Explanation Procedure used: Consider the distance of the mass beyond the equilibrium position x ( t ) . Write the expression for the Newton’s second law of motion. F n e t = m a Write the expression for the net force acting on the mass. F n e t = F 1 + F 2 + F 3 Here, weight of the mass is F 1 , restoring force is F 2 and damping force is F 3 . Considering damping force on the mass proportional to the instantaneous velocity of the mass. Calculation: Damping force of the mass is, F 3 = − β d x d t Restoring force is, F 2 = − K ( s + x ) Here, negative sign indicates force is acting opposite to weight of the mass. Stiffness of the spring is K and initial displacement is s . Weight of the mass is, F 1 = m g Substitute − K ( s + x ) for F 2 , − β d x d t for F 3 and m g for F 1 in the expression for the net force acting on the mass. F n e t = m g + ( − K ( s + x ) ) + ( − β d x d t ) = m g − K x − K s − β d x d t However, at the condition of equilibrium, m g = K s

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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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 1.3, Problem 20E

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The differential equation for the displacement of the system.

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Chapter 1.3, Problem 19E

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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.

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Ch. 1.1 - Prob. 1E Ch. 1.1 - Prob. 2E Ch. 1.1 - Prob. 3E Ch. 1.1 - Prob. 4E Ch. 1.1 - Prob. 5E Ch. 1.1 - Prob. 6E Ch. 1.1 - Prob. 7E Ch. 1.1 - Prob. 8E Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E

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The differential equation for the displacement of the system.

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