Question $2$: Energy in an Expanding Spring System

A block of mass $m$ is attached to a spring of natural length $L_0$ and spring constant $k.$ Initially, the spring is stretched to a length $L_0,$ and the block is at rest. At time $t=0,$ the spring begins to expand at a constant rate such that its length becomes:

$L(t)=L_0+vt$

where $v$ is the rate of expansion. Assume the block moves without friction on a horizontal surface.

$($a$)$ Write down the expression for the total energy of the system $($kinetic $+$ spring potential energy$).$ Is the total energy conserved as the spring expands? Why or why not?

$($b$)$ If the expansion is very slow $($small $v),$ the block adjusts smoothly to the changing spring length. Find how the block's kinetic energy and the spring's potential energy depend on $L(t).$

$($c$)$ If the spring expands quickly $($large $v),$ the block may oscillate more violently. Explain why this happens and how it affects the block's total energy.

$($d$)$ If the expansion becomes extremely slow $($as $v0),$ show that the block's motion remains steady and the total energy evolves smoothly with the spring's length. How does this compare to the case of rapid expansion?