Explanation Formula used: The newton’s second law is as follows: mass × acceleration = restoring force . The Hooks Law is as F = k x . Theorem used: If r 1 = α + j β and r 1 = α − j β are the two complex roots of the auxiliary equation a r 2 + b r + c = 0 , then y = e α x ( c 1 cos β x + c 2 sin β x ) is the general solution of a y ″ + b y ′ + c y = 0 . Calculation: The eight of the spring is 16 lb with initial stretch( x ) is 0.5 ft and the spring is pulled by 0.3 in longer than the equilibrium length and then released. From the Hooks Law, F = k x . Where k is spring constant, x is displacement of spring and F is force applied on the spring. Substitute the value of F and x in F = k x . F = k x 16 = k × 0.5 k = 16 0.5 k = 32 lb/ft Now, to get the mass of object: F = m a 16 = m × 32 m = 16 32 = 0.5 slugs So, the newton’s second law is as follows: mass × acceleration = restoring force m × d 2 x d t 2 = − 32 x m d 2 x d t 2 = − 32 x . Substitute the value of m in m d 2 x d t 2 = − 32 x . 0.5 d 2 x d t 2 = − 32 x D 2 x + 64 x = 0 The auxiliary equation is: m 2 + 64 = 0 m 2 = − 64 m = ± 8 j So, the general solution to the above differential equation is x = e 0 t ( c 1 sin 8 t + c 2 cos 8 t ) = c 1 sin 8 t + c 2 cos 8 t Substitute the initial conditions x = 0

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780135902912

Author: Allyn J. Washington

Publisher: PEARSON

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Chapter 31, Problem 12PT

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Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

Ch. 31.1 - Show that is a solution of . Is it the general...Ch. 31.1 - Prob. 1E Ch. 31.1 - In Exercises 1 and 2, show that the indicated...Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 4E Ch. 31.1 - In Exercises 3–6, determine whether the given...Ch. 31.1 - Prob. 6E Ch. 31.1 - In Exercises 7–10, show that each function y =...Ch. 31.1 - Prob. 8E Ch. 31.1 - In Exercises 7–10, show that each function y =...

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