quiz1_v2

For this problem, you may ignore altitude and assume the pilot will bring the plane level to the height of the runway (just assume no pitch). Define ship frame A = ( G, ⃗ p 1 , ⃗ p 2 , ⃗ p 3 ) and airplane frame B = ( C, ⃗ b 1 , ⃗ b 2 , ⃗ b 3 ) as shown. The airplane frame has a distance h p ( t ) ⃗ f 1 + l p ( t ) ⃗ f 2 from point O, and the ship frame has a distance h s ( t ) ⃗ f 1 + l s ( t ) ⃗ f 2 from point O. The airplane turns at rate ˙ β and the ship turns at rate ˙ α relative to F . Distances | −−→ CH | = r and | −→ GL | = d are constant. Also, −−→ CH = − r ⃗ b 2 . 1. For the airplane frame B : (i) Write ⃗ b 1 , ⃗ b 2 , ⃗ b 3 in terms of F . (ii) What is −→ ω B/F ? (iii) Find { −−→ CH } B . 2. For the ship frame A : (i) Write ⃗ p 1 , ⃗ p 2 , ⃗ p 3 in terms of F . (ii) What is −→ ω A/F ? (iii) Find { −→ GL } A . 3. Find ⃗v F ( L ) . (The velocity of the landing cable) 4. Find ⃗v F ( H ) . (The velocity of the hook) 5. We want the relative speed of the hook H to the cable L to be less than some constant v max . What’s a mathematical way to write this condition? (No need to compute it) 6. We want no sideways slipping of the hook relative to the ship deck. What’s a mathematical way to write this condition? (No need to compute it) Hint: think about dot or cross products. 2