Critical velocity at the bottom of a vertical circle is root of 5gl where l is length of string but in case we have a light rod instead of a string it is root of 4gl .Why does this difference arise?

no one was able to give me a satisfactory explanation it would be of immense help if someone could tell me the exact reason.

critical vel in string is calculatedby considering tension at topmost point to be zero. but in rod tension can never be zero. thus we consider vel at topmost point to be zero .  

morever it is interesting to note that u cannot give such a vel in string mass vertical circle such that vel at topmost point is zero.

This is because in the derivation of motion in vertical circle with string we assume tension never becomes zero even at the top.

In derivation of motion in vertical circle with light rod even if the tension becomes zero ,the rod won''t slack so velocity at highest point is assumed to be zero.

So it means in case of string the velocity at highest point is not assumed to be zero and incase of rod velocity at highest point is assumed to be zero.

So the difference is because of their derivations.

If you further have doubt then refer to H.C. VERMA Book