Question

A uniform chain of length $l$ is placed on the table in such a manner that length $l^{\prime }$ of it is hanging over the edge of the table without the chain sliding. If the coefficient of friction between the chain and the table is $\mu$ then find the maximum length of chain $l^{\prime }$ that can hang without the entire chain slipping.

• A. $0$
• B. $\frac{(\mu +1)l}{\mu }$
• C. $\frac{l}{\mu +1}$
• D. $\frac{\mu l}{1+\mu }$

Answer 1

The correct option is D

$\frac{\mu l}{1+\mu }$

Assume the mass of the chain to be M kg.
The linear density of the chain is given by $\pi =\frac{M}{l}$
The mass of chain hanging is $\pi l^{\prime }$ kg and the mass of chain on the table is $\pi (l-l^{\prime })$ kg.
The only force pulling the chain on the table is the weight of the chain that is hanging.

For maximum hanging length without sliding,
$f_{max}=\pi l^{\prime }g$ and
$f_{max}=\mu N=\mu \pi (l-l^{\prime })g$
$\Rightarrow \pi l^{\prime }g=\mu \pi (l-l^{\prime })g$
$\Rightarrow l^{\prime }=\frac{\mu l}{1+\mu }$