(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.

Loose Leaf for Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977206

Author: BEER, Ferdinand P., Johnston Jr., E. Russell, Mazurek, David, Cornwell, Phillip J., SELF, Brian

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…

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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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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