A partly hanging uniform chain of length L is resting on a...

A partly hanging uniform chain of length $L$ is resting on a rough horizontal table. $l$ is the maximum possible length that can hang in equilibrium. The coefficient of friction between the chain and table is
(a) $\frac{1}{L-1}$
(b) $\frac{L}{l}$
(c) $\frac{l}{L}$
(d) $\frac{A}{L+l}$

Key Concepts

Normal Force

The normal force is the reaction force exerted by a surface that supports the weight of an object resting on it. In this problem, the normal force on the chain is computed from the mass of the chain segment lying on the table. Since friction depends on this normal force, understanding its distribution is critical to analyzing the chain's equilibrium.

Uniform Distribution of Mass

A uniform chain has mass distributed evenly along its length, meaning that its weight per unit length is constant. This uniformity allows for the calculation of the weight of any portion of the chain, such as the hanging part versus the part on the table, which is essential for setting up the equilibrium condition.

Static Equilibrium

Static equilibrium refers to the state where all forces and moments acting on a system are balanced, resulting in no net movement. In the context of this problem, the chain is at rest when the gravitational force pulling the hanging part is exactly balanced by the frictional force resisting motion on the table.

Frictional Forces

Frictional forces are the resistive forces that occur when two surfaces interact. For this problem, the friction force acting on the chain is given by the product of the coefficient of friction and the normal force, which is derived from the part of the chain resting on the table. This concept is key to determining the maximum length of the chain that can hang without causing motion.

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