5, 22 kg 15 15,, 3 cm cm//s a)
645 ms
b)
c)
a) ω = ω2 =
2π = 9, 74 rad/ rad/s T
k , m
k = mω = mω 2 = 495 N/ N/m
b) vm´ax ax = Aω ,
A =
vm´ax ax = 1, 57 ω
× 10
2
−
m
c) f = (m = 1, 22 kg M
1 = 1, 55 Hz T
M = 8, 73 kg)
k = 344 N/ N/m m
M 0, 42 f
≤ ≤ µsN
f = µ s N = µ s mg
−+→ F = 0 −f + mam´axax = 0
x
am´ax = µ s g ax = µ
2 = am´ax ax = Aω
⇒
A =
am´ax µs g ax = (M + + m) ω2 k A = 0, 119 m
10 10,, 2 cm
I cm cm =
M 2 r + r22 2 1
6, 40 mm
2 I 1 = I cm cm + M r1
r1 I 2 = I = I cm (3 r1 + 2∆r 2∆ r)2 cm + M (3r ∆r I = 2I cm (0, 102 m)2 + M (0, (0, 3188 m)2 = 0, 134 m2 M cm + M (0,
dcg = 2r1 + ∆r T = 2π
I = 1, 60 s M gd cg
k = 2, 94 N/cm 23 23,, 9 cm
a)
b) c)
T = 2π 2π M
3M 2k E =
U =
1 kA2 2
K = K traslacional traslacional + K rotacional rotacional =
1 1 M vc2 + I ωc2 2 2
ωc
ωc =
vc R
1 I = M R2 2 1 1 K = M vc2 + 2 2
1 M R2 2
2/3
vc2 1 1 = M vc2 + vc2 2 R 2 2
1/3 E = = 8, 40 J K t = 23 E = 5, 60 J
a)
K r = 13 E = 2, 80 J
b) c)
3 M 2
3 M 2
m L h θ
M
E = 12 kx 2 + 12
3 M 2
v2
k
m
Mg kx
θ
x Mg
θ
M gL
L
θ + kxh cos θ =
x h
kx
s = hθ = hθ
h cos θ
−I α = −I θ¨
≈s θ
cos θ
≈1 θ¨ +
M gL + kh2 I
θ = 0
θ¨ + ω 2 θ = 0 2
ω =
M gL + kh2 I
ω ω 1 f = = 2π 2π L
M gL + kh2 M L2
≈ θ
k1
k2
x1 + x2 = x
k1 x1
k2
x2 F =
−k1x1 = −k2x2
x2 x2 =
F =
k1 x1 k2
(x1 + x2 ) −kefectiva x = −kefectiva (x
−k1x1 = −kefectiva (x (x1 + x2 )
k1 x1 = k efectiva x1 + k1 = k efectiva 1 + kefectiva =
mx ¨ =
k1 x1 k2
k 1 k2
1 1 + k1 k2
−kefectiva x
1
−
x − kefectiva m
x ¨ =
ω0 =
1 k1
+
1
−
1 k2
m
k1 k2 (k1 + k2 ) m
=
0, 6 kg 240N/m 240N/ 0, 6 m
m = 0, 6 kg 2m
k
ω0 1 f = = 2π 2π
k 1 = 2m 2π
240 N/m 240 2 (0, (0, 6kg)
f = 2, 25Hz
f =
ω0 1 = 2π 2π
k 1 = m 2π
240N/m 240N/ = 3, 3 , 18Hz (0, (0, 6kg) m
2m
→ m ⇒ K → → K 2 mv2
→ 12 mv2
K = E
K 1 2 = kA 2 4
E =
1 E = (240 (240 N/m)(0, m)(0, 6 m)2 = 21 21,, 6 J 4 E = 12 kA2
1 E = kA2 4 1 1 kA 2 = kA2 2 4 A A = 0, 42 m 2
√ ≈
k
A = A
E = E m
50 50,, 0 g 5, 0 Hz
t = 0 s v = 1, 0 m/s
x = 0 m
ω = 2πf = 10 10π π rad/ rad/s
x (t) = A cos(ωt cos(ωt + φ) v (t) =
−Aω
(ωt + φ)
x0 A cos φ = = v0 A φ
⇒
−
x20 + A =
cos φ π = 0, φ = rad φ 2
v02 = A 2 cos2 φ + 2 ω
±
x20 +
1 1 K = mv 2 = mω2 A2 2 2
2
v02 ω2
2
φ
≈ ±0, 032 m
1 (ωt + φ) = mω2 A2 cos2 (ωt) ωt ) 2
2
K = (0, (0, 025 025 J) cos cos
rad 10 10π π t s
x = 0 m
