Question

Assume that there's no damping, an object stretches a spring 4 inches in equilibrium. Find its displacement for t>0 if it's initially displaced 36 inches above equilibrium and given a downward velocity of 2 ft/s.

Answer 1

\[y''+\frac{ k }{ m }y=0\]

Answer 2

That above is the equation of motion but how do I find k and m?

Answer 3

@Kainui @perl @jim_thompson5910

Answer 4

the spring stiffness and object inertia simply determine the frequency of the response:
$$y''+\omega^2y=0;\quad\omega^2=\frac{k}m$$which has the general solution $$y(t)=A\cos(\omega t+\phi)$$

Answer 5

we're told \(y(0)=3, y'(0)=-2\) since 36 in is 3 ft which gives $$A\cos(\phi)=3\\-A\omega \sin(\phi)=-2$$ so $$A^2=(A\cos(\phi))^2+(-A\sin(\phi))^2=9+4/\omega^2\\\tan\phi=\frac{A\sin\phi}{A\cos\phi}=\frac2{3\omega}$$