$2.$ A mass slide down a smeoth $30$ degree indlind plane ag shown. $(20)$

$($a$)$ Derive the eguation of mation $($tise fhe static equilibrium position as the generalized cundinate.$)($S$)$

$($b$)$ Define the initial conditions: $(5)$

$($c$)$ Derive the free vibration response: $(5)$

$($d$)$ Determine the maximum acceleratom of the mass after spring attaclment. $(5)$