GM 2R = E + 2R
′
L =L m ′
E =
m 2 L2 r˙ + 2 2mr 2
m − GM 2r
L r
∗
V ef (r ) =
L2 2mr 2
m − GM 2r
2
2L rmin = Gm = 2rmin M m m/2 m/2 ∗
2
∗
V ef (rmin ) =
−
G2 m3 M 2 8L2
V ef (rmin ) = 14 V ef ef (rmin ) ∗
v
R = GM/R F g = GmM/(2 GmM/(2R R2 ) =
mv 2 /R
v2
F c = F c = GMm/R2
F c /2 ′
E = E + ′
E = 0
∆U =
−
3 GM 2 5
2 R
−
GMm GM m 2R
1 R
=
rmin
−
3 GM 2 5 R
≡R
6, 6
2 ms v 2 /r = GM T T ms /r
M T T
ms r = r RT ∗
2 = gR /r v2 = GM T T /r T
T = 2πr/v = 2πr
3/2
∗
RT /g
∗
m
R
v δm
mv 2 GM m = R R2 Rv 2 = GM R
M v 1 E = mv 2 2
− GMR m = − 12 mv2
m L = mRv
L = (m + δm) δm)Rv
′
′
v =
mv m + δm m + δm
1 E = (m + δm) δm)v 2 2 ′
′
−
GM (m + δm) δm) = E R
−
mδmv 2 2(m 2(m + δm) δm)
′
E < E ′
v
m V ef (r ) =
L L
L2 2mr
GM m − GMm r m + δm
V ef m+δm (r )
L2 = 2(m 2(m + δm) δm)r
GM (m + δm) δm) − GM ( r
− δmv 2
0.2
r
1
2
3
4
5
0.2
0.4
0.6
m
m + δm
m E
m+δm V ef (r )
m (r ) m + δm V ef m
rmin = R m
′
E
r m + δm ′
E
m + δm δm m+δm V ef (r )
m+δm dV ef (r ) = 0, dr
2
L → rmin = (m + δm) δm)2 GM
′
R = rmin L2 m2 R = = R (m + δm) δm)2 GM (m + δm) δm)2 ′
′
R
′
L = mRv = (m + δm) δm)R v
′′
′
R 2 v =
√
√
GM R
m R = (m + δm) δm) R
(v )2 = ′′
GM R ′
′
′
R =
m2 R (m + δm) δm)2
′′
v m + δm
m
M a p
R
mva ra = mv p r p 1 mva2 2
− G Mram = 12 mv p2 − G Mr pm ra = 60 r p = 60 R 120 GM 61 R
v p2 =
v p2 =
2GM R
rT v1 r2 v2 v v1 /v2
v12 =
2GM S S . rT (1 + rrT ) 2
M S S
v v
2
r T r
Sol 2
v1
2 mvT GmM S S = 2 rT rT
vT =
GM S rT
rT v1 = r2 v2
v1 r2 = . v2 rT
1 2 mv 2 1
1 GmM S S S S − GmM = mv22 − rT 2 r2
v1 = vT
2 1 + rrT 2
1 1 2 2 2 ∆E 1 = mvT + mvT = mvT . 2 2 2 2 1 1 2 r2 + rT ∆E 2 = mv12 + mv22 = mvT . 2 2 r22 + r2 rT
r2 > rT
2 r22 + rT < 1. r22 + r2 rT
∆E 1 > ∆E 2
m
vA , vB
OA = 12000km 12000km R = 3400km 3400km M = 6, 4551023 kg
11 N m kg 2
−
G = 6, 6710
2
E E = 0 e = 1
E total total
e
2 mvA =0= 2
m − GM OA
vA =
2GM m = 2678, 2678, 7 s OA
J 2 r= Gm2 M 1 J
−
1 1 e2 =a ecosθ 1 ecosθ
− −
e
a
E>0 0
rmax rmin OA OB = = 0,58 rmax + rmin OA + OB
−
−
a = AB/2 AB/ 2 = 7700km 7700km θ=0
θ=π J 2 1 a(1 OA = = 2 Gm M 1 e 1
− e2) = a(1 + e) − −e J 2 1 a(1 − e2 ) OB = = = a(1 − e) Gm2 M 1 + e 1+e J = cte OA =
(OAmvA )2 1 Gm2 M 1 e
− ⇒ vA = (OBmvB )2 1 OB = ⇒ vB = Gm2 M 1 + e J √ b = a 1 − e2
GM 1 e m = 1219, 1219,16 a 1+e s
−
GM 1 + e m = 4586, 4586,37 a 1 e s vA
−
T
S elipse πab J elipse = = T T 2m
⇒ T = t=
4π 2 a3 = 20459, 20459,9s GM
T = 2,84 84h. h. 2
≈ 5, 68 68h. h.
vB
m r (t) = kθ( kθ (t) d a = (¨r r θ˙ 2)ˆ r + 1r dt (r 2 θ˙)θˆ
−
F(r) =
m1 = m
m2 = 2m
−
α 2β r + 2 r r r α, β
r
(r, θ ) r˙
r
J
J =
2β 2 m α
ρ ω ω
d 27 27,, 3
7,0 5
× 109
× 103
×1030 mc
M s
10 km/ km/s
80 km/ km/s mT
30 km/ km/s
R 1,5
×
108
km
Ra
h p
ha ha , h p << R
R
h H T 1 T 2
R R0
m1
m2
k
r1 = (x1 , y1 , z1 ) r2 = (x ( x2 , y2 , z2 )
T =
m1 2 m2 2 (x˙ 1 + y˙ 12 + z˙12 ) + (x˙ 2 + y˙ 22 + z˙22 ) 2 2 rcm = (xcm , ycm , zcm )
r = r1
−r
2
= (x,y,z) x,y,z)
µ 1 1 1 = + µ m1 m2 rcm =
m1 r1 + m2r2 m1 + m2
m1 x1 + m2 x2 m1 + m2 m1 y1 + m2y2 ycm = m1 + m2 m1 z1 + m2 z2 zcm = m1 + m2 x = x1 x2
xcm =
y = y1 z = z1
− −y −z
2
2
m2 x m1 + m2 m2 ycm + y m1 + m2 m2 zcm + z m1 + m2 m1 xcm x m1 + m2 m1 ycm y m1 + m2 m1 zcm z m1 + m2
x1 = xcm + y1 = z1 = x2 = y2 = z2 =
T = = + = =
− − −
m1 2 m2 2 (x˙ 1 + y˙ 12 + z˙12 ) + (x˙ + y˙22 + z˙22) 2 2 2 m1 m2 m2 m2 [(x˙ cm + x˙ )2 + (y˙ cm + y˙ )2 + (z˙cm + z˙ )2 ] 2 m1 + m2 m1 + m2 m1 + m2 m2 m1 m m1 1 [(x˙ cm x˙ )2 + (y˙ cm y˙ )2 + (z˙cm z˙ )2 ] 2 m1 + m2 m1 + m2 m1 + m2 m1 + m2 2 m1 m2 2 2 (x˙ cm + y˙ cm + z˙cm )+ (x˙ 2 + y˙ 2 + z˙ 2 ) 2 2(m 2(m1 + m2) m1 + m2 2 µ 2 2 (x˙ cm + y˙ cm + z˙cm ) + (x˙ 2 + y˙ 2 + z˙ 2 ) 2 2
−
−
−
V g = m1 g r1 + m2 g r2 = m1 gz1 + m2 gz2
·
·
V g = m1 gz1 + m2 gz2 = m1g(zcm +
m2 z ) + m2g(zcm m1 + m2
− m m+ m 1
1
= (m1 + m2 )gzcm l0 V e =
k ( (x1 2
2
−x )
V e =
2
+ (y ( y1
2
−y ) 2
+ (z (z1
k ( x2 + y 2 + y 2 2
2
2
−z ) −l ) 2
0
2
−l ) 0
L = T V = T (V e + V g ) m1 + m2 2 µ 2 2 = (x˙ cm + y˙cm + z˙cm ) + (x˙ 2 + y˙ 2 + z˙ 2 ) 2 2 k ( x2 + y 2 + y 2 l0)2 (m1 + m2 )gz cm 2 m1 + m2 2 µ k = (r l0 )2 (m1 + m2 )g rcm r˙ cm + r˙ 2 2 2 2
−
−
−
−
−
−
r
r
−
−
·
2
z)
x + y + y − l ) x x +y +y y µy¨ = −k ( x + y + y − l ) x +y +y z µz¨ = −k ( x + y + y − l ) x +y +y
x ¨cm = 0
µx ¨=
2
2
y¨cm = 0 z¨cm =
2
−k (
2
2
−g
¨rcm = µ¨r =
2
2
2
2
0
0
0
2
2
2
2
2
2
2
2
2
−g −k(r − l ) rr 0
m M k
x
y k
m M y
M x
L = T
d dt
−
∂L ∂ ˙ ∂ x˙
¨ + my¨ = mx
m M
l0 m
y˙
M y˙2 m ˙2 V = + (x + y˙2 + 2x˙ y˙ ) 2 2
− k2 (x − l )
∂L = ∂x
∂L
d dt
−k(x − l )
∂ ˙ ∂ y˙
=
2
∂L ∂y
2x ¨) = 0 M ¨ M y¨ + m(¨ y + 2¨
0
M ˙ M y˙ =
0
−m(x˙ + y˙ ) m2
m1 s φ x1 = s; y1 = 0; x2 = l sen φ + s; y2 =
−l cos φ
m1s˙2 m2 ˙2 L= + (s + 2l 2ls˙ φ˙ cos φ + l2φ˙2 ) + m2 gl cos φ. 2 2
x˙ + y˙
m1
m2
s
ps =
∂L = (m1 + m2 )s˙ + m2lφ˙ cos φ. ∂ ˙ ∂ s˙ L m
L
θ
θ
θ
x θ xc = x + l/2sen l/2sen θ yc =
−l/2cos l/2cos θ
x˙ c = x˙ + l/2 l/2θ˙ cos θ y˙ c = l/2 l/2θ˙ sen θ ). vc2 = x˙ 2 + l2 /4θ˙2 + lx˙ θ˙ cos θ
1 1 T = mvc2 + I c θ˙ 2, 2 2 I c =
1 12
ml2 V =
m L= 2
−mgl/2cos mgl/2cos θ
1 x˙ + lx˙ θ˙ cos θ + l2θ˙2 + gl cos θ , 3 2
d dt
∂L m = 2 ∂ θ˙
∂L
=
∂ q ˙i
∂L , ∂q i
∂L m ∂L = 2x˙ + lθ˙ cos θ =0 ∂ ˙ ∂ x˙ 2 ∂x 2 2˙ ∂L m l θ + lx˙ cos θ = l sen θ θ˙ x˙ + g 3 ∂θ 2
−
l x˙ + θ˙ cos θ = 2
,
3 3g θ¨ + x ¨ cos θ + sen θ = 0 2l 2l θ l x ¨ + θ¨ cos θ 2
− 2l θ˙
2
sen θ =
,
1¨ 3g θ + 3θ˙2 sen θ cos θ + sen θ 4 2l θ
sen θ 1¨ θ= 4
≈ θ cos θ ≈ 1
θ˙2 θ
≈0
− 32gl θ
m k O
ω
r, θ 2
|v| T =
x = r cos θ, y = r sen θ = r˙ + r θ˙ 2
2 2
1 m(r˙ 2 + r 2θ˙ 2 ) 2
1 V = k(r 2
2
− a)
a
L = T
− V = 12 m(r˙
2
+ r 2θ˙ 2)
− 12 k(r − a)
2
m
k
mr¨
− mrθ˙
2
+ k (r
− a) = 0
d mr 2θ˙ = 0 dt θ˙ = ω = constante
−mrω
2
+ k(r
− a) = 0 =⇒ r = k −kamω
2
2L1
2L2
L1
L2
m1 m2
m3
xi x1 + x p = 2L1 x p x p = 2L1 x1
x2 +x3 = 2L2
−
z1 z2 z3
≡ ≡ ≡
x1 , x p + x2 = 2L1 x p + x3 = 2L1
−x −x
1
+ x2 ,
1
+ x3 = 2L1
−x
1
+ 2L 2L2
−x . 2
x3 = 2L2 x2
−
x1 , x2 1 1 m1 z˙12 = m1x˙ 21 , 2 2 1 1 m2 z˙22 = m2(x˙ 2 x˙ 1)2 , 2 2 1 1 m3 z˙32 = m3(x˙ 2 + x˙ 1)2 . 2 2
T 1 = T 2 =
−
T 3 = V i =
L =
−m gz T − V = T + T + T − V − V − V = 1 1 (m + m + m )x˙ + (m (m − m )x˙ x˙ + (m 2 2 (m − m )gx 1
+
2
1
2
2
3
3
i
j
3
1
2 1
2
3
3
2
1
2
2
+ m3 )x˙ 22 + (m ( m1
− m − m )gx 2
3
1
2
d ∂L dt ∂ ˙ ∂ x˙ i
∂L − ∂x
= 0,
i
i = 1, 2 (m1 + m2 + m3)¨ x1 + (m (m3 (m3
− m )¨x 2
1
− m )¨x 2
2
= g(m1
+ (m (m3 + m2 )¨ x2 = g(m2
− m − m ), −m ) 2
3
3
x ¨i
m m + m m − 4m m g, m m + m m + 4m 4m m 2(m 2(m m − m m )
x¨1 =
1
2
1
3
2
3
1
2
1
3
2
3
1
x¨2 =
2
1
3
m1 m2 + m1 m3 + 4m 4 m2 m3
g,
M = m1 + m2 + m3
x ¨1
x ¨2
m1 : z¨1 = x¨1 , m2 : z¨2 =
m m − 3m m + 4m 4m m −x¨ + x¨ = m m + m m + 4m g, 4m m m m − 3m m + 4m 4m m 1
1
1
m3 : z¨3 =
−x¨ − x¨ 1
2
=
1
1
3
2
3
2
3
1
3
1
2
2
2
3
3
m1m2 + m1 m3 + 4m 4m2 m3
m M
2
2
g.
r
α s
x T tras tras T rot rot
T =
M 2 M 2 m 2 I V + T tras V + vcm + ω 2 tras + T rot rot = 2 2 2 2
s
x
α
2 vcm
V = x˙ ω
I (xc , yc ) = (x + s cos α, cte cte s sen α) 2 2 2 vcm = x˙ + s˙ + 2 x˙ s˙ cos α θ s = rθ
−
ω = θ˙ = s/r ˙ T disco disco =
m 2 I (x˙ + s˙ 2 + 2x˙ s˙ cos α) + 2 s˙ 2 2 2r
V =
L = T
− V = 12 (M + m)x˙
2
−
I = s sen α mgs sen α
1 2
mr 2
3 + mx˙ s˙ cos α + ms˙ 2 + mgs sen α. 4 x
px =
px
∂L = (M + m)x˙ + ms˙ cos α = C T E ∂ ˙ ∂ x˙
E = T + V E = T + V =
1 3 (M + m)x˙ 2 + mx˙ s˙ cos α + ms˙ 2 2 4
x˙ = 0, s˙ = 0, s = 0
− mgs sen α.
E = 0
x˙ = m2 s˙ 2 cos2 α 2(M 2(M + m)
2
− (M m+ m) s˙ cos
2
3 α + ms˙ 2 4
−m/( m/(M + m)s˙ cos α
− mgs sen α = 0
C 2 s˙ = s 2 s(t) = m ω
1 2
C
t
x = ρ cos ωt y = ρ sen ωt z = z
x˙ = ρ˙ cos ωt ρ sen ωt y˙ = ρ˙ sen ωt + ρ cos ωt
−
z˙ = z˙ ρ
z L = T
L=
− V
m ˙2 (ρ + ρ2 ω2 + z˙ 2 ) 2 m1
− mgz m2
a π/4 π/4
m1
m2
XY x1 = 0 y1 = a sen φ x2 = a cos φ y2 = 0
a2 φ˙ 2 L= (m1 cos2 φ + m2 sen 2 φ) 2
φ
ga − √ (m 2
1
sen φ + m2 cos φ)
m1 xCM =
m2 a cos φ m1 +m2
yCM =
m2
m1 a sen φ m1 +m2
T t =
a2 φ˙ 2 (m21 cos2 φ + m22 sen 2 φ) 2(m 2(m1 + m2) ′
′
(X Y ) ′
x1 =
−R
1
′
′
′
X Y
cos φ y1 = R1 sen φ x2 = R2 cos φ y2 = R2 sen φ (0, 0, φ˙ ) I ω = (0,
T r = ′
zz I z z ′
′
R2 =
−
am1 m1 +m2
R1 = a
−R
2
1 T 1 ω Iω = I z z φ˙ 2 2 2 ′
′
′
m1m2 a2 = m1 (x 1 + y 1 ) + m2 (x 2 + y 2 ) = . m1 + m2 ′2
′2
′2
′2
a2 φ˙ 2 1 m1 m2a2 ˙ 2 a2 φ˙ 2 2 2 2 2 T = T t + T r = (m1 cos φ + m2 sen φ) + φ = (m1 cos2 φ + m2 sen 2 φ) 2(m 2(m1 + m2 ) 2 m1 + m2 2
d ∂L dt ∂ φ˙ ∂L ∂ φ˙ ∂L ∂φ
− ∂L =0 ∂φ
= a2 φ˙ (m1 cos2 φ + m2 sen 2φ) = a2 φ˙ 2 (m1
− m )a φ˙ (m 2
2
1
cos2 φ + m2 sen 2 φ)
a2 φ¨(m1 cos2 φ + m2 sen 2φ) + a2φ˙ 2 (m2
ga − √ (m 2
ga − m )sen φ cos φ − √ (m 2 1
1
cos φ
−m
2
sen φ)
1
cos φ
−m
2
sen φ) = 0
m1 = m2 φ¨
− 2ag√ 2 (cos φ − sen φ) = 0 √
a 2mg(sen mg(sen φ +cos φ) π/4 π/4 φ=0
φ = π/2 π/2
(r,θ,z) r,θ,z)
z
r 2 = az
T = 12 m(r˙ 2 + r 2θ˙2 +
mgz z˙ 2 ) r r
L = T
d dt d dt
∂L ∂L − ∂r ∂ ˙ ∂ r˙ ∂L ∂L ∂ θ˙
− ∂θ
θ
− V = 12 m(r˙ = =
r 2 = az
z
2
+ r 2θ˙ 2) + 4
d r 2r˙ (mr˙ + 4m 4m 2 ) dt a d (mr 2 θ˙) = 0 dt
−
r 2 r˙ 2 a2
2
− mg ra
r r˙ 2 mr θ˙ 2 + 4m 4m 2 a
−
r 2mg a
=0
θ mr 2θ˙ = mJ θ˙
r
r
m2 m1
α
L
s
m2
m1
q
m h k
l0
J
m
h
l
m1
m1
m2
z J z =
∂L , ∂ φ˙
φ
z
L
m1
m2
m α m 2α
m y = a tan2 (x/a) x/a)
a y
y
ω
m ys = y0 sen ωt ω
l
m2
F (r)
O r O
O
(r, θ) d2 u F (u 1) +u=− 2 2 dθ2 hu −
h
θ O u=r
−1
h
M
R
µ m m p
r p
M = 100 kg R = 0, 4 m m = 75 kg m p = 10 kg r p = 0, 2 m α = 30
◦
C D
Y
B
.
x
x
O
.
X G
θ A
S
α
s
l
m ω0 k
(40/3)ml2
ω
′
k k
A
k
B
l m y=a k a + l0
θ=0 l
θ=0 y
a
x k θ m
