Explanation Given information: The radius ( r ) of the disk is r . The length of the bar form AB is L 2 . The length of the bar form BC is L 2 . The mass of the rod and the disk is m Calculation: Show the displacement of the system as in Figure (1). Write the expression for the displacement ( x ) of spring: x = L θ 2 Write the expression for the displacement ( h ) of spring: h = L 2 ( 1 − cos θ ) ( 1 − cos θ = 2 sin 2 θ 2 ) = L 2 sin 2 θ 2 For small oscillation sin θ ≈ θ . h = L 2 θ 2 2 = L θ 2 4 Write the expression the velocity ( v s ) of spring: v s = L θ ˙ 2 Write the expression the velocity ( v r ) of rod: v r = L θ ˙ 2 r Write the expression for kinetic energy ( T ): T = 1 2 m v s 2 + 1 2 m r 2 2 ( v r 2 ) Substitute L θ ˙ 2 for v s and L θ ˙ 2 r for v r . T = 1 2 m ( L θ ˙ 2 ) 2 + 1 2 m r 2 2 ( L θ ˙ 2 r ) 2 = 3 m L 2 θ ˙ 2 16 Write the expression for potential energy ( V ): V = 1 2 k x 2 + m g h Substitute L θ 2 for x and L θ 2 4 for h . V = 1 2 k ( L θ 2 ) 2 + m g L θ 2 4 = k L 2 θ 2 8 + m g L θ 2 4 Write the expression for the total energy of the system: T + V = 0 Substitute 3 m L 2 θ ˙ 2 16 for T and k L 2 θ 2 8 + m g L θ 2 4 for V

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

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

Publisher: McGraw-Hill Education

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Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

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Chapter 19.3, Problem 19.77P

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The frequency (fn) of the resulting oscillation in terms of m, L, k, and g.

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Chapter 19.3, Problem 19.76P

Chapter 19.3, Problem 19.78P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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The frequency ( f n ) of the resulting oscillation in terms of m , L , k , and g .

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The frequency (fn) of the resulting oscillation in terms of m, L, k, and g.

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Chapter 19 Solutions