Explanation Given information: The mass ( m R ) of the rod AB is 4 kg. The mass ( m T ) of the tube CD is 6 kg. The angular velocity ( ω ) of the assembly is 5 rad/s. Calculation: Consider l is the length of the rod and the tube and O is the point of intersection of the tube and axle. Write the equation of the centroidal mass moment of inertia ( I ¯ T ) of the tube. I ¯ T = 1 12 m T l 2 Write the equation of the centroidal mass moment of inertia ( I ¯ R ) of the rod. I ¯ R = 1 12 m R l 2 Write the equation of the tangential equation of the tube using kinematics. ( v ¯ θ ) T = r ¯ T ω Substitute l 2 for r ¯ T . ( v ¯ θ ) T = l 2 ω Write the equation of the tangential equation of the rod using kinematics. ( v ¯ θ ) R = r ¯ R ω Substitute l 2 + x for r ¯ R . ( v ¯ θ ) R = ( l 2 + x ) ω Find the equation of angular momentum about point O . H O = m T r ¯ T ( v ¯ θ ) T + I ¯ T ω + m R r ¯ R ( v ¯ θ ) R + I ¯ R ω Substitute l 2 for r ¯ T , l 2 ω for ( v ¯ θ ) T , 1 12 m T l 2 for I ¯ T , l 2 + x for r ¯ R , ( l 2 + x ) ω for ( v ¯ θ ) R , and 1 12 m R l 2 for I ¯ R . H O = m T l 2 ( l 2 ω ) + 1 12 m T l 2 ω + m R ( l 2 + x ) ( l 2 + x ) ω + 1 12 m R l 2 ω = 4 12 m T l 2 ω + m R ( l 2 4 + 2 l 2 x + x 2 + 1 12 l 2 ) ω = 1 3 m T l 2 ω + m R ( 4 l 2 12 + l x + x 2 ) ω = [ 1 3 m T l 2 + m R ( 1 3 l 2 + l x + x 2 ) ] ω (1) Find the equation of kinetic energy. T = 1 2 m T ( v ¯ θ ) T 2 + 1 2 I ¯ T ω 2 + 1 2 m R ( v ¯ θ ) R 2 + 1 2 m R v r 2 + 1 2 I ¯ R ω 2 Substitute l 2 ω for ( v ¯ θ ) T , 1 12 m T l 2 for I ¯ T , ( l 2 + x ) ω for ( v ¯ θ ) R , and 1 12 m R l 2 for I ¯ R . T = 1 2 m T ( l 2 ω ) 2 + 1 2 ( 1 12 m T l 2 ) ω 2 + 1 2 m R ( l 2 + x ) 2 ω 2 + 1 2 m R v r 2 + 1 2 ( 1 12 m R l 2 ) ω 2 = 1 2 [ m T l 2 4 ω 2 + 1 12 m T l 2 ω 2 + m R ( l 2 4 + l x + x 2 ) ω 2 + 1 12 m R l 2 ω 2 ] + 1 2 m R v r 2 = 1 2 [ 1 3 m T l 2 + m R ( 1 3 l 2 + l x + x 2 ) ] ω 2 + 1 2 m R v r 2 (2) Substitute Equation (1) in Equation (2), T = 1 2 H O ω + 1 2 m R v r 2 (3) All the motion in the system is horizontal

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 17.2, Problem 17.88P

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Find the speed of the rod relative to the tube when x=400 mm.

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Chapter 17.2, Problem 17.87P

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Find the speed of the rod relative to the tube when x = 400 mm .

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