Consider a cylinder of mass M and radius R lying on a rough horizontal plane. It has a plank lying on its top as shown in figure. A force F is applied on the plank moves and causes the cylinder to roll. The plank always remains horizontal. There is no slipping at any point of contact then.

Since there is no slipping at any contact, net work done by friction is zero In time t, Work done by Applied force=F(cos$\mathrm{\theta }$ ) S_plank = KE of plank and cylinder
$\begin{array}{l}(\mathrm{Fcos}\mathrm{\theta })\left[\frac{1}{2}\left(2\mathrm{a}\right)\right]{\mathrm{t}}^2=\frac{1}{2}{\mathrm{mVp}}^2+\frac{3}{4}\mathrm{M}[\mathrm{at}{]}^2\\ \Rightarrow \mathrm{a}=\frac{4\mathrm{Fcos}\mathrm{\theta }}{3\mathrm{M}+8\mathrm{m}}\end{array}$
For plank,                                                
$\mathrm{Fcos}\mathrm{\theta }-{\mathrm{f}}_1=\mathrm{m}\left[2\mathrm{a}\right]$                            
$\Rightarrow {\mathrm{f}}_1=\frac{3\mathrm{MFcos}\mathrm{\theta }}{3\mathrm{M}+8\mathrm{m}}$

For cylinder,

${\mathrm{f}}_1-{\mathrm{f}}_2=\mathrm{Ma}$

$\Rightarrow {\mathrm{f}}_2=-\frac{\mathrm{MFcos}\mathrm{\theta }}{3\mathrm{M}+8\mathrm{m}}$

Negative sign means direction of f₂ is towards right side.