Develop Lagrangian for Frictionless Pulley System w/ Moment of Inertia

[big man]

Q. A block of mass m1 is connected to a second block of mass m2 by a light string that passes over a frictionless pulley. Mass m2 is connected to a light spring of spring constant s, as displayed below. Develop an expression for the Lagrangian of the system, assuming that the moment of inertia of the pulley $$I=\fraction{1/2}Mr^2$$ (ie it has mass M and radius r), and hence determine the natural frequency of vibration of the system.

http://img229.imageshack.us/img229/6502/lagrange2rs.png I can do the question without having to consider the moment of inertia. What I did is use the coordinate z, where z is the position of m2 relative to the pulley.
Now that works out just fine when neglecting the moment of inertia. However, I don't see how you can consider a moment of inertia for this problem?? I just don't see how the pulley is rotating about an axis if it is frictionless? Can someone tell me want I'm missing here please.

Thanks for any help

 

[Galileo]

The pulley has no friction with the axis around which it rotates, but there IS friction between the pulley and the rope, otherwise it wouldn't turn at all.

So there is rotational kinetic energy of the pulley in the system which you have to include. If the spring pushes up, it must not only push m2 and m1 up, it must also change the rotation of the pulley accordingly.

 

[big man]

ahhh ok I was thinking that it meant there was no friction between the rope and the pulley.
Ok now i can do the problem. Thanks for clearing that up man.