Solving Pulley Systems: Kinetic Energy and Lagrangian

[tom777]

Hello guys!
Another question appeared during the preparation for my exam .
For some reason I don't feel 100% comfortable when dealing with
systems including pulleys and masses.
In the given exercise I'm supposed to calculate the kinetic energy T of the system
(which ultimately leads to the Lagrangian)

Homework Statement

Scan of the problem:
http://img138.imageshack.us/img138/2064/question2.png

Homework Equations

The problem is solved by using methods of generalized coordinates
and Lagrangian functions.

The Attempt at a Solution

So the thing is...I sort of know where most of the terms come from.
For example the (I_2/ (R_2)^2 ) probably comes from (d/dt y_{1} ) = (d/dt \theta_{2}) * R_1. However I'm lacking a concrete, rigorous approach to the problem.
What I mean by that is:
Say in cartesian coordinates the formula for the kinetic energy is given by:
T = 0.5 * m_1 * y_{mass 1} + 0.5 * m_2 * y_{mass 2}
How do I rigorously solve this problem now? I'm sort of puzzled - especially about
the last term in the equation for T in the scan. It seems as though some kind
of binomial formula might have been applied. But I'm not sure though.

I'd be really really happy if you could help me or give me a hint
on home to tackle these pulley-problems.

Thanks a lot in advance!

 

[hikaru1221]

(d/dt y_{1} ) = (d/dt \theta_{2}) * R_1

Uhm... not right

The most challenging thing in this problem is to find the relations of $$\dot{\theta _1}, \dot{\theta _2} , \dot{y _1} , \dot{y _2}$$. I suggest you sit down and write down the relations without thinking about the final result of T, then write down T in its basic form, i.e. $$T_{linear}=\frac{1}{2}m\dot{x}^2$$ and $$T_{rotate}=\frac{1}{2}I\dot{\phi}^2$$.

Hint: There are 2 equations relating the coordinates. The 1st one contains $$\dot{\theta _1}$$ and $$\dot{y _2}$$ only. The 2nd one contains $$\dot{\theta _2} , \dot{y _1} , \dot{y _2}$$. The radii are included, of course.