Question

A uniform metal chain is placed on a rough table such that one end of chain hangs down over the edge of the table. When one-third of its length hangs down over the edge, the chain starts sliding. Then, the value of coefficient of static friction is-

• A. $\frac{3}{4}$
• B. $\frac{1}{4}$
• C. $\frac{2}{3}$
• D. $\frac{1}{2}$

Answer 1

The correct option is D $\frac{1}{2}$

The given situation is shown in figure.


Let the total mass of chain is $m$.
As chain is uniform,
$\therefore m_1=\frac{m}{3}\text{and}{m}_2=\frac{2m}{3}$

Here, $m_{1}g$ force is responsible for sliding the chain over table against friction force $f$, which is

$f=\mu m_{2}g$

For just sliding condition or static equilibrium,

$f=m_{1}g$

$\Rightarrow \mu m_{2}g=m_{1}g$

$\Rightarrow \mu =\frac{m_1}{m_2}=\frac{m/3}{2m/3}$

$\therefore \mu =\frac{1}{2}$

Hence, option $\left(d\right)$ is correct.

$$\begin{array}{c}\text{Why this question?}\\ \text{Concept-The gravitational force on the}\\ \text{hanging portion will try to pull the chain}\\ \text{down. Friction on the chain portion on}\\ \text{rough table will try to balance gravitational}\\ \text{pull. In critical case, frictional force and}\\ \text{gravitational force will be of equal magnitude}\\ \text{and chain will be in static equilibrium.}\end{array}$$