Question

Determine the maximum ratio $h/b$ for which the homogenous block will slide without tipping under the action of force $P$. The coefficient of static friction between the block and the incline is ${\mu }_k$.

Answer 1

Block has to slide without tipping, which implies that block is in translational as well as rotational equilibrium,
From conditions of equilibrium
Along sloping surface:
$F+\mu N=mg\sin \theta ...\left(i\right)$
Perpendicular to the sloping surface:
$N=mg\cos \theta ...\left(ii\right)$
Balancing toque about point $P$:
$F\times h=mg\cos \theta \frac{b}{2}...\left(iii\right)$
when the block is on the verge of tipping, the normal reaction will pass through the edge of the block
From Eqs. (i), (ii), (iii)
$F=mg\sin \theta -\mu mg\cos \theta ....\left(iv\right)$
On Substituting $F$ in Eq. (ii), we obtain
$(mg\sin \theta -\mu mg\cos \theta )h-mg\cos \theta \frac{h}{2}=0$
which on solving for $h/b$ yields $\frac{h}{b}=\frac{1}{\left(2{\mu }_s\tan \theta \right)}$