How Does Particle P Behave When Suspended on an Elastic String in SHM?

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    Elastic Gravity Shm
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Discussion Overview

The discussion revolves around the behavior of a particle suspended on an elastic string undergoing simple harmonic motion (SHM). Participants explore the conditions under which the motion is harmonic, the effects of the string's elasticity, and the transition between different states of motion as the particle moves through various positions relative to the equilibrium point.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the motion of particle P will be simple harmonic until it reaches the equilibrium point, questioning what happens afterward when the string becomes slack.
  • Another participant clarifies that the equilibrium position is where the weight of the particle balances the tension in the string, indicating that oscillation occurs harmonically past this point.
  • A different viewpoint introduces the idea that positive tension must be maintained for SHM to occur, proposing that the string behaves semi-Hookean and that SHM is valid as long as the displacement is less than a certain threshold.
  • Concerns are raised about the conditions under which the motion through different displacements mirrors SHM, with one participant expressing confusion about the equivalence of motion through different distances.
  • Participants discuss the need for an equation of motion to analyze the system, with one expressing uncertainty about how to derive it without a clear mathematical formulation.
  • There is a mention of a specific problem involving the speed of the particle as it transitions through different states, with references to energy conservation principles being used to derive results.

Areas of Agreement / Disagreement

Participants express various interpretations of the motion of the particle, with some agreeing on the conditions for SHM while others question the assumptions and equivalences presented. The discussion remains unresolved regarding the precise nature of the motion when the string becomes slack and the implications of different displacement scenarios.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the string's behavior, the definitions of displacement, and the mathematical steps required to fully model the motion. The participants do not reach a consensus on these aspects.

pc0019
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My first post here! I signed up a few minutes to correct someone... then deleted my post when I realized I was on about something entirely different. Ahem.

Anyway, have a look at this diagram:

http://img2.freeimagehosting.net/uploads/c8911ecf7e.jpg

Particle P is suspended on an elastic string fixed at O, natural length = 'l', equilibium point is ( l + d ) below O, and 'x' is the distance the particle has been pulled down to.

Correct me if this is wrong. If you release P its motion will be simple harmonic, at least until it reaches the equilibrium point. But what happens to its motion after that? Once " OP < l ", its motion is like a normal particle under gravity but how does it move when " l < OP < l + d "?

Thanks in advance!
 
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Welcome to PF!

pc0019 said:
Correct me if this is wrong. If you release P its motion will be simple harmonic, at least until it reaches the equilibrium point. But what happens to its motion after that? Once " OP < l ", its motion is like a normal particle under gravity but how does it move when " l < OP < l + d "?

Hi pc0019! Welcome to PF! :smile:

The equilibrium position is where the string is stretched, but the weight of the particle (particle?) exactly balances the tension in the string.

The equations are exactly the same on both sides of the equilibrium position (for OP > l, of course).

So it oscillates harmonically up and down past the equilibrium position. :smile:
 
Interesting. Being an elastic string and not a rigid spring, I'd think that positive tension must be maintained in order to get SHM. Assuming the string is semi-Hookean (restoring force proportional to positive displacement), you'd get SHM as long as x < d. (It will oscillate about the equilibrium point, as tiny-tim says.)
 
Thanks, tiny-tim!

OK, if I understand both of you correctly, the particle (A level maths student :"]) will act simple-harmonically when " l < OP < l + d + x ", ie motion through 'd' mirrors the motion through 'x' until (x - d) and if d = x, the motion of the particle would be exactly the same as a spring on a frictionless horizontal surface?

I think I must have got something horribly wrong because that doesn't add up (metaphorically speaking)? I was thinking that motion through 'd' would be similair to, but not precisely SHM.

Thanks for the link btw Doc Al :smile:
 
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pc0019 said:
Thanks, tiny-tim!

OK, if I understand both of you correctly, the particle (A level maths student :"]) will act simple-harmonically when " l < OP < l + d + x ", ie motion through 'd' mirrors the motion through 'x' until (x - d) and if d = x, the motion of the particle would be exactly the same as a spring on a frictionless horizontal surface?

I think I must have got something horribly wrong because that doesn't add up (metaphorically speaking)? I was thinking that motion through 'd' would be similair to, but not precisely SHM.

Thanks for the link btw Doc Al :smile:

(oh, it's a maths student on the end of the string … that makes more sense! :smile:)

I'm confused … what doesn't add up? :confused:

Have you written out the equation of motion? :smile:
 
tiny-tim said:
(oh, it's a maths student on the end of the string … that makes more sense! :smile:)


Well it will be soon, what with exams and all... :wink:

What I don't understand, is how my statement "if d = x, the motion of the particle would be exactly the same as a spring on a frictionless horizontal surface" could be true.

I don't know what you mean by 'equation of motion'. This is conjecture, although the question that got me thinking asked for the time between "OP = l + d + x ", to the point where the string becomes 'slack' - when "OP = l" I think...
 
pc0019 said:
What I don't understand, is how my statement "if d = x, the motion of the particle would be exactly the same as a spring on a frictionless horizontal surface" could be true.

Sorry … still can't see anything surprising about that. :confused:
I don't know what you mean by 'equation of motion'.

I mean an equation relating acceleration to forces and distances.

Or an equation beginning x'' = …

How will you solve this without an equation?
… the question that got me thinking asked for the time between "OP = l + d + x ", to the point where the string becomes 'slack' - when "OP = l" I think...

oh, you didn't mention that x > d.

But you still need an equation. :smile:
 
Sorry, the question actually asks for the speed, not the time.

tiny-tim said:
oh, you didn't mention that x > d.

x is not necceserally greater than d, although in the question it is. 'x' isn't displacement. I am using T = mg = [λ (x + d)]/l

So, for x > d, this is happening:

http ://img59.imageshack.us/img59/3192/shm2qa4.jpg

Where, when the displacement is 0, P is at the equilibrium position, and the blue lines are when the string is slack and P moves freely under gravity. And where x = d there is full SHM, and where x < d, chop off the graph somewhere above the equilibrium position and stick it together. To use the proper technical terms. Sorry if I am repeating myself but I want to make sure I am understood correctly.


In the question, the modulus of elasticity = 4g ~ 39.2, l = 0.8, d = 0.05, x-max = 0.1, T = (pi/7), m = 0.25.

Using initial energy = final energy, EPE(0.55125) = GPE(0.3675) + KE, therefore v = 0.7root3.

As such I don't need to model the motion, but I thought it would be useful to know.
 
pc0019 said:
Sorry, the question actually asks for the speed, not the time.
…
As such I don't need to model the motion, but I thought it would be useful to know.

If you're only asked for the speed, then just use the energy equation (which I think you have done).
 

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