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Nathanael

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## Main Question or Discussion Point

Imagine if I were to apply a constant force, F, to a spring of length L, spring constant K, mass m, and uniform density. The force causes the spring to accelerate with acceleration a.

For the sake of imagination, I will say I'm applying the force with my hand (which is pushing one end of the spring).

Surely the spring will compress, (how else would the other end of the spring, away from my hand, accelerate?) but, we shouldn't expect the spring to compress according to Hooke's law, because Hooke's law applies to when the force is acting on both sides of the spring, (right?) and in this case the force acts only on a single side.

So my question is, by how much would the spring be compressed? (What is ΔL?)

My approach:

Pretend the spring consists of infinitely many infinitesimal springs, each of identical length dx. I'm thinking that we can then treat each infinitesimal spring as having a spring constant of [itex]k\frac{L}{dx}[/itex] (right?).

Let's now put an x-axis along the direction of motion, with x=0 being the end of the spring where my hand is.

One of the infinitesimal springs at a distance x will have to apply a force of [itex]ma\frac{L-x}{L}[/itex] in order to accelerate the springs ahead of it. To then find the compression of that single spring we would now use Hooke's law and divide the force by it's spring constant, to get a change in length of [itex]-ma\frac{L-x}{kL^2}dx[/itex]

Then, to get the change in length of the entire spring, we would sum up all the length changes of all the infinitesimal springs to get [itex]ΔL=∫_0^L\frac{-ma(L-x)}{kL^2}dx=\frac{ma}{2k}=\frac{F}{2k}[/itex]

I might be making some simplifying assumptions. For example I am assuming the spring length is in equilibrium (the spring is not oscillating). I am not sure if this would be true or not (maybe someone can comment on if the spring would oscillate if you accelerated it from rest?) but I'm just trying to simply analyze a massive spring (because e

The answer seems reasonable to me: The spring would compress half as much as if the force was acting on both sides (i.e. half as much as if we used Hooke's law).

The reason this seems reasonable is because the density is uniform, and so the force applied by the infinitesimal pieces of the springs varies linearly, so we should be able to simply average it out to the middle, effectively treating the spring as if all of the mass were at the midpoint. (I know that this logic is not very precise, but it's just a post-rationalization).

I made up this problem, and so I have no way of checking my answer, so I am asking you helpful folks.

For the sake of imagination, I will say I'm applying the force with my hand (which is pushing one end of the spring).

Surely the spring will compress, (how else would the other end of the spring, away from my hand, accelerate?) but, we shouldn't expect the spring to compress according to Hooke's law, because Hooke's law applies to when the force is acting on both sides of the spring, (right?) and in this case the force acts only on a single side.

So my question is, by how much would the spring be compressed? (What is ΔL?)

My approach:

Pretend the spring consists of infinitely many infinitesimal springs, each of identical length dx. I'm thinking that we can then treat each infinitesimal spring as having a spring constant of [itex]k\frac{L}{dx}[/itex] (right?).

Let's now put an x-axis along the direction of motion, with x=0 being the end of the spring where my hand is.

One of the infinitesimal springs at a distance x will have to apply a force of [itex]ma\frac{L-x}{L}[/itex] in order to accelerate the springs ahead of it. To then find the compression of that single spring we would now use Hooke's law and divide the force by it's spring constant, to get a change in length of [itex]-ma\frac{L-x}{kL^2}dx[/itex]

Then, to get the change in length of the entire spring, we would sum up all the length changes of all the infinitesimal springs to get [itex]ΔL=∫_0^L\frac{-ma(L-x)}{kL^2}dx=\frac{ma}{2k}=\frac{F}{2k}[/itex]

I might be making some simplifying assumptions. For example I am assuming the spring length is in equilibrium (the spring is not oscillating). I am not sure if this would be true or not (maybe someone can comment on if the spring would oscillate if you accelerated it from rest?) but I'm just trying to simply analyze a massive spring (because e

*very*spring problem I've come across has treated the spring as massless...).The answer seems reasonable to me: The spring would compress half as much as if the force was acting on both sides (i.e. half as much as if we used Hooke's law).

The reason this seems reasonable is because the density is uniform, and so the force applied by the infinitesimal pieces of the springs varies linearly, so we should be able to simply average it out to the middle, effectively treating the spring as if all of the mass were at the midpoint. (I know that this logic is not very precise, but it's just a post-rationalization).

I made up this problem, and so I have no way of checking my answer, so I am asking you helpful folks.