Toppling of a Rigid body Consider a square block kept on a rough horizontal surface (friction is sufficient enough to prevent slipping/sliding). You apply a horizontal force F on the topmost point ,constant in magnitude and direction. The magnitude is adjusted so that the block doesn't gain significant kinetic energy. Due to rotation about the axis passing through the line of contact of the ground and block, the potential energy of the block increases as it loses contact with the surface. How does it gain potential energy? I'm confused which force is responsible for this gain. 
Thanks in advance.
 A: If I understood your question correctly, what gets the Center of Mass to rise (and hence gain potential energy) is the couple (torque) that F  creates:
Just like what happens in an imperfectly balanced wheel (let's say that more mass is in the lower part), where you get enough torque to get the wheel to start rotating on its axis. In this case, similarly, the Center of Mass rises, you just need enough Torque to overcome the torque from the gravitational force (which creates a torque as soon as the wheel rotates because the CM isn't exactly on the axis of rotation).

In a rigid-body, staticity is not only from $\sum\limits_{i=1}^{n} F_i = 0$
but also from the sum of all present torques  $\sum\limits_{i=1}^{n} R_i x F_i = 0$ which, in this case, is not happening
$\int\limits_{0}^{\theta_f} M_F(\theta) d\theta = W_{nonconservative} = \Delta E_{mec} = mgh + 1/2 I {\omega}^2$
with $h = lsqrt2 /2 - l/2$
and I from H.S. theorem
The integral actually solves out to constants, as $l\sqrt2 F$ applied along the arc of circumference
A: 
The normal and weight balance each other.

You are making an assumption here. While the block is stationary the weight and normal reaction balance. But to make the COM start moving upwards there must be a resultant force upwards. The only upward force is the normal reaction.

Also there is no displacement of the point of contact. Hence the work done by all the forces in vertical direction is zero.

If you squat on the ground then stand up, you have raised your COM. Work must be done by some force external to your body. That force can only be the normal reaction. 
But I still have not explained how this force can do work if the point of application does not move... 
