Question

A uniform rope of length $\ell$ lies on a table if the coefficient of friction is $\mu$, then the maximum length ${\ell }_1$ of the part of this rope which can over hang from the edge of the table without sliding down is

• A. $\frac{\ell }{\mu }$
• B. $\frac{\ell }{\mu +1}$
• C. $\frac{\mu \ell }{\mu +1}$
• D. $\frac{\mu \ell }{\mu -1}$

Answer 1

The correct option is C $\frac{\mu \ell }{\mu +1}$

Let mass of $chain=M$
Let the length of the hanging portion is x. Then the forces acting on the chain are
w = weight of the hanging part
$=\left(\frac{M}{l}x\right)g$
W = weight of the portion lying on the table
$=\left(\frac{M}{l}g\right)(l-x)$
$w=(f_s{)}_{max}={\mu }_{s}N$
or $\left(\frac{M}{l}g\right)x={\mu }_x\frac{M}{l}(l-x)g$
$\Rightarrow \frac{x}{l}={\mu }_x\left[\frac{l-x}{l}\right]$ or $\frac{x}{l}={\mu }_x[1-\frac{x}{l}]$
$\Rightarrow \frac{x}{l}=m{u}_s-{\mu }_s\frac{x}{l}\Rightarrow (1+{\mu }_s)\frac{x}{l}={\mu }_s$
$x=\frac{{\mu }_{s}l}{1+{\mu }_s}$ or $x=\frac{\mu l}{1+\mu }$