Question

A chain is placed on a rough table, partially hanging, as shown in the figure. The coefficient of static friction between the chain and the table is $\mu =0.4.$ If the maximum possible length of the hanging part is $x$ and the length of the chain is $L$ , then what is the value of $\frac{2 L}{x}$ ?

Answer 1

If $x=$ length suspended
$m_{1}=\lambda x$ (mass of hanging part)
$m_{2}=\lambda \left(\right.L-x\left.\right)$ (mass of the part kept on the table)
Then for balance,
$m_{1}g=\mu _{s}m_{2}g$
$\lambda x=0.4\lambda \left(\right.L-x\left.\right)$
$x=0.4L-0.4x$
$1.4x=0.4L$
$\left(\frac{L}{x}\right)=\frac{1 .4}{0 .4}=3.5\Rightarrow \frac{2 L}{x}=7$