Rotation dynamics, dealing with impulse and oscillation

[Andreas]

Homework Statement

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?

Homework Equations

I.ω = m.ω.d²

T= 2∏√(I/mgR)

2∏√(mgR/I) = 2∏√(l/g)

The Attempt at a Solution

First I take the conservation of angular momentum with the first relevant eq.

I.ω = m.ω.d²
where I of the rod is 1/3ml²

1/3ml²ω=mωd²

d²=1/3l²
d=1/3√3l

Please correct me if I'm wrong and also the solution for the second and third question

 

[BvU]

Hello Andreas, and welcome to PF.
Could you clarify a few things for me:
Is this the complete formulation of the exercise ?
I wonder how a minimum for d can be calculated if there is no further information about the "pulse".
And: what is the first relevant equation ?
And: what is R in the second and third ? (probably the answer to the third question?)

Then, in the solution: how does this show that the rod rotate at least 360 degrees ? And why is g absent in the answer ?

For the second question, what does "if the condition above is complete" mean ? The way I read it it would mean there is no harmonic motion at all: once it has passed the 180 degrees it will just keep spinning (in the absence of friction).