A block of mass 15kg is placed over a frictionless horizontal surface. Another block of mass 10kg is placed over it, that is connected with a light string passing over two pulleys fastened to the 15kg block. A force 80N is applied horizontally to the free end of the string. Friction coefficient between two blocks is 0.6. The portion of the string between 10kg block and the upper pulley is horizontal. Pulley, string and connecting rods are massless. (Take $\mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2$ )

The magnitude of acceleration of the 10kg block is 

First, let us check upto what value of F, both blocks move together. Till friction becomes limiting, they will be moving together. Using the FBDs 10kg block will not slip over the 15kg block till acceleration of 15kg block becomes maximum as it is created only by friction force exerted by 10kg  block on it.

$\begin{array}{l}\text{Let say both blocks have same acceleration.}\\ \frac{\text{F}-\text{f}}{10}=\frac{\text{f}}{15}\\ \Rightarrow f=\frac{3F}{5}\\ \text{put\hspace{0.17em}F}=80 \text{N}\\ \text{⇒f=48N}\\ {\text{f}}_{\text{maximum}}\text{ is }60 \text{N}.\\ \text{ Therefore, for F}=80 \text{N , both will move together. }\\ \text{ Their combined acceleration, by applying NLM using both as system, }\\ \text{F}=25\text{a}\\ \Rightarrow \text{a}=\frac{80}{25}=3.2 \text{m}/{\text{s}}^2\end{array}$