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My confusion is-

1) While taking limits, tension at end B of rod is taken 0, The function comes showing maximum Tension at end A.

But it seems to me that that at end A, T must be 0 as x=0 and at end B it must be max. Why?

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The tension force at any point in the rod provides the force that keeps the parts of the rod further from the centre of rotation moving in a circle. For points closer to the centre there is more mass to swing around i.e. a larger centripetal force is required, so the tension is larger.

I'm assuming your thinking of a rod rotating in a horizontal plane. In a vertical plane the tension will be more when B is below A and less when is above. You could even get compression in the rod if it's rotating slowly enough.

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