Explanation Formula used: Undamped free vibrations: If the actual damping is very small or there is no damping, then corresponding differential equation is given by m u ″ + k u = 0 and the general solution is u = A cos ω 0 t + B sin ω 0 t where ω 0 2 = k m . Calculation: Given that a mass hanging with an undamped vibration from a vertical spring. Let L be the elongation of the spring and g is acceleration due to gravity. Since there is no damping, the motion of mass u should satisfy the differential equation m u ″ + k u = 0 . The solution is given by u = A cos ω 0 t + B sin ω 0 t where ω 0 2 = k m . In order to solve the differential equation m u ″ + k u = 0 , compute the values of m and k as follows. The value of m can be computed from the equation ω = m g as follows. ω = m g m = ω g The value of k can be computed from the equation ω − k L = 0 as follows. ω − k L = 0 k = ω L Substitute the obtained values of m and k in the differential equation m u ″ + k u = 0 and get the new equation ω g u ″ + ω L u = 0

Elementary Differential Equations and Boundary Value Problems, 11e WileyPLUS Registration Card + Loose-leaf Print Companion

11th Edition

ISBN: 9781119336617

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley (WileyPLUS Products)

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Chapter 3.7, Problem 9P

To determine

To prove: The period of motion of an undamped vibration of a mass hanging from a vertical spring is T=2πLg.

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Chapter 3.7, Problem 8P

Chapter 3.7, Problem 10P

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Chapter 3 Solutions

Elementary Differential Equations and Boundary Value Problems, 11e WileyPLUS Registration Card + Loose-leaf Print Companion

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To prove: The period of motion of an undamped vibration of a mass hanging from a vertical spring is T = 2 π L g .

To prove: The period of motion of an undamped vibration of a mass hanging from a vertical spring is T=2πLg.

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Chapter 3 Solutions