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Unformatted text preview: v C PROBLEM 17.20 A rope is wrapped around a cylinder of radius r and mass m as shown. Knowing that the cylinder is released from rest, determine the velocity of the center of the cylinder after it has moved downward a distance s . SOLUTION Kinematics: Point C is the instantaneous center r v = r v = Energy: Position 1 . At rest. T 1 = 0 Position 2 . Cylinder has fallen through distance s . 2 2 2 2 1 2 1 I v m T + = 2 2 2 2 1 2 1 2 1 + = r v mr v m 2 4 3 v m = Work. mgs U = 2 1 Principle of Work and Energy. 2 2 1 1 T U T = + 2 4 3 v m mgs = + gs v 3 4 2 = = gs 3 4 v PROBLEM 17.22 A halfcylinder of mass m and radius r is released from rest in the position shown. Knowing that the halfcylinder rolls without sliding, determine ( a ) its angular velocity after it has rolled through 90 o , ( b ) the reaction at the horizontal surface at the same instant. [ Hint : Note that GO = 4 r /3 and that, by the parallelaxis theorem, ( 29 2 2 2 1 GO m mr I = .] SOLUTION Kinematics: Point C is the instantaneous center b v =  = 3 4 r r r . 5756 = Energy: Position 1 . T 1 = 0 V 1 = 0 Position 2 . Kinetic energy 2 2 2 2 1 2 1 I v m T + = ( 29 2 2 2 2 3 4 2 1 2 1 5756 2 1  + = r m mr r . m 2 2 3256 mr . = Potential energy ( 29 OG mg V = 2 r mg 3 4 = ( a ) Conservation of energy. 2 2 1 1 V T V T + = + mgr mr . 3 4 3256 2 2 = + r g . 3035 1 2 = r g . 142 1 = v O G C b ( b ) Kinematics: Translation + Rotation about O = Rolling motion Acceleration of C . xcomponent. component....
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 Spring '10
 Prof.V.H.Chu

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