A student sits on a freely rotating stool holding two dumbbells, each of mass $3\mathrm{.}09$ kg $($see figure below$).$ When his arms are extended horizontally $($Figure a$),$ the dumbbells are $1\mathrm{.}08$ m from the axis of rotation and the student rotates with an angular speed of $0\mathrm{.}749$ rad$/$s$.$ The moment of inertia of the student plus stool is $2\mathrm{.}78$ kg $$ m$2$ and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position $0\mathrm{.}297$ m from the rotation axis $($Figure b$).$

Two figures depict a man holding a dumbbell in each hand while spinning on a rotating stool.

In figure $($a$),$ the man's arms are fully extended so the dumbbells are far from his body. He is rotating with angular speed $$i$.$

In figure $($b$),$ the man's arms are pulled in so the dumbbells are close to his body. He is rotating with angular speed $$f $>$i$.$

$($a$)$ Find the new angular speed of the student.

rad$/$s

$($b$)$ Find the kinetic energy of the rotating system before and after he pulls the dumbbells inward.

Kbefore$=$ J Kafter$=$ J