For the given dimensions shown in figure, find critical value of coefficient of friction `mu`

Explanation

Condition of sliding
The block will slide if,
`FgtmuN`
but `N=mg` `(m=` mass of the block)
`thereforeFgtmumg` ..(i)
Condition of toppling
Block will topple about an axis passing through `O` and perpendicular to plane of paper if:
clockwise torque of `Fgt` anticlockwise torque of mg
`thereforef(2a)gt(mg)(a)`
or `Fgt(1)/(2)mg` ..(ii)
From eqs (i) and (ii) we can see that
`mu_(cr)=(1)/(2)`
if given value of `mu` is less than `mu_(cr)` (say it is `(1)/(4)`) then Eq. (i) is
`Fgt(1)/(4)mg`
So, Eq (i) is satisfied before Eq. (ii). Therefore, the block will slide before toppling
if given value of `mu` is greater than `mu_(cr)` (say it is `(3)/(4)`) then Eq. (i) is
`Fgt(3)/(4) mg`
so, Eq (ii) is satisfied before Eq. (i) hence the block will topple before sliding