Question

A uniform rope of total length l is at rest on a table with fraction f of its length hanging (see figure). If the coefficient of friction between the table and the chain is $\mu$ then which of the following relation is true:

• A. $f=\mu$
• B. $f=1/(1+\mu )$
• C. $f=1/(1+\frac{1}{\mu })$
• D. $f=1/(\mu +1/\mu )$

Answer 1

The correct option is C $f=1/(1+\frac{1}{\mu })$

Weight of portion hanging from table $\lambda fLg$ where $\lambda$ is mass per unit length

weight of portion table is $\lambda (1-f)Lg$

Frictional force $F_{fr}=\lambda (1-f)Lg\mu$

equating weight of hanging portion with frictional force , $\mu =\frac{f}{1-f}$

$\Rightarrow f=\frac{\mu }{1+\mu }$