A mass $m$ is attached to the end of a light inextensible string, which is wound around a cylindrical wheel of radius $a$. The moment of inertia of the wheel, about an axis perpendicular to the wheel, is $I$. The axis of the wheel is horizontal, gravity acts downwards with magnitude g.
Let $q$ be the angle rotated by the wheel. Calculate the kinetic energy and potential energies of the system in terms of $q$ and $\dot{q}$. Hence calculate the Lagrangian and derive the equation of motion.
This was our homework for last week but we haven't gotten the answer to it yet and I just don't know how to go about it. Diagram of the situatuion given.
Rotating the wheel through an angle $q$ raises the mass through a distance $qa$, so the GPE is $mgaq$ (up to an additive constant, which is arbitrary in our Lagrangian). We can take $L=T-U$ with$$T=\frac12(I+ma^2)\dot{q}^2,\,U=mgaq.$$The EOM is$$(I+ma^2)\ddot{q}=\frac{d}{dt}\frac{dT}{d\dot{q}}=\frac{d}{dt}\frac{dL}{d\dot{q}}=\frac{dL}{dq}=-\frac{dU}{dq}=-mga.$$