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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
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