Question

A chain with length l and  mass m hangs over the edge of a frictionless table by an amount e. If the chain starts with zero initial velocity, find the position of the end of the chain as a function of time

Answer 1

All the links of the chain have the same displacement, velocity and acceleration. The corner only produces a change of direction. We therefore consider the chain to be a single mass with an applied force that depends on the length x of the overhanging part. Thus, with a = ¨x,the equation of motion is

 This differential equation of second order with constant coefficients has the solution 

we calculate the integration constants from the intial conditions :

This solution is valid only for x ≤ l. We may also solve the problem by making an imaginary section cut at the corner. then the equations of motion for each part of the chain are 

if we eliminate S we again obtain the differential equation