A uniform chain of length L and mass M is lying on a smooth table and (1 / 4)^th of its length

Step 1: The work done to pull the hanging part of the chain onto the table is given by the formu

A uniform chain of length L and mass M is lying on a smooth...

A uniform chain of length $\mathrm{L}$ and mass $\mathrm{M}$ is lying on a smooth table and $(1 / 4)^{\text {th }}$ of its length is hanging vertically down over the edge of the table. If $g$ is acceleration due to gravity, the work required to pull the hanging part on to the table is (A) MgL (B) $\mathrm{MgL} / 9$ (C) $\mathrm{MgL} / 18$ (D) $\mathrm{MgL} / 32$

A uniform chain of length $\mathrm{L}$ and mass $\mathrm{M}$ is lying on a smooth table and $(1 / 4)^{\text {th }}$ of its length is hanging vertically down over the edge of the table. If $g$ is acceleration due to gravity, the work required to pull the hanging part on to the table is
(A) MgL
(B) $\mathrm{MgL} / 9$
(C) $\mathrm{MgL} / 18$
(D) $\mathrm{MgL} / 32$

Key Concepts

Work-Energy Principle

The work-energy principle connects the work done on an object to its change in energy. In the context of this problem, the work done in pulling the chain onto the table is exactly the energy required to raise the chain’s center of mass, thereby changing its gravitational potential energy.

Uniform Mass Distribution

A uniform chain has a constant mass per unit length. This property is important because it allows the use of simple linear relationships when determining how much mass is present over any segment of the chain, and it greatly simplifies the calculations of both the center of mass and the potential energy.

Gravitational Potential Energy

This concept explains the energy possessed by an object due to its position in a gravitational field. In problems like this, the work required to change the position of a mass is equivalent to the change in its gravitational potential energy, typically calculated by multiplying mass, gravitational acceleration, and vertical displacement.

Center of Mass

The center of mass is the point where the total mass of an object or system can be considered to be concentrated. For uniformly distributed objects, like a chain, determining the center of mass is crucial when calculating potential energy changes since it simplifies the task of integrating over the object’s entire distribution.

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