Physics

To find the change in gravitational potential energy of the chain as it slides off the table, we need to determine the initial and final gravitational potential energies and calculate the difference between them.

 

Initial Gravitational Potential Energy

At the start, of the chain's length is hanging off the table, and of it is still on the table.

 

The hanging portion of the chain has a length of , and its center of mass is located at the midpoint of the hanging portion, i.e., at a height of below the table. Thus, the gravitational potential energy of the hanging portion is:

 

The portion on the table has a length of , and its center of mass is at a height of 0 (since it is on the table). So, the gravitational potential energy of the portion on the table is:

 

Thus, the total initial gravitational potential energy is:

 

Final Gravitational Potential Energy

When the chain just leaves the table, the entire chain is hanging vertically.

 

The center of mass of the entire chain is at a height of from the table (since the center of mass of a uniform chain is at the midpoint of the chain's length). Therefore, the final gravitational potential energy is:

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