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I've got a rather humiliating question considering newton's third law

"If an object A exterts a force on object B, then object B exerts an equal but opposite force on object A" -> $F_1=-F_2$

Considering that, why is there motion at all? Should not all forces even themselves out, so nothing moves at all?

When I push a table using my finger, the table applies the same force onto my finger like my finger does on the table just with an opposing direction, nothing happens except that I feel the opposing force.

But why can I push a box on a table by applying force ($F=ma$) on one side, obviously outbalancing the force the box has on my finger and at the same time outbalancing the friction the box has on the table?

I obviously have the greater mass and acceleration as for example the matchbox on the table and thusly I can move it, but shouldn't the third law prevent that from even happening? Shouldn't the matchbox just accommodate to said force and applying same force to me in opposing direction?

I've found a lot of answers considering that question but none was satisfying to an extend that I had an epiphany solving my fundamental problem I've got understanding it.

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There are excellent answers below. I wanted to add that on the system scale (i.e. all objects together) the forces DO cancel out---that's why momentum is conserved. – zhermes Dec 2 '12 at 2:49
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Here's a point of view that helped me to "get" this question: If the matchbox didn't push back on your finger with equal force, your finger would go right through it as if it were a ghost! – wim Dec 3 '12 at 0:45
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Note that the acceleration of the object (i.e. matchbox) depends on its mass and the net sum of forces acting upon it. Crucially, it does not depend on forces which the object exerts upon other things (i.e. finger). – wim Dec 3 '12 at 0:47

5 Answers

up vote 37 down vote accepted

I think it's a great question, and enjoyed it very much when I grappled with it myself.

Here's a picture of some of the forces in this scenario.$^\dagger$ The ones that are the same colour as each other are pairs of equal magnitude, opposite direction forces from Newton's third law. (W and R are of equal magnitude in opposite directions, but they're acting on the same object - that's Newton's first law in action.)

Finger Prodding Matchbox

While $F_{matchbox}$ does press back on my finger with an equal magnitude to $F_{finger}$, it's no match for $F_{muscles}$ (even though I've not been to the gym in years).

At the matchbox, the forward force from my finger overcomes the friction force from the table. Each object has an imbalance of forces giving rise to acceleration leftwards.

The point of the diagram is to make clear that the third law makes matched pairs of forces that act on different objects. Equilibrium from Newton's first or second law is about the resultant force at a single object.

$\dagger$ (Sorry that the finger doesn't actually touch the matchbox in the diagram. If it had, I wouldn't have had space for the important safety notice on the matches. I wouldn't want any children to be harmed because of a misplaced force arrow. Come to think of it, the dagger on this footnote looks a bit sharp.)

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This answer is completely awesome. My question to you is how on earth did you decide it that it was better to write in an entire warning label instead of removing the word "match"? – Steven Lu Dec 2 '12 at 5:41
Just a little thing: the reaction force from the table shouldn't be paired with the matchbox's weight. – Javier Badia Dec 2 '12 at 7:54
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@StevenLu because I found it funny, particularly "May cause fire.". – AndrewC Dec 2 '12 at 9:28
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@JavierBadia Fixed now. Thanks for pointing out my silly (and ironic) but key mistake. My answer is better now because of your comment. – AndrewC Dec 2 '12 at 21:20
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A nice exercise is to draw the table, matchbox, person and earth and find as many third law matched pairs you can (remember to make sure they're acting on different objects). There's an answer to be found in the revision history of my answer (click the link after the word edited), but I hid it because I feel it distracts from the main part of the answer. – AndrewC Dec 2 '12 at 21:32
up vote 9 down vote

Good! This question implies that you're thinking hard and questioning the laws. It turns out that you are misunderstanding Newton's 2nd Law though. Motion on a body is due to an external force. F1 acts on your box, but not F2. An object can never act on itself.

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Think you mean "motion of a body". – Eugene Seidel Dec 1 '12 at 23:48
up vote 6 down vote

Forces related to Newton's third law apply to different bodies, therefore they cannot cancel each other out.

For example, the reaction to Earth's gravitational pull on the Moon is the Moon's pull on Earth. That force won't have any relevante to the Moon.

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Interestingly, this question is one of the foundations of Thomas Aquinas' proof of G-d. God set the world into motion ("prime mover"), providing just a little initial "first cause", hence they all do not cancel each other out.

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action and reaction Depends on the frame eg: if u push matchbox with ur fingers kept on a table, from matchbox's frame, we have to see for only those force acting ON the matchbox, not those forces that matchbox applies i.e the reaction force to ur fingers So, from matchbox's frame, forces acting on matchbox are: ur push, downward mg and normal reaction from table, that is why it moves

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Try to avoid abbreviations ("ur") and run-on sentences. It makes your answer hard to read. – Javier Badia Dec 2 '12 at 7:55
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Aside from that, you can "see" all forces from a reference frame, so this answer is wrong. The frames don't matter, it is just that an object moves only due to the forces acting on it. – Manishearth♦ Dec 2 '12 at 14:16

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