Classical Mechanics Problem

1. The problem statement, all variables and given/known data

Give the equations of motion of the following system:

http://www.jelp.org/imagenes/mech.jpg


2. Relevant equations

So, i assume the following cases (the diagram i so deficient).

1) Black Point (x1) is fixed
2) There's a force applied at x1 (black dot)
3) The position of the black dot is a function of time f(t).


3. The attempt at a solution

For 1) I think the equation of motion is:

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bx%7D%2Bkx%2BG%5Cdot%7Bx%7D%3D0.gif [Broken]

It's a dampeda harmonic oscillator, Right?

2) If a force f(t) is applied to the system, then we have :

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bx%7D%2Bkx%2BG%5Cdot%7Bx%7D%3DF%28t%29.gif [Broken]

like the Damped and driven harmonic oscillator, am i right?

3) If by example, the full system (the damper too) is moving to the left (i.e. negative X).

For this, i tried the following:

The position of the mass is X=f(t)+L+lo , where L is the natural lenght of the spring, lo is the elongation or strecht of the spring. Then, taking the time derivatives of X, and substituing in the damped harmonic oscillator equation (Case 1), i get to:

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bf%7D%2Bk%28f%28t%29%2BL%2Bl_0%29%2BG%5Cdot%7Bf%7D%3D0.gif [Broken]

Please, tell me if i am right in the whole problem. Thanks