Past exam of the physics 1 paper of the University of Oxford / 2015 / 10 / a / Solution

Let the pivot's prescribed horizontal position be $x(t)$ and measure $\theta$ from the downward vertical. The bob's coordinates are $X=x+l\sin \theta$ and $Y=-l\cos \theta$. ThereforeWith zero gravitational potential energy at pivot height, the bob's Lagrangian isHere $x(t)$ is a prescribed input, not yet a free coordinate. Terms depending only on that prescribed time function can be dropped without changing the $\theta$ equation. The moving block's own kinetic energy becomes relevant when its motion is free in part (d).