Explanation It is given that the differential equation stating the motion of the circular disk of mass m and radius r when twisted at an angle θ is 1 2 m r 2 d 2 θ d t 2 = − k θ where k is a constant. The initial conditions are θ = θ 0 and θ ′ ( 0 ) = ω 0 . Apply Laplace transform and obtain the result. 1 2 m r 2 L { d 2 θ d t 2 } + k L { θ } = 0 1 2 m r 2 ( s 2 L { θ } − s θ ( 0 ) − θ ′ ( 0 ) ) + k L { θ } = 0 1 2 m r 2 ( s 2 L { θ } − s θ 0 − ω 0 ) + k L { θ } = 0 L { θ } ( 1 2 m r 2 s 2 + k ) = 1 2 m r 2 ( s θ 0 + ω 0 ) On further simplification, L { θ } = 1 2 m r 2 ( s θ 0 + ω 0 ) 1 2 m r 2 s 2 + k = s θ 0 + ω 0 s 2 + 2 k m r 2 = s θ 0 s 2 + ( 2 k m r 2 ) 2 + ω 0 s 2 + ( 2 k m r 2 ) 2 = θ 0 ( s s 2 + ( 2 k m r 2 ) 2 ) + ω 0 2 k m r 2 ( 2 k m r 2 s 2 + ( 2 k m r 2 ) 2 ) Apply inverse Laplace transform and obtain the result

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 31, Problem 98RE

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The equation of motion where θ=θ0 and dθdt=ω0 when the time is zero.

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Chapter 31, Problem 97RE

Chapter 31, Problem 99RE

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Chapter 31 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

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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The equation of motion where θ = θ 0 and d θ d t = ω 0 when the time is zero.

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The equation of motion where θ=θ0 and dθdt=ω0 when the time is zero.

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Chapter 31 Solutions