homework and exercises - Rod sliding on a frictionless surface

A uniform rod$(m,l)$ is standing vertically on a horizontal frictionless surface. Gravity is downwards and uniform. I give its upper end a little push and off it goes. I want to find the Normal Force(and all other variables) as a function of time.

Here's what I do :

$mg-N=ma$

Where a is downward acceleration(and the only) of centre of mass wrt ground.

Next, I think of conservation of energy. I assume it rotates $\theta$ from vertical

$mg\frac{l}{2}(1-\cos\theta)=K.E.$

I have problem writing expression of its Kinetic Energy. I have studied about instantaneous centre of rotation.

So, I can write its speed and rotational kinetic energy about it as $K.E.$? Also How do I find relation between $\omega$ (angular velocity about that instantaneous centre) and velocity of rod?

Answer

This is what I did and I think is simple and right :

Assume $v$ linear speed of centre of mass downwards and $\omega$ angular speed around it. Use the fact the bottom point has no vertical speed to find relation between $v$ and $\omega$. And I am done.