Explanation Given Information: When the spring is connected in series to a block A vibrates with a period ( τ s ) of 5 s. When the spring is connected in parallel to a block A vibrates with a period ( τ p ) of 2 s. Calculation: Write the expression for displacement ( δ 1 ) of the spring at top: δ 1 = P 1 k 1 Here, P 1 is load applied on spring 1. Write the expression for displacement ( δ 2 ) of the spring at bottom: δ 2 = P 1 k 2 Here, P 1 is load applied on spring 1. Write the expression for total displacement ( δ ) both the spring in bottom and top: δ = P 1 k s Here, k s is spring constant of a sing equivalent of spring 1 and 2. Calculate the total displacement of spring at top and bottom using the relation: δ = δ 1 + δ 2 Substitute P 1 k s for δ , P 1 k 1 for δ 1 , and P 1 k 2 for δ 2 . P 1 k s = P 1 k 1 + P 1 k 2 k s = k 1 k 2 k 1 + k 2 Write the expression for pull ( P 1 ) in spring 1: P 1 = k 1 δ Write the expression for pull ( P 2 ) in spring 2: P 2 = k 2 δ Write the expression for pull ( P ): P = k δ Calculate the total pull ( P ) in the system using the relation: P = P 1 + P 2 Substitute k P δ for P , k 1 δ for P 1 , and k 2 δ for P 2 . k P δ = k 1 δ + k 2 δ k P = k 1 + k 2 Write the expression for frequency ( ω s ) of series spring using the relation: ω s = k s m Write the expression for frequency ( ω p ) of parallel spring using the relation: ω p = k p m Write the expression for time period ( τ s ) of series spring using the relation: τ s = 2 π ω s Substitute k s m for ω s . τ s = 2 π k s m (1) Write the expression for time period ( τ p ) of parallel spring using the relation: τ p = 2 π ω p Substitute k p m for ω p

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 19.1, Problem 19.23P

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The ratio (k1/k2) of the two spring constants.

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Chapter 19.1, Problem 19.22P

Chapter 19.1, Problem 19.24P

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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 ratio ( k 1 / k 2 ) of the two spring constants.

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Chapter 19 Solutions