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I am new to the site and donot have the reputation to comment, and hence I repost the question What is the force exerted by a spring when pulled by a force $F$? I was not satisfied with the answer there because my doubt is about the case where there is NO MASS attached to the spring. If there is some mass attached to the spring, we can analyze forces between the spring and the mass, and that between the mass and the other force pulling it (say the gravitational force pulling a mass downwards with the mass attached to a spring which in turn is attached to something rigid, say a ceiling.)In this case, one can, with some thinking understand how Newton's 3rd law holds. The force applied by the spring on the mass (and also that applied by the mass on the spring) will be equal to kx, and is independent of the force due to gravity on the mass (in the sense that the spring force is dependent only on its extension, which does depend on the force with which the mass, which pulls the spring is pulled, though)

However, If there is NO MASS attached to the ideal spring, how can I apply Newtons 3rd law? Say a force of 10 newtons is suddenly applied to the end of an unstretched ideal spring (Lets assume there is some mechanism to do so ).How will Newton's third law hold till the string gets stretched to some length?

Or, is it wrong to think of such a case? I mean, am I wrong in thinking of the spring as an object on which force can be applied? is it just a "source of force" and not an object on which force and the force laws can be applied?
Edit: To be more precise, will I be right in saying that a real spring can be thought of as an 'ideal spring + the mass it has', with its mass in a way playing the role of the object on which the ideal spring('the source of force' and not an object) exerts force? Applying force on the ideal spring is hence meaningless?

If it can be considered an object on which force can be applied, what will be its reaction to an external force ? will it infinitely stretch since its massless? ( and since we do not know the source of the external force as well...it can be a non-contact force. There are only two elements here...the ideal spring and the source of external force)

Could you please add clarity to my understanding of springs?

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  • $\begingroup$ If you are OK with doing some math, you might have a look at the derivation of the time evolution of a massive body held by an ideal spring. (It should be on the internet.) I suppose all you had to change in the solution is to replace the gravitational force with an arbirtary constant force (e.g. in realty it could be a charged body in homogenous electric field), and look at the edge case where the mass of the body goes to 0. $\endgroup$
    – fanyul
    Commented Mar 22, 2022 at 17:05
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    $\begingroup$ It's impossible to "suddenly" apply a 10 N force to a massless spring. The force applied to the spring will equal the force that the spring pushes back with (Newton's 3rd law), so that force will start at 0 N, and it will linearly increase until it reaches 10 N as the spring is compressed or stretched. $\endgroup$
    – David White
    Commented Mar 23, 2022 at 2:33
  • $\begingroup$ @David White I think that should make sense. John Doty's answer also tells me that such a situation as I described is impossible, wrong. However replying to your answer, I think my problem lies with the fact that it is very difficult to agree/digest that I cannot apply a force that does not start with 0 and increase gradually, because, If I apply the force, I must be able to control the amount of (external) force...isn't it? and the object that I apply force on must react with an equivalent force? Is there some way I can understand why I cannot control external force? $\endgroup$
    – Eternal Learner
    Commented Mar 23, 2022 at 8:27
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    $\begingroup$ @EternalLearner, Newton's 3rd law states that as you apply force to the spring, it applies an equal force directed in the opposite direction. Since the spring starts unstretched or uncompressed, the applied force will start at 0 N, and will linearly increase until it reaches 10 N (or whatever force you finally apply to it). $\endgroup$
    – David White
    Commented Mar 23, 2022 at 13:18
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    $\begingroup$ @EternalLearner I assume that by "apply a force on the spring" you mean apply the force on the end of the spring (e.g. on a body mounted to the spring). With a charged body and an electric field that you can easily controll, you can apply an arbitary force on the spring. (Although it takes a some time for the electric field to form/change in realty, maybe in the order of milli- or microseconds.) $\endgroup$
    – fanyul
    Commented Mar 23, 2022 at 18:51

2 Answers 2

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Physics is an experimental science, so get yourself a massless spring, apply a force to it, and see what happens ツ.

Seriously, idealizations are not necessarily compatible with each other. You have colliding idealizations: a massless object and a force that doesn't depend on acceleration. You can't get a sensible answer from that combination.

Edit in an attempt to answer comments:

Consider what happens if there's a massive body at the end of the ideal spring. Ignore friction. Start with displacement x=0, at equilibrium with no external force. Now, apply a constant external force to the body. The body accelerates until, at some displacement d, the net force on the mass is zero. At this time, the body is in motion, so it continues beyond point x=d. It continues to move until x=2d (you may work out the math yourself, or, better, do an experiment). The motion reverses, and the body moves back to x=0, where the process repeats. The body thus oscillates between x=0 and x=2d.

Note that I haven't mentioned the mass of the body. That doesn't influence where the body goes, it only influences how rapidly it moves. The body always oscillates between x=0 and x=2d, but the smaller the mass, the higher the oscillation frequency. So, consider the limit of a zero mass body. In this counterfactual case, the oscillation cycle takes zero time. So, the body occupies every point between x=0 and x=2d at all times.

I don't know how this can help you understand springs.

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  • $\begingroup$ Ahh sorry! But I had to do all this to understand the CONCEPT of springs better...I find difficulties while applying Newton's third law in such cases...I do understand that it is impossible to have a massless spring though :) $\endgroup$
    – Eternal Learner
    Commented Mar 21, 2022 at 0:44
  • $\begingroup$ @EternalLearner While reasoning about impractical experiments is often educational, reasoning about impossible experiments is treacherous. $\endgroup$
    – John Doty
    Commented Mar 21, 2022 at 11:21
  • $\begingroup$ Could you please clarify whether my thinking about it in para 3 of my question "or, is it wrong to ..." is right? Is that why you call it "impossible" and not just "impractical"? I think I need more clarity on why you call it "impossible" and not " impractical" probably because I have a wrong understanding of what exactly a spring is... a wrong picture of it... . $\endgroup$
    – Eternal Learner
    Commented Mar 22, 2022 at 11:41
  • $\begingroup$ Please do check out the edit(addition) that I made to the 3rd paragraph in order to make my thoughts clear, and please do comment on whether that is the right way to think about it. $\endgroup$
    – Eternal Learner
    Commented Mar 22, 2022 at 12:00
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so it's like just pulling regular spring apart, just because it's not attached to something hooks law doesn't apply. Hooke's law is a property of the spring so if we have a force acting on the spring it will act in accordance with hooks law till a physical limit is reached. so pulling with some random force will just pull the spring in accordance with Hooke's law. $$F_{springforce}=-F_{pulling force}=kx$$ adding a mass just bring the coolness of SHM

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  • $\begingroup$ However, the pulling force is completely IN OUR CONTROL and the spring force isn't- it is always equal to the spring constant times the extension (of course, within the range, in which Hooke's law is applicable). So how can the third law be valid in this case? I agree that it will reach a point when the 2 forces do become equal...but that requires time (however small ) to elongate till the required length, so we cannot say that the third law in this case holds at every point in time...(?) $\endgroup$
    – Eternal Learner
    Commented Mar 21, 2022 at 3:03

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