Assignment 10 Solution 2009

Assignment 10 Solution 2009 - PROBLEM 17.20 A rope is...

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v ω 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 ϖ r v = r v = ϖ Energy: Position 1 . At rest. T 1 = 0 Position 2 . Cylinder has fallen through distance s . 2 2 2 2 1 2 1 ϖ 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 0 v m mgs = + gs v 3 4 2 = = gs 3 4 v
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PROBLEM 17.22 A half-cylinder of mass m and radius r is released from rest in the position shown. Knowing that the half-cylinder 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 π and that, by the parallel-axis theorem, ( 29 2 2 2 1 GO m mr I - = .] SOLUTION Kinematics: Point C is the instantaneous center ϖ b v = ϖ π - = 3 4 r r ϖ r . 5756 0 = Energy: Position 1 . T 1 = 0 V 1 = 0 Position 2 . Kinetic energy 2 2 2 2 1 2 1 ϖ I v m T + = ( 29 2 2 2 2 3 4 2 1 2 1 5756 0 2 1 ϖ π ϖ - + = r m mr r . m 2 2 3256 0 ϖ mr . = Potential energy ( OG mg V - = 2 r mg π 3 4 - = ( a ) Conservation of energy. 2 2 1 1 V T V T + = + mgr mr . π ϖ 3 4 3256 0 0 0 2 2 - = + r g . 3035 1 2 = ϖ r g . 142 1 = ϖ v ω O G C b
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( b ) Kinematics: Translation + Rotation about O = Rolling motion Acceleration of C . x -component. 0 = - α r a O α r a O = Acceleration of G . x -component.
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