Why is the Gravitational Potential Energy calculation requires half the total length in case of the falling chain? 
In this case, please note that it's written: $\displaystyle\frac{my}{l}g\frac {y}{2}$. Isn't the length supposed to be the total height of the falling chain, i.e., $y$ instead of $\dfrac {y}{2}$ where the height is written?
Please explain. 
 A: The potential energy of the hanging part of the chain is the potential energy of its centre of mass, which is $y/2$ below the table. The mass remaining on the table (before and after) can be ignored, because the PE of this part does not change.
There is no need to integrate along the hanging part of the chain to find its potential energy because the mass per unit length of the chain is constant and gravity is constant. You get the same answer using the total mass of the hanging part and the position of its centre of mass.
A: A chain of distinct length when hanging will possess different potential energy at different points. The lowermost point will possess the lowest and the topmost point will have the highest potential energy among all other points.This is because the position of the chain is not well defined from the datum level. In these situations to find the potential energy we can take the help of two methods.
Method 1
In this situations centre of mass always comes into the scene. If we concentrate the mass along the length of the string at a point the potential energy can easily be found out (as now position is very much defined). For a chain of uniform density it will lie at the middle point of the chain.     
Method 2
Though a bit longer approach than Method 1, conceptually it is a better method. Since the potential energy along the string is not constant take an infinitesimally small bit of the chain. Find the potential energy of the chain at that point and then integrate it to find the potential energy for whole of the chain.
Both the methods will eventually fetch you the same result.
