Consider a single$-$degree$-$of$-$freedom model of a disk connected to two springs as shown in

below figure. The friction in the system is such that the disk rolls without slipping.

$($a$)$ Derive the equation of motion using $($i$)$ Newton's the second law, $($ii$)$ Energy method, $($iii$)$

Lagrange's method.

$($b$)$ Use Rayleigh's method to calculate the natural frequency given by ${}_n=2\frac{r+a}{r}\sqrt[\phantom{2}]{\frac{k}{3m}}.$