Rotation dynamics, dealing with impulse and oscillation

A homogene rod with length "l" is placed vertically, and a nail is stabbed on the top of the rod (now the rod has an axis). And then an impulse is given on the rod with the separation between the impulse given to the rod's axis is "d". earth gravitational acc is represented as g, the mass of the rod is m. Now, calculate the minimum value of d to make the rod rotate 360°.

Now if the condition above is complete, and the rod make a harmonic movement (oscillation) what is the period?

And what is the length of a mathematical pendulum should be to make the same period with the oscillating rod?