A uniform rope of length L lies on a table. If the coefficient of friction is $\text{μ}$ , the maximum fractional length of the hanging part of the rope from the edge of the table without sliding down is

let 'x' be the length of the rope hanging and '$\lambda$' be the linear mass density.

The frictional force offered by the table due to the weight of the rope on the table must balance the weight of the rope hanging.

$\mathrm{\lambda xg}=\mathrm{\mu }\left(\mathrm{L}-\mathrm{x}\right)\mathrm{\lambda g}\Rightarrow (\mathrm{\mu }+1)\mathrm{x}=\mathrm{\mu L}$

$\Rightarrow \frac{\mathrm{x}}{\mathrm{L}}=\frac{\mathrm{\mu }}{\mathrm{\mu }+1}$