Problem $4($HW $+2$ p$.)$

Ships can be loaded by overhead cranes. A simplified model of such crane is shown in the figure. The crane has a hoist mechanism inside a car $($mass $M)$ that is moving on rails. The car is supported in lateral direction by supports having stiffness $k($modelled as linear springs, which are at rest when $x=0).$ The lifted load $($mass $m)$ hangs in the hoist cable, modelled here as massless rigid rod $($length $l).$ Derive the Lagrangian $L$ and equations of motion for the system, using generalized coordinates $x$ and $.$

Solution:

$(M+m)x^{}+ml(({}^{})cos-{}^{}{?}^{2}sin)+2kx=0$

$ml(x^{}cos+l{}^{}+$gsin$)=0$