Explanation Given information: The length of rod AB and CD is l . The mass of rod AB and CD is m . Calculation:; Show the free body diagram equation as in Figure (1). The external forces in the system are forces due to mass of the rods. The effective force in the system is m ( a ¯ ) A B and m ( a ¯ ) C D and the effective restoring couple is I ¯ θ ¨ . Take moment about A in the system for external forces. ( ∑ M A ) e x t e r n a l = ( m g × l 2 sin θ ) c l o c k w i s e + ( m g × l sin θ ) c l o c k w i s e = ( m g l 2 sin θ ) + ( m g l sin θ ) = ( l 2 + l ) m g sin θ = 3 2 m g l sin θ As the angular displacement is very small, take the value of sin θ as θ . Thus, the moment of external forces is 3 2 m g l θ . Take moment about A in the system for effective forces. ( ∑ M A ) e f f e c t i v e = ( ( m ( a ¯ ) A B × l 2 ) + I ¯ A B θ ¨ ) a n t i c l o c k w i s e + ( ( m ( a ¯ ) C D × l ) + I ¯ C D θ ¨ ) a n t i c l o c k w i s e = − m l 2 ( a ¯ ) A B − I ¯ A B θ ¨ − m l ( a ¯ ) C D − I ¯ C D θ ¨ The linear acceleration of the rod AB ( a ¯ ) A B is equal to the product of distance at which the weight of the rod acts and angular acceleration ( l 2 θ ¨ ) . The linear acceleration of the rod CD ( a ¯ ) C D is equal to the product of distance at which the weight of the rod acts and angular acceleration ( l θ ¨ ) . Thus, express the moment about A due to the effective forces: ( ∑ M A ) e f f e c t i v e = − m l 2 ( l 2 θ ¨ ) − I ¯ A B θ ¨ − m l ( l θ ¨ ) − I ¯ C D θ ¨ = − m l 2 4 θ ¨ − m l 2 θ ¨ − I ¯ A B θ ¨ − I ¯ C D θ ¨ = − ( m l 2 4 + m l 2 + I ¯ A B + I ¯ C D ) θ ¨ = − ( 5 4 m l 2 + I ¯ A B + I ¯ C D ) θ ¨ Equate the moment about A in the system for external and effective forces

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 19.2, Problem 19.61P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The frequency ( f n ) of small oscillation of the assembly.

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