In the figure, a block of weight $60\ N$ is placed on a rough surface. The coefficient of friction between the block and the surfaces is $0.5$. What minimum can be the weight $W$ such that the block does not slip on the surface?

Since, block does not slip, hence, $\mathrm{T}_{1}<\mathrm{f}_{\max }$

$\Rightarrow \mathrm{T}_{1}<\mu \mathrm{mg}$

Now, for equilibrium-

$\begin{array}{l} \frac{\mathrm{T}_{1}}{\sin 135^{\circ}}=\frac{\mathrm{T}_{2}}{\sin 90^{\circ}}=\frac{\mathrm{W}}{\sin 135^{\circ}} \\ \Rightarrow \mathrm{T}_{1}=\mathrm{W} \\ \Rightarrow \mathrm{W}<\mu \mathrm{mg}=0.5 \times 60 \times 9.8 \end{array}$

$\Rightarrow \mathrm{W}<294 \mathrm{~N}=30 \mathrm{~kg}-\mathrm{wt} \text { for block does not slip. }$