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the differential equation that describes the motion.

the differential equation that describes the 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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Consider a vertical spring-mass system. A long spring with spring constant k = 8 g/s² has a mass attached that stretches the spring 245 cm. The damping coefficient is d = 8 g/s. At time t = 0, the mass is at the equilibrium position (stretched due to gravity) and has a velocity of 3 cm/s downward. Use a vertical coordinate system where x = 0 is the equilibrium length of the spring stretched due to gravity. Let x be positive downwards (corresponding to stretching the spring). Set up the differential equation that describes the motion. Solve the differential equation. State whether the motion of the spring-mass system is harmonic (with no damping), underdamped, critically damped, or overdamped. If the motion is harmonic or an underdamped oscillation, rewrite in amplitude-phase form.
Transcribed Image Text:Consider a vertical spring-mass system. A long spring with spring constant k = 8 g/s² has a mass attached that stretches the spring 245 cm. The damping coefficient is d = 8 g/s. At time t = 0, the mass is at the equilibrium position (stretched due to gravity) and has a velocity of 3 cm/s downward. Use a vertical coordinate system where x = 0 is the equilibrium length of the spring stretched due to gravity. Let x be positive downwards (corresponding to stretching the spring). Set up the differential equation that describes the motion. Solve the differential equation. State whether the motion of the spring-mass system is harmonic (with no damping), underdamped, critically damped, or overdamped. If the motion is harmonic or an underdamped oscillation, rewrite in amplitude-phase form.
Expert Solution
Step 1: Given the data:

Spring constant k=8g/s2

 and the mass stretched the spring by,  l=245 cm

The damping coefficient δ=8g/s

Also at time t= 0 ,the mass is at the equilibrium position i.e. at t=0; x(0)=0 and has a velocity 3cm /s downward

           d(x=0)/dt=3  ......eq(1)

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