Let both blocks move together with acceleration a. making \(FBD\) of \(m-2\) block for vertical balance
\(M=M_2\,g\)
\(\Rightarrow fr =\mu M _2\,g\)
\(\Rightarrow a =\frac{ fr }{ M _2}=\mu g \ldots \ldots .(1)\)
making \(FBD\) of second block
\(F-f r=M_1 a\)
\(F-\mu m_2 g=M_1(\mu g )\)
\(\Rightarrow F=\mu\left[M_1+M_2\right]\,g\)