A spring-mass system has mass \(2\) kg, frictional constant \(10\text{,}\) and spring constant \(12\) kg/s\(^2\text{.}\)
Model this with a 2nd-order linear homogeneous ODE and solve for the position function.
The spring is stretched to a length of \(1\) m from rest and then released. Describe in words what happens over time, and then solve this IVP analytically. What is the long-run behavior of this system, from your solution?
Now we connect a motor to our system that provides a force \(f(t) = 20 \sin(t)\) to our system. Describe in words what happens over time, and then solve this forcing IVP analytically. What is the long-run behavior of this system from your solution?
2.
A spring-mass system has mass 1 kg, no frictional forces, and a spring constant \(k = 9\) kg/s\(^2\text{.}\)
Model this with a 2nd-order linear homogeneous ODE and solve for the position function.
The spring is stretched to a length of \(1\) m from rest and then released. Describe in words what happens over time, and then solve this IVP analytically. What is the long-run behavior of this system, from your solution?
The laboratory in which we are analyzing this spring-mass system is on a research boat, and the motion of the waves provides a force \(f(t) = \sin(t)\) to our system. Describe in words what happens over time, and then solve this forcing IVP analytically. What is the long-run behavior of this system, from your solution?
The next day the motion of the waves provides a force \(f(t) = \cos(3t)\) to our system. Describe in words what happens over time, and then solve this forcing IVP analytically, if possible. What is the long-run behavior of this system? (We will look more closely at situations like this in the Section 4.4 on resonance.)
