Question

can it be stated that the result obtained does not depend on the choice of the reference frame.

Answer 1

In part (a) we have calculated work done from the ground frame of reference. Now, let us take plank as the reference frame.
Acceleration of plank $a_0=\frac{f}{M}=\frac{\mu mg}{M}$

Free body diagram of disc with respect to plank is shown in figure. Here, $m{a}_0$ = pseudo force.

$\therefore$ Retardation of disc w.r.t. plank.

$a_r=\frac{f+m{a}_0}{m}=\frac{\mu mg+\frac{\mu m^{2}g}{M}}{m}\mu g+\frac{\mu mg}{M}$

$=\left(\frac{M+m}{M}\right)\mu g$

The disc will stop after travelling a distance $S_r$ relative to plank, where

$S_r=\frac{v_r^2}{2a_r}$ $(0=v_r^2-2{\alpha }_r{S}_r)$

$=\frac{Mgh}{(M+m)\mu g}$

$\therefore$ Work done by friction in this frame of reference

$W_{fr}=-f{S}_r=\left(\mu gh\right)\left[\frac{Mgh}{(M+m)\mu g}\right]$

$=-\frac{Mmgh}{(M+m)}$ which is same as part (a)
Note: Work done by friction in this problem does not depend upon the frame of reference, otherwise in general work depends upon reference frame.