The system is pushed by a force F as shown in Fig. (a). All surfaces are smooth except between B and C. Friction coefficient between B and C is. Minimum value of F to prevent block B from downward slipping is

Horizontal acceleration of the system is

$\mathrm{a}=\frac{\mathrm{F}}{2\mathrm{m}+\mathrm{m}+2\mathrm{m}}=\frac{\mathrm{F}}{2\mathrm{m}}$

Let N be the normal reaction between B and C
Free body diagram of C gives

$\mathrm{N}=2\mathrm{ma}=\frac{2}{5}\mathrm{F}$

Now B will not slide downward if

$\mathrm{\mu N}\ge {\mathrm{m}}_{\mathrm{B}}\mathrm{g}$

$\text{ or }\mathrm{\mu }\left(\frac{2}{5}\mathrm{F}\right)\ge \mathrm{mg}\text{ or }\mathrm{F}\ge \frac{5}{2\mathrm{\mu }}\mathrm{mg}$

$\text{ So }{\mathrm{F}}_{min}=\frac{5}{2\mathrm{\mu }}\mathrm{mg}$