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6. A 1 kg mass is attached to a spring whose constant is 20 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 4 times the instantaneous velocity. (a) Write the second-order linear differential equation to model the motion. (b) Convert the second-order linear differential equation from part (a) to a first-order linear system. (c) Classify the critical (equilibrium) point (0,0). (d) Sketch the phase portrait. (e) Indicate the initial condition X(0) = ( v(0)) ) = ( ˜¯} 0 on the phase portrait you drew in part (c) and sketch. the trajectory that passes through this point. Use your sketch to describe the behavior of the spring-mass given this initial condition. (f) What type of motion does the solution of the initial value problem describe? (free undamped (simple harmonic). motion, free overdamped motion, free critically damped motion, or free underdamped (oscillatory) motion)

6. A 1 kg mass is attached to a spring whose constant is 20 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 4 times the instantaneous velocity. (a) Write the second-order linear differential equation to model the motion. (b) Convert the second-order linear differential equation from part (a) to a first-order linear system. (c) Classify the critical (equilibrium) point (0,0). (d) Sketch the phase portrait. (e) Indicate the initial condition X(0) = ( v(0)) ) = ( ˜¯} 0 on the phase portrait you drew in part (c) and sketch. the trajectory that passes through this point. Use your sketch to describe the behavior of the spring-mass given this initial condition. (f) What type of motion does the solution of the initial value problem describe? (free undamped (simple harmonic). motion, free overdamped motion, free critically damped motion, or free underdamped (oscillatory) motion)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. A 1 kg mass is attached to a spring whose constant is 20 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 4 times the instantaneous velocity. (a) Write the second-order linear differential equation to model the motion. (b) Convert the second-order linear differential equation from part (a) to a first-order linear system. (c) Classify the critical (equilibrium) point (0,0). (d) Sketch the phase portrait. (e) Indicate the initial condition X(0) = ( v(0)) ) = ( ˜¯} 0 on the phase portrait you drew in part (c) and sketch. the trajectory that passes through this point. Use your sketch to describe the behavior of the spring-mass given this initial condition. (f) What type of motion does the solution of the initial value problem describe? (free undamped (simple harmonic). motion, free overdamped motion, free critically damped motion, or free underdamped (oscillatory) motion)
Transcribed Image Text:6. A 1 kg mass is attached to a spring whose constant is 20 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 4 times the instantaneous velocity. (a) Write the second-order linear differential equation to model the motion. (b) Convert the second-order linear differential equation from part (a) to a first-order linear system. (c) Classify the critical (equilibrium) point (0,0). (d) Sketch the phase portrait. (e) Indicate the initial condition X(0) = ( v(0)) ) = ( ˜¯} 0 on the phase portrait you drew in part (c) and sketch. the trajectory that passes through this point. Use your sketch to describe the behavior of the spring-mass given this initial condition. (f) What type of motion does the solution of the initial value problem describe? (free undamped (simple harmonic). motion, free overdamped motion, free critically damped motion, or free underdamped (oscillatory) motion)
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