Show that the distance of 2 masses oscillates

Question

Consider spring-mass system given below .

I have calculated that we could model the system by : $$m\frac{d^2x}{dt^2}=-k(x-l) + k(d-x-y-l)$$ $$m\frac{d^2y}{dt^2}=-k(y-l) + k(d-x-y-l)$$ for left mass and right mass where k is the spring constant and $l$ is unstretched length of the spring. Also, all the spring is identical, so they have same spring constant and unstretched length. The question is, how do I show that distance between left mass and right mass oscillates. The hint is I should define a new variable for this distance.

So, let say $z=d-x-y$. I don't know what to do here. How would I show that $z$ oscillates? Thanks in advance.

Answer 1

We have for each mass

$$ m\ddot x = k(d-x-y)-k x\\ m\frac{d^2}{dt^2}(d-y) = -k(d-x-y)+k y $$

if $z = d - x - y$ then subtracting the first from the second

$$ m\frac{d^2}{dt^2}(d-x-y) = -2k(d-x-y)-k(d-x-y)+ kd $$

or

$$ m\ddot z +3kz = kd $$

or

$$ z = C_1 \sin \left(\sqrt{\frac{3k}{m} t}\right)+C_2 \cos \left(\sqrt{\frac{3k}{m} t}\right)+\frac{d}{3} $$