Motion of a cylinder attatched to a spring

[polarcheese]

Homework Statement

Determine the equation of motion and the natural frequency of vibration for A uniform cylinder of mass M, rolling without slipping on a horizontal plane and constrained by a spring of stiffness K, as shown.

http://img638.imageshack.us/img638/613/cylinder.png

2. The attempt at a solution

So I am now in full on revision mode, and I have the worked solutions to the problem:

http://img571.imageshack.us/img571/9987/cylinder1.png

Although I can follow most of the solution, I don't understand why the spring extends by 2x if the centre moves by x?

Thanks very much

 

[gneill]

However much x the center of the cylinder shifts, since it rolls without slipping it must turn through an angle so that it rolls along an equal length x of its circumference. Where the spring attaches experiences both the linear shift and the corresponding amount of arc length. If you draw a diagram showing the profile of the cylinder in both positions, you'll see that the result is a total length of 2x.

 

[polarcheese]

ahhhhhhhhhhhhhhhhhh, amazing amazing.

Thank you so much, was getting myself really worked up about that.

 

[alice.w]

I'm really sorry to bring this up, but I have found out where I have been going wrong in this problem - and it is the 2x bit, but I still don't understand why the spring stretches by 2x?
So the spring moves horizontally x, so therefore the spring stretches by x, but since the cylinder is rolling... ah I think I understand having posted this, sorry!