A uniform chain of mass , and length is held onto a horizontal frictionless table. The table has a large hole in which a segment of length is hanging over the hole. The chain is then released from rest, and the chain begins to descend. How long will it take the chain to completely slide through the hole?
Define as the constant linear mass density of the chain. Let represent the variable length hanging over the hole.
Since , the equation for force become the differential equation
Since the chain was initially at rest, and the initial length hanging over the hole is , the solution for velocity is
Since , the above equation can be written as
Integrate the above differential equation with the integrating bounds and .
The solution for time is