The minimum acceleration with which a fireman can slide down a rope, if the breaking strength of the rope is $\frac{2}{3}$ of  his weight, is

If the fireman slides down with some acceleration, his apparent weight decreases. For critical condition, rope can bear only  $\frac{2}{3}$ of his weight. If a is the minimum acceleration, then,

Tension in the rope Breaking stress $=m(g-a)$

$m(g-a)=\frac{2}{3}mg\Rightarrow a=g-\frac{2g}{3}=\frac{g}{3}$