Does Really Energy is Conserved..??

$Concept$ :

No energy is not Conserved since For a short time There is Current is flowing in Metallic disc when Free electrons are moving with respect to metal . Finally they get stops moving with respect to metal. So current will also becomes Zero since it's duration is very small so there is no significant energy loss.

And it is understood That this rod is Rotating By an external agent so that it can move with constant angular velocity , Otherwise Rod will never be rotated with constant angular velocity.

$Note$ :

Energy is loss when current is flowing is explained by
" $Joules$ $Law$ $of$ $Heating$ "

$Solution$ :

It is very easy by equating Pseudo Force on electron to the Electric Force on the electron After The Equilibrium is set-up with respect to metal disc.

Let

System (electron)

&

Reference frame( metallic disc)

${ F }_{ net }=m{ \omega }^{ 2 }r\quad \\ eE=m{ \omega }^{ 2 }r\\ E=\frac { m{ \omega }^{ 2 }r }{ e } \\ { \E }_{ motional\quad emf }=\quad \int _{ 0 }^{ R }{ E.dr\cos { 0 } } \\ \quad \quad \quad \quad \quad \quad \quad \quad =\quad \int _{ 0 }^{ R }{ \frac { m{ \omega }^{ 2 }r }{ e } .dr } \\ \quad \quad \quad \quad \quad \quad \quad \quad =\frac { { m }_{ e }{ \omega }^{ 2 }{ R }^{ 2 } }{ 2e }$ .

$Rotational\quad kinetic\quad energy\quad will\quad be\quad equal\quad to\quad Vq\quad \\ \frac { I{ w }^{ 2 } }{ 2 } =Vq\\ using\quad the\quad given\quad values,\\ V=8.008$