Motion of mass connected to a spring at constant velocity

[Holmez2_718]

Homework Statement

We have a mass $m$ at $x = 0$ attached to a spring with spring constant $k$ which is moving at constant velocity $v$ such that the position of the spring is described by $X = l + vt$ where $l$ is the equilibrium length of the spring. Solve for the motion of the mass.

Homework Equations

We have $$F_s = -kd$$ where $d$ is the displacement from equilibrium, and $$F = m\frac{d^2x}{dt^2}$$.

The Attempt at a Solution

$$d = X - x - l = vt - x$$, so $$F = F_s = m\frac{d^2x}{dt^2} = k(d-vt)$$. Trouble is, I don't think the differential equation is separable and don't know how to deal with it.

 

[Simon Bridge]

You want to find x(t) for the mass ... where the X(t) is the bulk motion of the spring?

Which part of the spring? Is it the part of the spring that would normally be attached to a wall or the center of mass of the spring or what?
It looks like you will need to draw a free body diagram for that bit of the spring: in order to move at constant velocity, it must be acted on by a varying force... but that may not be a problem.

I take it the surface is without friction and the initial extension of the spring is not zero?
I see that the spring is moving to the right (X increases with time) ... Is the mass to the right or to the left?

[edit] looking at the description you gave me, it looks like you start with a mass+spring setup at rest, and the end of the spring not attached to the mass is pulled away from the mass at a constant velocity.

 

[Simon Bridge]

Anyway - that's a second order non-homogenious linear equation with... constant coefficients right?
You'll know techniques for dealing with them.

BTW: You don't want a "d" in your equation though, it will vary with time: you want only x, t, and constants in there.