$(20$ points$)$ As shown in the figure, the motor drives the taut conveyor belt at a speed of $v_0=2\frac{m}{s}.$ The angle

between the conveyor belt and the horizontal plane is $=30.$ Now a workpiece with a mass of $m=10kg$ is

gently placed on the bottom of the belt. After a period of time, the workpiece is sent to a platform at a height of

$h=2m.$ The dynamic friction coefficient between the workpiece and the belt is known to be $=\frac{\sqrt[\phantom{2}]{3}}{2}.$ Apart from

this, no other dissipation is considered. $($Note that the dynamic friction is $f_k=F_n,$ where $F_n$ is the normal force

exerted by the workpiece on the conveyor belt. The static friction is equal to or less than the dynamic friction.$)$

$($a$)$ Calculate the speed of the workpiece when it reaches the platform $(8$ points$).$

$($b$)$ Calculate the change in mechanical energy of the workpiece $(6$ points$).$

$($c$)$ Find the extra electric energy consumed by the motor due to the transmission of the workpiece. $($Hint: In

the presence of friction $f$ bewteen two objects, some mechanical energy is transformed to thermal energy

$W=|fs|,$ where $s$ is the relative displacement of the two objects.$)(6$ points$)$