Question

A chain of mass $10kg$ and length $8m$ is resting on a rough horizontal surface $(\mu =0.2)$. A force $F=15N$ is applied as shown. Find the length (in $m$) of the chain for which no friction force acts.

• A. $2$
• B. $6$
• C. $8$
• D. $0$

Answer 1

The correct option is A $2$

For the entire chain,

Limiting friction $f_{max}=\mu mg=20N$

Since $F=15N<f_{max}$, the frictional force can balance the external force and the chain will not move.

Friction force per unit length of the chain $=\frac{20}{8}=2.5N$

Now, consider a segment of the chain starting from the right end, of length $x$. FBD for the segment will be


$F=T+f$
$T=F-f=15-2.5\times x$

When $F-f=0$, tension will also be zero.
$F-f=0\Rightarrow 15-2.5x=0\Rightarrow x=6m$

Thus, for the portion of the chain for which $x>6m$, tension will be zero. Because tension is zero, there will be no tendency for relative motion and hence friction will also be zero.

i.e friction is zero for $8-6=2m$ of the chain.