(a) Explanation Theorem used: Hooke’s law: “The strain in a solid is proportional to the applied stress within the elastic limit of that solid.” Formula used: If m 1 = − λ + λ 2 − ω 2 and m 2 = − λ − λ 2 − ω 2 are the roots of the auxiliary equation m 2 + 2 λ m + ω 2 = 0 , then there are three different cases for which the solution exists. If λ 2 − ω 2 > 0 ,then x ( t ) = c 1 e m 1 t + c 2 e m 2 t .(Overdamped) If λ 2 − ω 2 = 0 ,then x ( t ) = e − λ t ( c 1 + c 2 t ) .(Critically damped) If λ 2 − ω 2 < 0 , m 1 = − λ + ω 2 − λ 2 i and m 1 = − λ − ω 2 − λ 2 i ,then x ( t ) = e − λ t ( c 1 cos ω 2 − λ 2 t + c 2 sin ω 2 − λ 2 t ) .(underdamped) Calculation: The elongation of the spring after the mass is attached is 7 − 5 = 2 ft . Convert the units of weight given in pounds into units of mass, since here the engineering system of units is used. The weight 8 pounds is converted to mass as follows. Here, m is the unit of mass, W is the weight in pounds and g is the acceleration due to gravity, that is g = 32 ft/s 2 . m = W / g = 8 / 32 = 1 / 4 slug Now, consider the initial mass 10 pounds . By Hooke’s law, 10 = k ( 2 ) k = 5 lb/ft The differential equation of a free damping motion is given as, d 2 x d t 2 + 2 λ d x d t + ω 2 x = 0 where, 2 λ = β m and ω 2 = k m ( or ) m d 2 x d t 2 + β d x d t + k x = 0 Here, β is the positive damping constant and the negative sign indicates the damping force acts in a direction opposite to the motion. Since the mass is released 1 2 foot below the equilibrium position with the downward velocity of 1 ft/s and the damping force is equal to the instantaneous velocity, the value of β = 1 and x ( 0 ) = 1 2 , x ′ ( 0 ) = 1 (b) To determine The equation of the motion given in ( 23 ) . (c) To determine The number of times at which the mass passes through the equilibrium position heading downward. (d) To determine To sketch: The graph of the equation of the motion.

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Question

Chapter 3.8, Problem 30E

(a)

To determine

The equation of the motion.

(b)

To determine

The equation of the motion given in (23).

(c)

To determine

The number of times at which the mass passes through the equilibrium position heading downward.

(d)

To determine

To sketch: The graph of the equation of the motion.

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Chapter 3.8, Problem 29E

Chapter 3.8, Problem 31E

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Chapter 3 Solutions

Advanced Engineering Mathematics

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The equation of the motion.

Solution Summary: The author explains Hooke's law: "The strain in a solid is proportional to the applied stress within the elastic limit of that solid."

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The equation of the motion.

The equation of the motion given in (23).

The number of times at which the mass passes through the equilibrium position heading downward.

To sketch: The graph of the equation of the motion.

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Chapter 3 Solutions