(a) Explanation Given information: The mass ( m ) of the slender rod is 6 kg. The spring constant ( k ) is 1.5 ft ⋅ lb/rad . The rod is rotated through 180 ° about the vertical and released. Calculation: Show the free-body-diagram of the slender rod as in Figure (1). The external force in the system is the torque due to the angular displacement. The effective restoring couple is I ¯ θ ¨ . Take moment about G in the system for external forces. ( ∑ M G ) e x t e r n a l = ( k θ ) Here, the moment about G is M G . Take moment about G in the system for effective forces. ( ∑ M G ) e f f e c t i v e = − I ¯ θ ¨ Equate the moment about O in the system for external and effective forces. ( ∑ M G ) e x t e r n a l = ( ∑ M G ) e f f e c t i v e k θ = − I ¯ θ ¨ θ ¨ = − k I ¯ θ θ ¨ + k I ¯ θ = 0 (1) Compare the differential Equation (1) with the general differential equation of motion ( x ¨ + ω n 2 x = 0 ) and express the natural circular frequency of oscillation ( ω n ) : ω n 2 = k I ¯ ω n = k I ¯ (2) Convert the unit of length of the rod from in. to ft. l = 8 in . = 8 in . × 1 ft 12 in . = 0.6667 ft Calculate the mass ( m ) of the rod: m = W g Substitute 6 lb for W and 32 .2 ft / s 2 for g (b) To determine The maximum velocity ( v m ) of end A of the rod.

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 19.59P

(a)

To determine

The period (τn) of oscillation.

(b)

To determine

The maximum velocity (vm) of end A of the rod.

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Chapter 19.2, Problem 19.58P

Chapter 19.2, Problem 19.60P

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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 period ( τ n ) of oscillation.

Solution Summary: The author shows the free-body-diagram of the slender rod as in Figure (1).