Problem
4. [2360071423 (33 pts)] A 2-kg mass is attached to spring with restoring/spring constant of $2 \mathrm{Nt} / \mathrm{m}$. The apparatus is aligned horizontally with a damping constant of $5 \mathrm{~N} / \mathrm{m} / \mathrm{sec}$, and is forced by $f(t)=3 e^{-t}+4 \mathrm{Nt}$. Initially, $x(0)=-4$ and $\dot{x}(0)=-3$. (a) (2 pts) Where is the mass with respect to its equilibrium position when $t=0$ and in what direction is it moving at that time? (b) (3 pts) Is the oscillator over-, under-, or critically damped? Justify your answer. (c) (3 pts) Is the oscillator in resonance? Justify your answer. (d) (15 pts) Find the position of the mass at any time $t$, that is, solve an appropriate initial value problem. (e) (2 pts) From your answer to part (d), identify the transient and steady state solutions. (f) $(8 \mathrm{pts})$ Write the initial value problem from part (d) as a system of differential equations/IVPs, using matrices and vectors in your answer.
Solution
Step 1 ：The equilibrium position of a mass-spring system is the position where the spring force balances the gravitational force, which is usually defined as \(x=0\).
Step 2 ：The initial position of the mass is given as \(x(0)=-4\), which means the mass is 4 units to the left of the equilibrium position at \(t=0\).
Step 3 ：The initial velocity of the mass is given as \(\dot{x}(0)=-3\), which means the mass is moving to the left at \(t=0\) (since negative velocity indicates leftward motion in this context).
Step 4 ：Final Answer: The mass is \(\boxed{4}\) units to the left of its equilibrium position when \(t=0\) and it is moving to the \(\boxed{left}\) at that time.
