A pendulum swings back and forth about a pin joint on a fixed surface through an angle $.$ The pendulum can be considered as a lumped mass, $mkg,$ at the end of a rigid, massless link of length $Lm.$

$($a$)$ Derive the equation of motion of this system using the Euler$-$Lagrange equation.

$($b$)$ In this case, the pendulum has a lumped mass at the end of its length. What differences would be observed if the system was considered as a compound pendulum $($with its mass distributed evenly along its length$)?$

Ans: ${}^{}+\frac{g}{L}sin()=0$

Figure $2$: The simple pendulum system described in Q$2.$