Question

A heavy uniform chain lies on a horizontal tabletop. If the coefficient of friction between the chain and the table surface is $0.25$ , then the maximum fraction of the length of the chain that can hang over one edge of the table is

• A. $20\%$
• B. $25\%$
• C. $35\%$
• D. $15\%$

Answer 1

Answer: (A)

The force of friction on the chain lying on the table should be equal to the weight of the hanging chain. Let
$\rho =$ mass per unit length of the chain
$\mu =$ coefficient of friction
$l =$ length of the total chain
$x =$ length of hanging chain
$Now, \, \mu \left(\right.l-x\left.\right) \, \rho g=x\rho g \, \, or \, \, \mu \left(\right.l-x\left.\right)=x \\ \Rightarrow \, \, \mu l=\left(\right.\mu +1\left.\right)x \, \, or \, \, x=\frac{\mu l}{\left(\right. \mu + 1 \left.\right)} \\ \therefore \, \, x=\frac{0.25 l}{\left(\right. 0.25 + 1 \left.\right)}=\frac{0.25 l}{1.25}=0.2l \\ \therefore \, \, \frac{x}{l}=0.2=20\%$