Q$.3$ A $10-$kg cube was initially held at rest by a massless string at the top of an incline of height $\backslash ($ h $\backslash ),$ as shown in the figure below.

$($a$)$ If the friction between the cube and the incline cannot be ignored and the static friction coefficient $\backslash (\backslash$mu$\_\{\backslash$mathrm$\{$s$\}\}=0\mathrm{.}1\backslash ),$ what is the tension in the string? $(\backslash (\backslash$theta$=30^\{\backslash$circ$\}\backslash )$; assume that the string is fixed.$)$

$($b$)$ Assuming the kinetic friction coefficient $\backslash (\backslash$mu$\_\{\backslash$mathrm$\{$k$\}\}=0\mathrm{.}05\backslash )$ and $\backslash ($ h$=2\backslash$mathrm$\{$~m$\}\backslash ),$ if the string is cut and the cube starts to slide down the incline, what are the vertical and horizontal components of the acceleration of the mass $\backslash ($ m $\backslash )?$

$($c$)$ How much mechanical energy will have been lost due to kinetic friction by the time the cube reaches the bottom of the incline?

$(5$ marks$)$

$(6$ marks$)$

$(4$ marks$)$

$($d$)$ What is the kinetic energy of the cube once it reaches the bottom of the incline?

$(4$ marks$)$

$($e$)$ If the string is not cut $($as it was in part $($b$))$ but instead the string is connected via a massless rope and five low friction pulleys to a large cube of mass $100$ kg $,$ where three of the pulleys are attached to the large cube and two of the pulleys are attached to the wedge and the end of the rope is fixed to the wedge $($as shown in the figure below$),$ calculate the magnitude of the acceleration of small cube down the incline. Note: Assume all other parameters are the same as in part $($b$).$ Also, assume that coefficients of friction between the large cube and the ground are the same as those between the small cube and the incline.