PSQ$4.$ Block $()/(($ A$(2\mathrm{.}75()/($mathrm$\{$~kg$\}))()/()))$ is connected to a light spring with a force constant of $()/((44\mathrm{.}0()/($mathrm$\{$~N$\})()/()()/($mathrm$\{$m$\})()/()))$ and oscillates freely on a frictionless, horizontal surface. Block B $(1\mathrm{.}54$ kg $)$ is dropped vertically on top of block $()/(($ A $()/()))$ while it passes through its equilibrium point. Both sticks together afterward and oscillate with an amplitude of $24\mathrm{.}7$ cm $.$

$($a$)$ Find the system's angular frequency before and after the collision.

$($b$)$ Determine the speed of the blocks immediately after the collision. Determine the speed of block A immediately before the collision.

$($c$)$ Determine the original oscillating amplitude of block $()/(($ A $()/()).)$

$($d$)$ Find the total mechanical energy change of the system during this collision.