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PME2100 Gabarito da 2a Prova 29/10/2004
Departamento de Engenharia Mecaˆnica - PME Escola Polite´cnica da USP
1 Questa˜o (3,0 pontos)
Uma forc¸a F e´ aplicada na pec¸a em forma de U. Esta pec¸a pode deslizar ao longo da
guia. No ponto de contato em A o coeficiente de atrito e´ nulo, e no ponto de contato em
B o coeficiente de atrito e´ µ.
a) Desenhe o diagrama de corpo livre da pec¸a em forma de U.
b) Em func¸a˜o de a, F e µ, determine o intervalo dos valores de b compat´ıvel com o
equil´ıbrio. � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � �
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F
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Peça em
forma de U
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Figura 1: Pec¸a em forma de U
1.1 Soluc¸a˜o
� � � � � � � � � �
� � � � � � � � � �
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F
b
a
µΑ = 0
 
µΒ = µ
A
B
Peça em forma
 de U
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NA
NB
TB
Figura 2: Diagrama de Corpo Livre (1,0)
Equac¸o˜es do equil´ıbrio da pec¸a em forma de U: (1,0)
vertical: TB = F
horizontal: NA = NB
momento com polo B: NA · b = F · a ⇒ NA = NB = a
b
F
Atrito sem escorregamento:
TB ≤ µ NB (0,5) ⇒ F ≤ µ a
b
F ⇒ b ≤ µ a (0,5)
1
2 Questa˜o (3,5 pontos)
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C
L
θ
ωBC
O=CIRBC
vC
vB
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ω2
C D=CIRC
vC
Figura 4: Movimento dos treˆs so´lidos
a) Disco B
~vB = ~vA︸︷︷︸
~0
+ ~ω1︸︷︷︸
−ω1 ~k
∧ (B − A)︸ ︷︷ ︸
R1~j
⇒ ~vB = ω1R1 ~i (0,5)
2
b) Barra AB
~vC = ~vB + ~ωBC ∧ (C −B) ⇒ vC~j = ω1R1~i+ ωBC~k ∧ L
(
cos θ~i+ sin θ~j
)
{
0 = ω1R1 − ωBCL sin θ
vC = ωBCL cos θ
⇒ ωBC = ω1R1
L sin θ
(0,5)
vC =
ω1R1 cos θ
sin θ
(0, 5)
c) Disco C
~vC = ~vD︸︷︷︸
~0
+ ~ω2︸︷︷︸
−ω2 ~k
∧ (C −D)︸ ︷︷ ︸
−R2~i
⇒ ω2 = ω1R1 cos θ
R2 sin θ
(0,5)
ω˙2 =
dω2
dt
=
−ω1R1θ˙
R2 sin
2 θ
como
θ˙ = ωBC ⇒ ω˙2 = −ω
2
1R
2
1
R2L sin
3 θ
(0,5)
d) Disco C
~aC =
d~vC
dt
=
−ω21R21
L sin3 θ
~j (0,5)
e
~aD = ~aC + ~˙ω2 ∧ (D − C) + ~ω2 ∧ [~ω2 ∧ (D − C)]
~aD =
−ω21R21
L sin3 θ
~j +
ω21R
2
1
R2L sin
3 θ
~k ∧R2~i+ ω1R1 cos θ
R2 sin θ
~k ∧
[
ω1R1 cos θ
R2 sin θ
~k ∧R2~i
]
~aD = −ω
2
1R
2
1 cos
2 θ
R2 sin
2 θ
~i (0,5)
3
3 Questa˜o (3,5 pontos)
A placa ABCD pode girar em torno do eixo Ox, e sua velocidade angular em relac¸a˜o
ao garfo e´ ω2 (constante). O garfo (referencial mo´vel) gira em torno do eixo Oz com
velocidade angular ω1 (constante) em relac¸a˜o ao solo (referencial fixo). No instante em
que a placa ABCD esta´ na vertical, conforme mostra a figura, determine, em func¸a˜o de
ω1, ω2, a e b, e na base (~i,~j,~k) que orienta o sistema de coordenadas Oxyz (solida´ria ao
garfo):
a) As velocidades relativa ~vA,rel, de arrastamento ~vA,arr e absoluta ~vA,abs do ponto A.
b) As acelerac¸o˜es relativa ~aA,rel , de Coriolis ~aA,cor e absoluta ~aA,abs do ponto A.
c) O vetor velocidade angular absoluta ~ωabs da placa ABCD, e seu vetor acelerac¸a˜o
angular absoluta ~˙ωabs.
Solo
(referencial fixo)
Garfo
(referencial
móvel)
Placa
ABCD
O y
z
x
b
b
a
a
A
B
C
D
k
ji
ω2
ω1
Figura 5: Composic¸a˜o de movimentos de rotac¸a˜o
3.1 Soluc¸a˜o
a) Item (a):
~vA,rel = ~v0,rel + ~ω2 ∧ (A−O) = ~0 + ω2~i ∧
(
b~i+ a~k
)
= −ω2a~j ⇒ ~vA,rel = −ω2a~j (0,4)
4
~vA,arr = ~v0,arr + ~ω1 ∧ (A−O) = ~0 + ω1~k ∧
(
b~i+ a~k
)
= ω1b~j ⇒ ~vA,arr = ω1b~j (0,4)
~vA,abs = ~vA,rel + ~vA,arr = −ω2a~j + ω1b~j ⇒ ~vA,abs = (ω1b− ω2a)~j (0,3)
b) Item (b):
~aA,rel = ~a0,rel +
(
~˙ω2
)
rel
∧ (A−O) + ~ω2 ∧ [~ω2 ∧ (A− 0)]
= ~0 +~0 ∧
(
b~i+ a~k
)
+ ω2~i ∧
[
ω2~i ∧
(
b~i+ a~k
)]
~aA,rel = ω2~i ∧
[
−ω2a~j
]
= −ω22a~k ⇒ ~aA,rel = −ω22a~k (0,4)
~aA,arr = ~a0,arr + ~˙ω1 ∧ (A−O) + ~ω1 ∧ [~ω1 ∧ (A− 0)]
= ~0 +~0 ∧
(
b~i+ a~k
)
+ ω1~k ∧
[
ω1~k ∧
(
b~i+ a~k
)]
~aA,arr = ω1~k ∧
[
−ω1b~j
]
= −ω21b~i ⇒ ~aA,arr = −ω21b~i (0,4)
~aA,cor = 2~ω1 ∧ ~vA,rel = 2ω1~k ∧
(
−ω2a~j
)
= 2ω1ω2a~i ⇒ ~aA,cor = 2ω1ω2a~i (0,4)
~aA,abs = ~aA,rel + ~aA,arr + ~aA,cor = −ω22a~k − ω21b~i+ 2ω1ω2a~i
~aA,abs =
(
2ω1ω2a− ω21b
)
~i− ω22a~k (0,4)
c) Item (c):
~ωabs = ~ωrel + ~ωarr = ω2~i+ ω1~k ⇒ ~ωabs = ω2~i+ ω1~k (0,4)
~˙ωabs =
(
~˙ωrel
)
rel
+~˙ωarr+~ωarr∧~ωrel = ~0+~0+ω1~k∧ω2~i = ω2ω1~j ⇒ ~˙ωabs = ω2ω1~j (0,4)
5

Outros materiais