(a) Explanation Given Information: The period of vibration ( τ n 1 ) of the barrel is 0.58 sec. The period of vibration ( τ n 2 ) of the barrel when it is filled ( W ) with 55 gallons of crude oil is 1.8 sec. The density ( ρ ) of the oil is 900 kg/m 3 . The diameter ( D ) of the barrel is 572 mm. Calculation: Consider when the barrel is empty: Calculate the natural circular frequency ( ω n 1 ) using the relation: τ n 1 = 2 π ω n 1 Rewrite the above expression: ω n 1 = 2 π τ n 1 Substitute 0.58 s for τ n 1 . ω n 1 = 2 π 0.58 s = 10 .833 rad / s Write the expression for natural circular frequency of vibration ( ω n 1 ) : ω n 1 = k m b Here, k is the equivalent spring constant of the salt water and m b for the mass of the barrel when it is empty. Substitute 10 .833 rad / s for ω n 1 . 10 .833 rad / s = k m b k m b = ( 10 .833 ) 2 (1) When the barrel is filled with crude oil: Calculate the natural circular frequency ( ω n 2 ) using the relation: τ n 2 = 2 π ω n 2 Rewrite the expression for time period of vibration: ω n 2 = 2 π τ n 2 Substitute 1.8 sec for τ n 1 . ω n 2 = 2 π 1.8 s = 3 .4907 rad / s Write the expression for natural circular frequency of vibration ( ω n 2 ) : ω n 2 = k m b + m o i l Here, m oil is mass of the oil. Substitute 3 .4907 rad / s for ω n 2 (b) To determine The density ( ρ s w ) of the salt water.

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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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

Question

Chapter 19.1, Problem 19.26P

(a)

To determine

The mass (mb) of the empty barrel.

(b)

To determine

The density (ρsw) of the salt water.

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Chapter 19.1, Problem 19.25P

Chapter 19.1, Problem 19.27P

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Vector Mechanics for Engineers: Statics and Dynamics

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The mass ( m b ) of the empty barrel.

Solution Summary: The author calculates the natural circular frequency (m_b) of the empty barrel.

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The mass (mb) of the empty barrel.

The density (ρsw) of the salt water.

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Chapter 19 Solutions