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my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied orce in Newtons is 30 F(t) = - {3⁰ if 0 ≤ t ≤ π/2, if t > π/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t) remain continuous when the applied force is discontinuous. For 0 ≤ t ≤/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1-80 zero. If not, enter a function that approximates y(t) for very large positive values of t. For very large positive values of t, y(t) ≈

my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied orce in Newtons is 30 F(t) = - {3⁰ if 0 ≤ t ≤ π/2, if t > π/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t) remain continuous when the applied force is discontinuous. For 0 ≤ t ≤/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1-80 zero. If not, enter a function that approximates y(t) for very large positive values of t. For very large positive values of t, y(t) ≈

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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Consider the initial value problem my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is F(t) = {30 if 0 ≤ t ≤ π/2, if t > π/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t) remain continuous when the applied force is discontinuous. For 0 ≤ t ≤ π/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1→∞0 zero. If not, enter a function that approximates y(t) for very large positive values of t. For very large positive values of t, y(t) ~
Transcribed Image Text:Consider the initial value problem my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is F(t) = {30 if 0 ≤ t ≤ π/2, if t > π/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t) remain continuous when the applied force is discontinuous. For 0 ≤ t ≤ π/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1→∞0 zero. If not, enter a function that approximates y(t) for very large positive values of t. For very large positive values of t, y(t) ~
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