Question

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

• A. $3/4$
• B. $1/4$
• C. $2/3$
• D. $1/3$
• E. $1/2$

Answer 1

The correct option is E

$1/2$

Step 1. Given data:

Uniform metal chain of length $L$ hangs$\frac{1}{3}$below the table,

Let $\frac{1}{3}$ of length $L$is, $x$ that is $x=\frac{1}{3}L$,

Step 2. Finding the force equation and Solving

Assuming the mass of the chain to be $m$,

Since one-third of the chain is hanging, the mass of the hanging chain is $\frac{m}{3}$,

Two- third of the chain is on the table, the mass of the chain on the table is $\frac{2m}{3}$,

The balancing equation of force on the chain will be,

$$\begin{gathered} \frac{m}{3}g-\mu \frac{2m}{3}g=0 \\ \Rightarrow \frac{m}{3}g=\mu \frac{2m}{3}g \\ \Rightarrow \mu =\frac{\frac{m}{3}g}{\frac{2m}{3}g} \\ \Rightarrow \mu =\frac{1}{2} \end{gathered}$$

On solving, we get $\mu =\frac{1}{2}$

Hence, option E is correct.