Past exam of the theoretical physics I course of the University of Cambridge / 2015 / Paper 1 / 1 / b

The derivatives needed in the Euler-Lagrange equation areThe terms proportional to ${\dot{a}}_i\dot{\theta }$ cancel after differentiation, leavingWith $\mathbf{l}=l(\sin \theta ,-\cos \theta ,0)$ and $\mathbf{g}=-g\hat{\mathbf{y}}$,Hence the vector equation is $l^2\ddot{\theta }\hat{\mathbf{z}}=\mathbf{l}\times (\mathbf{g}-\ddot{\mathbf{a}})$. In the translating support frame, $-m\ddot{\mathbf{a}}$ is the inertial force. The rod's tension passes through the hinge and contributes no torque about it. The same torque equation therefore has an immediate mechanical interpretation.