Suppose a uniform chain of length L and mass M suspended vertically from a fixed point O.
Suppose that each point of the chain is given a small horizontal displacement at time and then the chain is released.
Choose a rectangular coordinate system having origin at O and with positive x and y axis.
The equilibrium position of the chain is coincident with the x axis.
The horizontal displacement of each point x of the chain at time t can be denoted by or Y.
Since the chain has mass, there will be tension at each point x which is equal to the weight of that portion of the chain below x. If we denote by the mass per unit length of the chain, then since the length of that portion of the chain below x is ,
The mass of this portion is .
Then weight or tension is
In this case the equation of motion is
Since end is fixed,
Assume that initially each point x of the chain is given displacement,
, …… (3)
Also, the chain is released so that each point has velocity zero leads to
Let in (1), and separate the variables, to obtain
Thus the possible solution for equation (5), is given by
Since the solution must be bounded, in particular at , choose in (6).
Also by condition (4) choose
Thus solution is
The condition (2) gives
Denote positive roots of this equation by
This leads to the solution
By the principle of superposition,
By condition (3)
To find the coefficients in (11), let
Then (11) becomes
Substitute and use the result and , to obtain
Substitute value of in (10),