229_Dynamics 11ed Manual

229_Dynamics 11ed Manual - Engineering Mechanics - Dynamics...

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Unformatted text preview: Engineering Mechanics - Dynamics θ 1 = 90 deg Chapter 13 g = 32.2 ft W M= 2 g s θ = θ1 Solution: r= r'' = a r' = 1 − cos ( θ ) −a sin ( θ ) ( 1 − cos ( θ )) 2 θ'' + −a sin ( θ ) ( 1 − cos ( θ )) 2 −a cos ( θ ) θ' 2 ( 1 − cos ( θ )) 2 θ' 2a sin ( θ ) θ' 2 + 2 ( 1 − cos ( θ )) 3 Find the angle ψ using rectangular coordinates. The velocity is parallel to the path x = r cos ( θ ) x' = r' cos ( θ ) − rθ' sin ( θ ) y = r sin ( θ ) y' = r' sin ( θ ) + rθ' cos ( θ ) x'' = r'' cos ( θ ) − 2r' θ' sin ( θ ) − rθ'' sin ( θ ) − rθ' cos ( θ ) 2 y'' = r'' sin ( θ ) + 2r' θ' cos ( θ ) + rθ'' sin ( θ ) − rθ' sin ( θ ) 2 ψ = atan ⎛ ⎜ y' ⎞ ⎟ ⎝ x' ⎠ Given ψ = 45 deg P cos ( ψ) + H sin ( ψ) = M x'' ⎛P ⎞ ⎜ ⎟ = Find ( P , H) ⎝H⎠ Guesses P = 1 lb H = 1 lb P sin ( ψ) − H cos ( ψ) = M y'' ⎛ P ⎞ ⎛ 12.649 ⎞ ⎜ ⎟=⎜ ⎟ lb ⎝ H ⎠ ⎝ 4.216 ⎠ Problem 13-110 The tube rotates in the horizontal plane at a constant rate θ'. If a ball B of mass M starts at the origin O with an initial radial velocity r'0 and moves outward through the tube, determine the radial and transverse components of the ball’s velocity at the instant it leaves the outer end at C. 2 Hint: Show that the equation of motion in the r direction is r'' − rθ' = 0. The solution is of the form r = Ae-θ't + Beθ't. Evaluate the integration constants A and B, and determine the time t at r1. Proceed to obtain vr and vθ. 227 ...
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