So after encountering a physics problem I had bumped into the following issue.

Let's say I want to push a skyscraper with my hands.

I will push the building with a constant force of F = 10 N let's say

The building of course, it does not move because I am too weak so this theoretically means that the acceleration of the building is 0.

https://postimg.cc/QKGr1R8x here is the drawing that I made for a better visualization of what I am saying

Newton's second law says that :

`The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.`

This means(in my humble opinion), that the building's acceleration is equal to the resulting force divided by its mass.

The resultant force would be F - Ff(friction force)

The thing is that even though I change the force to let's say F = 20N the building will not move, and the building's acceleration will still be 0.

After writing down the forces on the sky scraper object I reached the conclusion that F = µmg, in order for the acceleration be 0 on the X axis.

But this does not feel right, I mean they are clearly not equal. And the formula does not really express the reality. I must have done something wrong, I am sure of that but what though?

My intuition says that the formula should hold as an inequality, thus

the force F shoud be less then the friction force( F < Ff = µmg, m = the mass of the building)

This is my first post. I want to learn physics and understand every detail of it.

I posted a paint showing the steps I took to reach the formulas.

Thanks for reading and I hope you don't laugh at the silliness of the question