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If I push against a brick wall I'm exerting a force against the brick wall the the brick wall will exert the same force against me. These will cancel each other out and neither myself or the wall will move.

Since I am not moving and therefore have no acceleration how do I calculate the force I exert on the wall given $F = ma$?

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You can't really calculate that from principles, since you can just push as hard as you like. Do you mean you want to know how to measure the force? –  Colin K Jan 12 '12 at 2:37
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You have some misunderstandings apparent here - the forces you and the wall exert on each other do not cancel each other out, they are cancelled by forces the ground or other supports exert on each of you. Imagine an astronaut in free fall or a person floating in a swimming pool pushing on a wall - they will move away. Also, the equation $F=ma$ should be written $\Sigma F = ma$ - you need to add up all of the forces acting on a body to find it's acceleration. –  user2963 Jan 12 '12 at 2:56
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To measure the force you need some kind of instrument. For example, one way would be to put a bathroom scale between your hand and the wall - what ever "weight" is registered on the scale is the force you are exerting. –  FrankH Jan 12 '12 at 3:39
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I think I understand: the F in F = ma is not just force but 'net force' therefore F = ma does not apply to the force I exert on the wall. –  Chris Herring Jan 12 '12 at 3:47
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Yep, that last comment is exactly true and is a very important point. That's why I always recommend that people write $\sum F = ma$ instead of just $F = ma$. –  David Z Jan 12 '12 at 6:13
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2 Answers

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Put a scale between your hand and the wall.

The reason you have no acceleration is that the sum of force vectors is zero.

You are pushing, and the wall is pushing back in the opposite direction, adding up to zero.

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That last paragraph is wrong. –  Colin K Jan 12 '12 at 4:47
    
-1 this is very misleading though I suppose all the sentences are technically true. –  user2963 Jan 12 '12 at 4:50
    
@Colin I think (hope?) Mike is talking about the forces acting on the scale, not the forces that the person and wall would exert on each other directly. –  David Z Jan 12 '12 at 6:12
    
@zephr: Maybe I didn't pick my words carefully enough. Anyway, the comments on the question are saying exactly what I meant. (This ain't rocket science :) –  Mike Dunlavey Jan 12 '12 at 12:51
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What is the name of the force exerted on a body that doesn't move?

Pretty much every known kind of force is often exerted on a body that "doesn't move" (relative to the ground) -- gravitational force from the Earth and the Moon and the Sun, tidal force, magnetic force, electric force, spring force, tension, compression, etc.

Static friction is a kind of force that is always exerted on a body that doesn't move.

How do I measure or calculate the force exerted on a body that doesn't move?

One of many ways to measure how much of one kind of force from some source is applied to a body is to assume ∑F = ma , and then measure the force and the acceleration of an object, and multiply them together to get the net force.

Alas, that only works when all the other forces are carefully balanced so they cancel out, so the net force is entirely due to the one source we are interested in. When you push against a brick wall, all the other forces on you are not cancelled out.

A more practical way to measure the force of a hand against a brick wall is to put a spring scale between them, which will measure the force your hand exerts against the wall, and also the (equal and opposite) force the wall exerts on your hand.

When I'm in a swimming pool and push against a wall, I accelerate -- the sum of the forces on me is not zero, even though (as always) the force of my hand on the wall is equal and opposite to the force the wall exerts on my hand.

When I stand on the ground and push against a brick wall, I am stationary -- the vector sum of the sideways push of the wall on my hands, the vertical force of gravity on my body, and the diagonal force of the ground on my feet, all balance to zero net force.

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