PROBLEM STATEMENT:
A mass weighing $16$ pounds stretches a spring $\frac{8}{3}$ feet. The mass if initially released from rest from a point $4$ feet below the equilibrium position and the subsequent motion takes place in a medium that offers a damping force is $\frac{1}{2}$ times $x^{'}$ where $x=x(t)$. The mass is driven by external force given by $f(t)=20cos(3t)$. Convert mass = $\frac{weight}{32}$, Hook's constant Weight = k (length of stretch). Also the position below equilibrium is positive, and the downward motion is positive. Find the equation of motion.
My Working
Since the position below equilibrium is positive and the downward motion is positive we have:
$$x(0)=4$$ $$x'(0)=0$$
Now I have figured out the above initial conditions but can't seem to form the resulting differential equation of this problem. Once I do it, I can solve for the $x(t).$ Can anyone help in it. I will really appreciate that.
Best Answer
Hint.
After the $k$ determination from static considerations, we have
$$ m \ddot x = -\mu \dot x - k x + f(t),\ \ \ x(0)=4,\ \dot x(0) = 0 $$
here $\mu = \frac 12$