Problem $5$: $(2$ points$)$

In the figure shown, a disk is initially rotating on a frictionless shaft of negligible radius. Another disk, initially not rotating, is then dropped freely down onto it$.$ The two disks then rotate together at a common rotational speed. Which of the following statements is true? $($only one answer is needed$)$

$-$ The total mechanical energy is conserved throughout the process.

$-$ The total angular momentum is conserved throughout the process.

$-$ Both the mechanical energy and angular momentum are conserved throughout the process.

$-$ Neither mechanical energy nor angular momentum is conserved throughout the process.