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

A uniform chain of length $L$ and mass $M$ is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If $g$ is acceleration due to gravity, work required to pull the hanging part on to the table is

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy that an object possesses due to its position in a gravitational field. In problems involving the lifting of objects, the work done against gravity is directly related to the increase in the gravitational potential energy of the object or system, making it a central concept in calculating the energy required for such movements.

Center of Mass of a Continuous Body

The center of mass is the effective point at which the mass of an object is concentrated and where gravitational forces can be considered to act. In the analysis of continuous and uniformly distributed objects, identifying the center of mass simplifies the process of calculating changes in potential energy, especially when only part of the object is being moved.

Uniform Mass Distribution

Uniform mass distribution means that the mass per unit length (or area or volume) is constant throughout the object. This concept allows for straightforward proportional calculations, such as determining the mass of any segment of the object based on its length, which is crucial when evaluating systems like chains or ropes in physics problems.

Work

Work is defined as the process of energy transfer when a force is applied to move an object over a distance. In the context of gravitational systems, the work done by or against gravity is calculated by the product of the force (typically weight) and the displacement, which corresponds to the change in gravitational potential energy.

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