# Forces Acting on a Spring

I have consulted other pages on this site with regards to this question, but I have not found a satisfactory answer yet. Take a smooth, horizontal surface, and two blocks, A and B, on the surface. Assume the two blocks are connected by a spring. If I pull on both masses, the spring stretches. As the spring wants to return to its equilibrium position, there is a restoring force acting on both the masses.

Now comes the bit that confuses me:

(a) If I pull on the spring, with a force, what is its action-reaction pair, and how does the spring exert this force?

(b) If the action-reaction pair of the pulling force is the restoring force, why are they not equal in magnitude? (I can pull with any force, while the restoring force is confined to a magnitude of $$kx$$.)

(c) When I release the spring, it returns to its equilibrium position. This can be thought of as a conversion of energy (i.e. a conversion of elastic potential energy to kinetic energy). But when the spring returns to its equilibrium position, there must be some unbalanced internal force causing this change in structure. What is this internal force, and how does the spring return to its equilibrium position.

• The reaction force of A pulling on B is B pulling on A. The reaction force of you pulling on the spring is the spring pulling on you. This is very confusing for students. They always seem to think (you included) that the "reaction force" is a second force that keeps something from moving. No. It's very simple. Just rearrange the sentence. If you said "I pull on the spring" the reaction force is "spring pulls on me" Commented Feb 6 at 2:39
• Voting to reopen. Clearly a conceptual question and not a homework question. And a good answer below. Commented Feb 6 at 7:13

Your situation is slightly more complicated to put in words because you pull on both masses. This ends up with an action-reaction pair between your hand and block A and a second action-reaection pair between your other hand and block B. Then there's reactions with the spring itself, and it ends up getting a little challenging to book-keep it.

For simplicity, I'd like to offer a different but equivalent problem. Instead of having a spring between two blocks, let's have a spring between block A and the wall, which does not move. Once this scenario is clear to you, it will be easy to extend it to the scenario in your question. But this way I don't have to disambiguate one hand from the other while typing.

So, you pull on the spring. The action reaction pair is always between two objects, and the reaction always has them in the opposite order from the action. So if the action is the hand applies a force on block A (pulling it), the reaction is an equal and opposite force applied by block A to the hand. We can also speak of the action-reaction pair between the block and the spring. The action is the block applying a force to the spring, and the reaction is the spring applying a force to the block.

(As a note: the order doesn't matter. You can assign either one as the action and the other as the reaction. What matters is that they come in pairs.)

Now, you talk of unequal actions and reactions. This cannot occur in Newtonian physics. If you think it's happening, its a good time to slow down and analyze the situation more carefully. You give an important argument for this:

I can pull with any force, while the restoring force is confined to a magnitude of $$kx$$.

Good. You chose an example to back up why you think the rules don't make sense. We can work with this example.

Rememeber that you're pulling on a block. Let's say you pull on the block with a force $$F_{hand}$$. As an action reaction pair, the block naturally pulls back on you with a force of $$F_{hand}$$ (same magnitude, opposite direction). The block is also pulled on by the spring, with a force of $$kx$$. It too has an action-reaction pair, so it pulls on the spring with a force of $$-kx$$.

So if we do the force body diagram on the block, we see that it's being pulled on by $$F_{hand}$$ in one direction and $$kx$$ in another. These are not action-reaction pairs, so they can be unequal. If they are equal, $$\Sigma F=ma$$ says that the block will begin to accelerate. Which is exactly what happens. You pull on the block hard enough, and you succeed at moving it outward while stretching the spring.

So what if we make this really tricky, and let you grab the spring directly. Then you apply $$F_{hand}$$ to the spring. Well, it must apply $$F_{hand}$$ back at you. But the spring only pulls with a force of $$kx$$ right?

The trick to this is that you cannot pull on a massless spring with any force greater than $$kx$$. If you did so, it would extend infinitely fast to the correct length to match your force. Now you'll never see this in real life. All real springs have mass, and don't quite follow Hooke's simple law when stretched really quickly.

This, by the way, is why it hurts so terribly bad to fail to break a board in Karate. If the board breaks, it is not possible for you to have applied a force higher than the breaking strength of the board. But if it doesn't break, you're really only limited by how hard you can punch... which is usually pretty hard because you've been training for it.

And hopefully by this point you're comfortable with the answer to the third question. Take three particles, L (left), C (center), and R (right). By the rules of how forces work, there's an action-reaction pair between L and C (L pulls on C, and C pulls on L). There's an action-reaction pair between C and R (C pulls on R, and R pulls on C). And in both cases, the action and reaction have exactly the same magnitude. They are always balanced. The imbalance you are looking for is that the force of L on C may be unbalanced with the force of R on C, which will cause C to move.

• Thanks for the insightful answer! I have one question though - I can still apply a force greater than the breaking strength of the Karate board to break it right? Commented Feb 8 at 10:37
• @VTNaveenMugundh You can apply a force equal to the breaking strength of the Karate board, and you can apply a force that would be greater if the board did not break. But when the board breaks, it gives way before you can apply a larger force. Any further force above that point gets put into accelerating the board away from your fist. Boards are pretty light, so it takes very little force to accelerate them very quickly. It is definitely unintuitive, until you're comfortable with how forces work. The reality is that what's in our head is an approximation of what happens. Commented Feb 9 at 23:41
• For a sense of just how different the world is from our intuition, consider this video of a golf ball striking a golf club at 150mph. What that ball does is quite different from what I think balls do when I hit them with golf clubs. Commented Feb 9 at 23:44
• Ah, that makes sense. It’s quite counterintuitive. Thanks for the explanation and video! Commented Feb 10 at 9:00