Here's the problem I have to solve. I can do the math just fine once I get the IVP set up, but getting it set up is what I don't know how to do.
A 2 kg (20 N) mass is attached to a spring, thereby stretching it 0.5 m beyond its undisturbed length. The system is then suspended horizontally in a hydraulic fluid that provides a resistive force of $40\sqrt{5}$ N for every 5 m/s of velocity. Assume no external forces act on this system.
(a) Find a general solution for the equation modeling the horizontal displacement $x$ of the mass from its equilibrium position $t$ seconds after it is set in motion.
(b) Suppose the mass is set in motion from equilibrium with an initial rightward velocity of 2 m/s. Find the particular displacement model given these initial conditions.
My attempt.
(a) Supposedly this system is modeled using the ODE $$mx''+bx'+kx=F_{\text{ext}}(t)$$ where $m$ is inertia (? or maybe mass of the spring? or both?), $b$ is the damping coefficient, $k$ is stiffness of the spring, and $F_{\text{ext}}$ is the external force function. So I would use $m=2$ and $F_{\text{ext}}(t)=0$. I think (but am not sure) that we find $b$ by taking $b=(40\sqrt{5})/5$ and $k$ by taking $k=20/0.5$. Is that correct? Once I have $m,b,k,F_{\text{ext}}$, I can solve it easily.
(b) Clearly $x'(0)=2$---or is it negative 2? But I am confused about the system setup. I take the 0.5 m stretching to be vertical. In its horizontal position, equilibrium is just 0 m, right? So that would mean $x(0)=0$. Is that correct? Again, once I have the IVP I can take it from there.
Thanks guys!
