# How do we account for the non-net accelerations for forces in static systems?

I've googled the heck out this (as a non-physicist) and found a ton of similar questions answered but did not find anything that made the light bulb go on. I don't think I'm a crank, so here it goes.

## Overview

When one object exerts a force an another and they do not move, the net force is zero. I agree. But intuitively there must be non-zero forces somewhere in this system (if not what do we call these "energies"?). To the observer, however, there doesn't appear to be any non-zero force in any component of the system because there's no acceleration to be found. In other words, one might think: I don't see motion anywhere, so there's definitely no acceleration, so definitely there's no force. But that doesn't line up with reality and intuition, in which we know some sort of energy interaction is taking place.

## Example

I put my back up against a brick wall of a building. I'm poised to push to push the wall but have not begun to exert myself. I'm at rest and static. Then, after a few seconds I begin to push. While I'm pushing/exerting let's assume that there's no movement visible from a macro-observer level of my posture or body when I transition from non-exertion to exertion. Let's also assume that I'm pushing perfectly perpendicularly to the wall and that the wall cannot be moved by my puny legs. The net force is zero. No argument. But there's no acceleration to be found anywhere that the eye can see. The wall has not moved, I have not moved. If you need a more ideal pusher please feel free to replace my body with whatever.

## Hunches

Clearly energy is being expended somewhere here. I figure that either there's some other energy exchange going— one that's not considered force in the $$F = ma$$ sense. Or maybe that we're now talking about how force is transformed into something chemical or some material phenomenon of my pushing body and the brick wall.

## Why I need another perspective

I've spent decent time googling this. I've read several responses to similar questions in which the responders talk say things like "no! $$F = ma = 0$$ means net force of zero" — okay, I get it, I agree with it, but I don't see any non-zero non-net forces here either, because I haven't observed any acceleration in any component of the system. If the net force is the vector sum of all the vector forces then where are these vector forces with non zero accelerations (assuming there's a non-zero force somewhere)?

The selected answer here for example is not satisfying to me. https://physics.stackexchange.com/a/19401/271396

## Thank you

For all those who make an effort to understand where my density lies, thank you!

• you compress the atoms of the wall a bit if you push it quite hard – Buraian Aug 2 '20 at 22:05

There are several places the energy could go.

But the most intuitive and easily understandable part that I like to think of is that, consider the wall (in your case ).

For the sake of further simplicity let's assume it is perfectly elastic, meaning if you push it, the atoms get slightly compressed and on releasing they return to their original positions.

So when you push the wall, you are increasing the potential energy of a lot of particles, by, in this case, pushing them against the electric field of the neighbouring atoms.

Such systems can be thought of as a spring atached between the atoms, playing the role of the electric force on expanding or compressing.

So the energy you spend on the wall goes into the potential energy of these particles. On releasing these perform an oscillatory motion, and the motion gets damped out by frictional and other forces.

Yes the energy goes into other places, like potential energy of the ground particles if you exert a force on them, heat due to your body's metabolism, etc-You could go on giving examples.

Hope you got an intuition behind this?

Cheers.

In a static system, for each force there is an equal and opposite reaction force. If you push on the wall, the earth (to which it is attached) pushes back. As you push on the wall, the wall pushes back on you. You push backward on the earth and the earth pushes forward on you. There is no net force on you, the wall, or the earth. Energy is defined as the ability to do work, and work requires a force acting through a distance. In a static system, nothing moves, and no work is done. The only energy involved is in the metabolic process that maintains tension in your muscles.

• "There is no net force on you, the wall, or the earth." Agreed, but are there any non-net non-zero forces? – Zduff Aug 4 '20 at 14:35
• Isn't there a net force on earth? – Danny LeBeau Aug 4 '20 at 15:14