Density of a Spring

Here is the following excersise one teacher posed as a challenge to solve:

A spring has one side attached to the wall and the other one free.The spring has at rest
length L, and linear mass density ρ(x)= constant. Suppose we pull the spring to a new length L'.
1)Find the new mass density of the spring ρ'(x).

Now after trying it at home with a spring it is clear to me that the mass density of the spring becomes smaller as we get from the wall to the other end (after i have streched it). So there must be an explanation to this...

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Baluncore
When you apply a force to a spring you change the radius of curvature of the material.
One side gets shorter, the other gets longer. The effects cancel, the volume remains the same, as does the density.

Simon Bridge
Homework Helper
Now after trying it at home with a spring it is clear to me that the mass density of the spring becomes smaller as we get from the wall to the other end (after i have streched it). So there must be an explanation to this...
... how did you determine the mass density so that it was clear to you?
This is a startling result, please be specific about the kind of spring and what measurements you made.
Be careful to define your terms.

Note: the spring constant depends on the density of the material - if density changed so easily, then the spring constant would not be, um, constant...

Chestermiller
Mentor
This problem is very ambiguous. The answer depends on how you define the mass density. If the mass density is defined as the mass of the spring divided by the superficial length L, then as you increase L, the mass density decreases. But, if you define the mass density as the mass of the spring divided by the developed length of the spring along the helix, then, since the length of wire along the helix doesn't change (as baluncore pointed out), the mass density doesn't change.

Chet

Technically doesn't the density have to change somewhat in order to transmit intermolecular forces? Granted this is a very small quantity relative to the displacement of the spring itself so we can ignore it and assume density is constant. But the volume still does change ever so slightly...

Simon Bridge
Homework Helper
You can imagine a spring that is a single chain molecule - fix one end and pull the other and the "zig-zag" of the chain gets flatter ... the atoms mean spacing gets wider, so the linear mass density gets smaller.

This is not the kind of thing that a student is likely to notice in an improvised experiment at home.

We can also imagine the spring is not made of coils - we just have a straight length of metal, fix one end, pull (very hard) on the other end - the metal gets longer and the sides "suck in" ... mass stays the same so the linear mass density is lower.

Maybe it is a torsional spring, or a bent-beam spring?

Of course we could use a rubber band - but is that a good "spring" for these purposes: the slightest pull may pass it's elastic limit?

All this is why I want to see the details of the experiment before commenting properly.

256bits
Gold Member
I understand the question as the OP stating linear mass density, that should be in units of mass/length (kg/m), and not mass/volume.

Suppose you take a steel spring of given length and number of coils/unit length.
The linear mass density of the spring is the number of coils per unit length.
Now if you stretch the spring then the number of coils per Δχ is not the same everywhere in the spring. (There are more coils per unit length at the wall than at the other end which means the linear density drops)
In other words the spring appears more stretched at the other end than at the end attached to the wall.
I have included a picture which shows that a stretched spring has the same number of coils per length everywhere (maybe this is idealized but still has to be explained theoretically) but this is NOT what you see if you do the experiment at home with a normal spring.

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Nugatory
Mentor
I have included a picture which shows that a stretched spring has the same number of coils per length everywhere (maybe this is idealized but still has to be explained theoretically) but this is NOT what you see if you do the experiment at home with a normal spring.
It's fairly easy to calculate the number of coils per unit length at each point along the spring's length as a function of the spring's stiffness at each point. If the spring has the same stiffness all along its length, the coils will be evenly spaced. If it doesn't, they won't be.

I conclude that your spring is not uniform along its length.

Suppose you take a steel spring of given length and number of coils/unit length.
The linear mass density of the spring is the number of coils per unit length.
Now if you stretch the spring then the number of coils per Δχ is not the same everywhere in the spring. (There are more coils per unit length at the wall than at the other end which means the linear density drops)
In other words the spring appears more stretched at the other end than at the end attached to the wall.
I have included a picture which shows that a stretched spring has the same number of coils per length everywhere (maybe this is idealized but still has to be explained theoretically) but this is NOT what you see if you do the experiment at home with a normal spring.
Flip your spring around and do the experiment again. Is the spring density now higher away from the wall?

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The spring density is again bigger at the wall even if i flip it around.
The spring has the same density when it is not stretched and it doesn't matter which side i stick to the wall..

I think you're pulling our legs. Are you keeping the spring still? How can a still hand be different than a still wall?

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AlephZero
Homework Helper
it is clear to me that the mass density of the spring becomes smaller as we get from the wall to the other end (after i have streched it).
Please explain why "it is clear to you".

The simplest explanation for all this is "you believe something which is not true", but unless you tell us WHY you think it is true, we can't explain why you are wrong.

Is the spring vertical, by any chance?

Nugatory
Mentor
Is the spring vertical, by any chance?
Y'know, I had been wondering the same thing
although the original attachment shows the springs drawn horizontally....

the spring is not vertical but horizontal and my hand stretching the spring does not move..

What dauto said how does the spring know the difference between a still hand and the wall is true, meaning it shouldnt know... Although if you do the experiment for example with a rubber band of a given color ( blue) then the side near to the wall is darker than the side of your hand ( because it is more dense) ....

Simon Bridge
Homework Helper
A real spring is well modeled by a series of small ideal massless spring-elements connecting idealized small mass-elements. A FBD is drawn for each mass etc.

A horizontal real spring is (usually) supported by a surface which will add drag when the spring is extended slightly. This can produce a varied coil density with horizontal distance. Perhaps in the home experiment
mkarydas had the spring lying on the ground or a table-top?

If it is supported only at the ends, then the spring will form a catenary - will will also show a changing coil density with horizontal length.

If the spring is suspended vertically, then the total mass below the higher spring elements is greater - resulting in a greater extension the farther up you go. (I'm looking for an unambiguous picture of this online btw.)

But if we define mass-per-unit-length in terms of the path around the coils, that is a different story.
It seems that this was not what OP intended by linear mass density. <phew>.

Breaking the spring into small elements should be able to deal with the case that there are no drag or weight forces acting on it - but this is not a situation you can easily test at home.

AlephZero
Homework Helper
If the spring is suspended vertically, then the total mass below the higher spring elements is greater - resulting in a greater extension the farther up you go. (I'm looking for an unambiguous picture of this online btw.)
Just the first few seconds. Apart from the fact that they have a computer simulation of something, I've no idea what the rest of the video is about.

This effect doesn't show up for most real springs, because the tension required to stretch them significantly is much greater than their weight.

Simon Bridge