Do two springs compressed springs placed against each other obey Newton's third law?

Question

I've been mulling over the following scenario. Say I take two springs whose force is given by some well-defined function of position (for example, Hooke's law F=kx will do). Using this law, I choose to compress spring A by a distance x that will guarantee me that the spring will exert a force of 10N. I then choose to compress spring B by a distance such that it will exert a force of 7N. I put the two springs next to each other, release them and allow each to exert its force against the other. It's easier to imagine along the horizontal axis, and we can ignore gravity and friction and any other potential meddlers. Spring A should exert a force of 10N on spring B according to its force law and compression, but spring B should exert force of 7N back on spring A according to ITS force law and compression. Thus the force that spring A exerts on spring B is NOT equal and opposite to the force spring B exerts on spring A. This is a violation of Newton's Third law. What am I missing here?

Answer 1

The error is that the two forces you describe are not a Newton's third law action-reaction pair. Say that the first spring you mention is compressed so that is exerts a 10 N force and then you place it next to an uncompressed second spring. The first spring will exert a force of 10 N on the second spring. What Newton's third law says is that, solely because of that 10 N force, the second spring -- even uncompressed -- will exert a 10 N force back on the first spring.

The result of this is that, even if the second spring is uncompressed, and you let loose the first spring next to it, the first spring will also shoot away in the opposite direction from the second spring (absent any other forces), due to the reaction force alone.

The same thing would be true if the second spring were compressed so that it exerted a 7 N force, and was placed next to an uncompressed first spring. The first spring would still exert a 7 N force back on the second spring BECAUSE the second spring exerted a force on the first.

Now, if both springs are compressed, then each one will receive both the reaction force from the other spring AND the force the other spring exerts due to its compression. So the net force on each will be 17 N (in opposite directions, as Newton's third law prescribes).

The way I usually paraphrase Newton's third law is: "If System 1 exerts a force on System 2, then System 2 exerts an equal force, in the opposite direction, on System 1." Note that conditional phrase at the start. Newton's third law ties a reaction force to the presence of an instigation force: IF System 1 exerts a force on System 2, then there is a reaction force. The reaction force is always in response to what System 1 exerts.

I hope that helps! If you have any follow-up questions, please do not hesitate to ask.

Answer 2

The answer is that the momentum is conserved (according to Newton's third law. What matters is the net force