(a) To determine The period ( τ n ) of oscillation if the equilibrium position of CD is vertical as shown. Explanation Given information: The mass ( m ) of the rod CD is 5 kg. The length of the rod CD is 0.7 m. The spring constant ( k ) is 24 N ⋅ m/rad Assume the acceleration due to gravity ( g ) is 9.81 m/s 2 . Calculation: Show the free body diagram and kinetic diagram as in Figure (1). The external forces in the system are forces due mass of the rod and the torque to overcome the angular displacement. The effective force in the system is m a ¯ c and m a ¯ t and the effective restoring couple is I ¯ θ ¨ . Take moment about C in the system for external forces. ( ∑ M C ) e x t e r n a l = ( m g × l 2 sin θ ) c l o c k w i s e + ( K θ ) c l o c k w i s e = m g l 2 sin θ + K θ As the angular displacement is very small, take the value of sin θ as θ . Thus the moment of external forces is m g l 2 θ + K θ . Take moment about C in the system for effective forces. ( ∑ M C ) e f f e c t i v e = ( m a ¯ t × l 2 ) a n t i c l o c k w i s e + ( I ¯ θ ¨ ) a n t i c l o c k w i s e = − m l 2 a ¯ t − I ¯ θ ¨ The tangential component of acceleration for the rod a ¯ t is equal to the product of distance at which the weight of the rod acts and angular acceleration ( l 2 θ ¨ ) . Thus express the moment about C due to the effective forces: ( ∑ M C ) e f f e c t i v e = − m l 2 ( l 2 θ ¨ ) − I ¯ θ ¨ = − ( m l 2 4 + I ¯ ) θ ¨ Equate the moment about C in the system for external and effective forces. ( ∑ M C ) e x t e r n a l = ( ∑ M C ) e f f e c t i v e ( m g l 2 + K ) θ = − ( m l 2 4 + I ¯ ) θ ¨ θ ¨ = − m g l 2 + K m l 2 4 + I ¯ θ θ ¨ + m g l 2 + K (b) To determine The period ( τ n ) of oscillation if the equilibrium position of CD is horizontal as shown.

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 19.65P

(a)

To determine

The period (τn) of oscillation if the equilibrium position of CD is vertical as shown.

(b)

To determine

The period (τn) of oscillation if the equilibrium position of CD is horizontal as shown.

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Chapter 19.2, Problem 19.64P

Chapter 19.2, Problem 19.66P

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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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