Damped harmonic motion with one end without weight free

[nomadreid]

Homework Statement

A block on a horizontal surface is attached to two springs whose other ends are fixed to walls. A light string attached to one side of the block initially lies straight across the surface. The other end of the string is free to move. There is significant friction between the block and the surface but negligible friction between the surface and the string. The block is displaced in the direction of one of the walls and released from rest. What is the shape of the string a short time later?
That is: a top view is:
Wall
Spring
Weight_______________________ (string)
Spring
Weight

This is a conceptual, not a numerical, question, so that the choices would be the string's amplitude constant, decreasing from left to right, increasing from left to right, or first increasing then decreasing.

Homework Equations

Since it is a conceptual problem, I don't think that the equations are relevant.

The Attempt at a Solution

It is clear that the amplitude of the block's motion decreases with time, and so as the wave travels from left to right, so should its amplitude. This therefore would, in my view, give the form of the string's amplitude increasing from left to right. The fly in the ointment comes from the arrangement with an open-ended string. All treatments of damped harmonic motion that I find are either dealing with the amplitude of the block, or of a string with one end fixed (or at least not free: having a weight at the end). Therefore I am not sure whether this might cause the end configuration of the string to have an amplitude that first increases then decreases.

 

[Matterwave]

Imagine you attached the string to another weight at the end of it. What do you think the string will do? (You can look at the treatments where this weight is there.)

Imagine you made the weight very light, and eventually try to imagine that weight approaching 0, how does this change your answer?

 

[nomadreid]

Thank you, Matterwave.

Imagine you attached the string to another weight at the end of it. What do you think the string will do? (You can look at the treatments where this weight is there.)

I would like to look at treatments where another weight (call it W) is on the other end (the right-hand side of my drawing) , but all the treatments I find only concentrate on graphing the position/velocity/acceleration of the block (on the left hand side) itself. I do not find a treatment whereby a string is attached perpendicularly. If you could give me a link, that would be helpful.

To be consistent with the conditions of the original question, W would have to have negligible friction with the surface. (So a treatment where W is fixed would not be relevant here.)

However, the choices for the problem (which I did not include) do show the entire length of string having been affected, although it is ambiguous whether this is just an initial section of the string. Since the question stated that we are looking at the string a short time after, I would assume that the traveling wave would not have had time to reflect off of W, so we can discount interference from a reflected wave. I would like to discount the case where the wave reaches W and hence moves it, hence transferring some of the energy of the wave to kinetic energy of W, hence damping it the wave that transferred the energy, but on the other hand increasing to amplitude of the right-hand end. This case would argue for the scenario of decreasing-increasing-decreasing amplitude with a sudden "tail" at the right-hand side (but none of the choices give this isolated "tail"). I am not sure whether I can discount this case given the ambiguous statement "a short time after" combined with the ambiguous drawings. If I could discount it, then this would argue for the scenario of a decreasing amplitude from right to left, that is increasing from left to right.

Imagine you made the weight very light, and eventually try to imagine that weight approaching 0, how does this change your answer?

As the mass of W decreased, then if the wave did reach it, the motion of W would increase, which would argue for an even greater "tail" as just outlined. Since there was no such sudden "tail", I would assume that the wave did not reach W.
As W hit 0, then this would simply be an increased "tail" (if the wave reached it) or simply be irrelevant (if the wave did not reach it).

However, my uncertainty about my analysis is the reason for which I am turning to this forum. (If it's any help, although this is in the "homework problem" rubric, it is not an assigned homework (I am not a student), and will not be turned in anywhere. So you don't have to worry that you are unethically "doing my homework for me".) I will be grateful for any further help.

 

[Matterwave]

1) I thought you had some sources that talked about what would happen if a mass was attached at the other end. I have no such sources, sorry. If you have no such sources, actually this analogy might not be a great one to solve this problem from. I assumed you had a source, so you could take a simple limit.

2) You can't assume that the string doesn't transfer energy to the weight. Energy transfer is what waves do! Waves transfer energy from one side to the other. If there is really no friction on the string, this energy transfer should be actually quite efficient shouldn't it?

3) Since you don't have a source for a problem with a weight attached, maybe let's just look at this from a different perspective.

What do you think would happen if there were no friction at all? Only the springs pushing on the block, and then the block making the string wave. Where's the energy? Can energy be lost?

By the way, this problem is a somewhat difficult conceptual question. One might try it out for oneself to build some intuition. Take a string, and pretend that your hand is the block. Make some waves and see what happens. Of course, in this case, the string WILL have friction with the ground.

 

[nomadreid]

The answer to this question on the site from which I took this from is a simple sinusoidal wave with amplitude increasing as the string got further away from the block, and I can envision this arrangement if one does not worry about what happens to the energy at the end of the string. So if we are talking only about an initial section of the string, then this answer seems reasonable. But if the energy gets to the end and there is no friction there, this means that the energy would not be able to be transferred, and this would be the point I was wondering about: would this energy then act in the same way as a reflecting wall, traveling in the opposite direction along the string, or would it translate into extra kinetic energy of the end of the string, giving it a sort of tail? Or would the two give the same result?
As you say, an experiment with a real string has not only the problem of real friction but also to simulate a damped harmonic motion generator is not easy with household materials. I just end up with my string all over the place. :-(
As you say, it can be tricky conceptually, which is why I am posing it. If equations would be helpful, I would not be amiss to having them included in an explanation.

 

[Matterwave]

If the string were mass-less as is the case of most treatments, the energy would reflect back towards the block. The difference between an open string and a string attached to a wall is that the waveform that is reflected back if the string were attached to the wall would be upside down, while the wave form for an open string would be reflected right side up (think for a moment of a hump moving down the string). In both cases the energy would be reflected back towards the block.

If the string has finite mass, then certainly there can be a whip-like action at the end of the string happening - kinetic energy will be transferred from one side of the string to the other.

 

[nomadreid]

Thanks, Matterwave. Makes sense.