A small block of mass `m` is placed on a plank of masss `M` .The block is connected to plank with the help of a light string passing over a light smooth pulley shown in fig. The co-efficient of static friction between the plank is `mu ` The co-efficient of friction between the plank and the maximum horizontal force `F` applied on the block of mass `m` can make block not to slide relatively?

Explanation

The force diagram and the FBD are shown in fig respectively taking the ground as reference frame

`N_(2) = mg` ....(i)
`F - T - f = 0` ...(ii)
From figure,
`N_(1) = N_(2) + Mg`....(iii)
`T - f = 0`...(iv)
Again `f_(max) = mu N_(2)`
i.e. `f_(max) = mu mg`...(v)
For `m` not to slide relatively on `M` .
`f le f_(max)`....(vi)
Using Eq. (i) ti (v) and the condition (vi) `f_(max)` can be found out as `f_(max) = 2 mu mg`