Figure $2$ presents the potential energy profiles for two distinct systems. In both systems, a particle with mass $m$ undergoes oscillatory motion between positions $x=0$ and $x=L.($a$)$ Derive the period of the particle's motion for the potential energy function depicted in Fig. $2($a$).($b$)$ Derive the period of the particle's motion for the potential energy function depicted in Fig. $2($b$).($c$)$ In the limit where $U{E}_{\text{m}\text{e}\text{c}},$ determine the difference in the periods derived in parts $($a$)$ and $($b$).(20$ points$)$