Explanation Given information: The weight ( W d i s k ) of the disk is 8 lb. The weight ( W A B ) of the rod AB is 20 lb. The radius ( r ) of the semicircular plate is 6 in . Assuming the acceleration due to gravity ( g ) as 32.2 ft/s 2 . Calculation: Show the position 1 and position 2 of the system with velocity as in the Figure (1). Calculate the mass ( m d i s k ) of the disk: m d i s k = W d i s k g Substitute 8 lb for W d i s k and 32.2 ft/s 2 for g . m d i s k = 8 32.2 = 0.24845 lb ⋅ s 2 / ft Calculate the mass ( m A B ) of the disk: m A B = W A B g Substitute 20 lb for W d i s k and 32.2 ft/s 2 for g . m A B = 20 32.2 = 0.6211 lb ⋅ s 2 / ft Calculate the mass moment of inertia ( I ¯ d i s k ) of both disk: I ¯ d i s k = 1 2 m d i s k r Substitute 0.248 lb ⋅ s 2 / ft for m d i s k and 6 in . for r . I ¯ d i s k = 1 2 ( 0.24845 ) ( 6 in . × 1 ft 12 in . ) 2 = 0.03106 lb ⋅ s ⋅ ft Write the expression for maximum velocity ( v m ) using the relation: v m = r θ ˙ m Substitute 6 in . for r . v m = ( 6 in . × 1 ft 12 in . ) θ ˙ m = 0.5 θ ˙ m Write the expression for velocity of rod AB ( v A B ) by referring Figure (1): v A B = ( 2 in . × 1 ft 12 in . ) θ ˙ m = 1 6 θ ˙ m Substitute 6 in . for r . v m = ( 6 in . × 1 ft 12 in . ) θ ˙ m = 0.5 θ ˙ m For position 1: (Maximum displacement): Write the expression for the kinetic energy ( T 1 ) : T 1 = 0 Write the expression for the potential energy ( V 1 ) : V 1 = − W A B ( ( 4 in . × 1 ft 12 in . ) cos θ m ) Substitute 20 lb for W A B . V 1 = − 20 ( ( 4 in . × 1 ft 12 in . ) cos θ m ) = − 80 12 cos θ m For position 2: (Maximum speed) Write the expression for the kinetic energy ( T 2 ) : T 2 = 2 ( 1 2 m d i s k v m 2 + 1 2 I ¯ d i s k θ ˙ m 2 ) + 1 2 m A B v A B 2 Substitute 0

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 19.3, Problem 19.91P

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The frequency (fn) of small oscillations of the system.

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Chapter 19.3, Problem 19.90P

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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 oscillations of the system.

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The frequency (fn) of small oscillations of the system.

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Chapter 19 Solutions