A cylinder is sandwiched between two planks. Two constant horizontal forces F and 2 F are applied on the planks as shown. Determine the acceleration of the centre of mass of cylinder

Equations of motion are,

$$\begin{gathered} \mathrm{F}+{\mathrm{f}}_1={\mathrm{Ma}}_1 \\ {\mathrm{f}}_1+{\mathrm{f}}_2={\mathrm{Ma}}_2 \\ 2{\mathrm{F}}^{-}-{\mathrm{f}}_2=2{\mathrm{Ma}}_3 \\ \mathrm{\alpha }=\frac{\left({\mathrm{f}}_1-{\mathrm{f}}_2\right)\mathrm{R}}{\frac{1}{2}{\mathrm{MR}}^2} \\ \mathrm{\alpha }=\frac{2\left({\mathrm{f}}_1-{\mathrm{f}}_2\right)}{\mathrm{MR}} \end{gathered}$$

or

For no slipping condition,

${\mathrm{a}}_2+\mathrm{R\alpha }=-{\mathrm{a}}_1 {\mathrm{a}}_2-\mathrm{R\alpha }={\mathrm{a}}_3$

We have six unknowns, ${\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$ and $\mathrm{\alpha }$. Solving the above six equations, we get

$$\begin{gathered} {\mathrm{a}}_1=\frac{2\mathrm{LF}}{26\mathrm{M}} \\ \text{ and } {\mathrm{a}}_2=\frac{\mathrm{F}}{26\mathrm{M}} \end{gathered}$$