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What is this effect called?

  1. Mar 3, 2014 #1
    Hi guys,

    I am not a physicist or a mathematician but I have a question relating to physics. I am trying to find a simple word to describe an effect so I can research it a bit more. As it stands at the moment, I don't know what to search for in Google.

    Imaging a small circle, and then it begins to expands rapidly. But the bigger it gets the slower its growth is.

    In physics, what is this called? Please keep it simple as I said I am not a physicist.

    Thanks in advance
  2. jcsd
  3. Mar 3, 2014 #2
    Growth of what? Surface area of the disk? Length of the circle? Curvature?
  4. Mar 3, 2014 #3
    hmm I don't know.

    An exploding star maybe. It explodes and rapidly expands outwards and then the expansion slows down as it gets larger and farther out. Is that a good way to describe what I am after?
  5. Mar 3, 2014 #4
    Or you have a balloon being filled with helium and the gas going in remains constant. It begins to inflate fast and then slows down as it gets bigger?

    I'm not looking for a specific object. Just a general term to describe this sort of rapid expansion and then slow down
  6. Mar 3, 2014 #5
    Hmmm. There's nothing to describe it. It's best explained mathematically as the relationship between volume and radius for a sphere.

    We typically know that the volume of a sphere, V = 4πR3/3


    R = 3√(3V/4π)

    Which looks something like this


    As you can see, the radius increases at a lesser rate as the volume gets higher.There's no special name for that, other than it being a cubic relation between volume and radius.

    I know you were talking of a more general example, but it's all the same kind of idea.

    The only alternative is when you consider something that expands to a certain point but then slows down such that it never actually exceeds that point, but it also never stops getting closer and closer to that point. This is called 'asymptotic' and it looks a bit like this (note that the blue line will never cross the red line, no matter how far you go along the x-axis):

    http://ajillgalloway.com/wp-content/uploads/2010/04/paperless-asymptote-286x300.png [Broken]

    However an expanding balloon/star/object/whatever is not asymptotic, it is the cubic relation I already mentioned, which goes on forever.
    Last edited by a moderator: May 6, 2017
  7. Mar 3, 2014 #6
    Are you referring to something like exponential decay?
  8. Mar 3, 2014 #7


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    What is the origin of this, i.e. where did you get the idea that such a behavior exists AND has a unique name given to it?

  9. Mar 4, 2014 #8
    I think exponential decay might be what I am looking for.

    In what natural occurrences could exponential decay be seen in?
  10. Mar 4, 2014 #9


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    Really? It doesn't fit at all with what you originally described. The rate of decay uniformly slows down, instead of doing that non-monotonic change you illustrated.

    Radioactive decay.

  11. Mar 4, 2014 #10
    This is a nebulous ill defined phenomenon we have here. What makes you think there would be a specific name for that?
  12. Mar 4, 2014 #11
    The topic that seems to be invoked here is that of Related Rates.

    As the radius increase, the relative rate of growth of its area decrease.
    Let A[r] = Pi r^2 be the area function of a circle of radius r.

    Let A'[r] be its derivative.

    I am referering to the function

    A'[r] / A[r] which is monotonically decreasing. Perhaps it is what is being searched.

    Best Wishes,

    Last edited: Mar 4, 2014
  13. Mar 4, 2014 #12
    I'm basically writing a report on finance. To be precise it's to do with growth of a company.

    In the beginning a company can grow faster, but once the company reaches a very large position in its industry, the same growth rate cannot be maintained and hence the growth starts to slow down.

    I am trying to describe this effect using visuals and physics hence the circle description. I'm probably way off but this is what I am trying to achieve.
  14. Mar 4, 2014 #13
    One might call that the "law of diminishing returns"
  15. Mar 4, 2014 #14


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    It appears that this is more of a math question and has very little to no physics.

  16. Mar 4, 2014 #15
    Yeh but I need to use physics to describe it. Like an analogy
  17. Mar 4, 2014 #16
    If you want to say it in a physicsy way, you could say that the rate of change of growth decays exponentially with time. I'm pretty sure that's what you're getting at!

    The graph I showed a few posts ago visualises it quite well. Here it is again:


    I'm reasonably sure this is the point you're trying to make. The y-axis represents the size of the 'thing' (company, balloon, whatever), and the x-axis could be considered as time. Clearly as time goes on, the growth of the company slows.

    You can further visualise it with this:


    , where the y-axis is 'growth rate' and the x-axis is time. Again you can see the growth rate decays exponentially with time.

    I could explain it in a more mathsy way if needs be. It kind of depends on what you know of basic calculus though.

    There's no single word for it, other than to say that the growth rate slows with time, which is surely enough? If you're just looking for a big word to score points with, you won't have any luck!
  18. Mar 4, 2014 #17


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    It can be also related to scaling. Say, if volume grows linearly with time, radius grows as a cube root of time.
  19. Mar 4, 2014 #18


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    That's a bit odd. Usually, when people do an analogy, it has to be something familiar. Physics is usually the one where we explain IT using other things as an analogy. You are doing it the opposite way, trying to explain something in finance using a..... physics example? Is this wise? Are you trying to explain finance to a bunch of physicists?

  20. Mar 4, 2014 #19
    That would be asymptotic growth. A company may grow rapidly from 1% to 80% market share and then continue to grow indefinitely without ever achieving 100% market share.
  21. Mar 4, 2014 #20
    It's a mental visualisation exercise. Using an animation to show the growth of a company. I wanted to use an animation of an exploding star or something similar to show how a company grows rapidly and then growth or expansion slows down once it reaches a certain large enough point.

    SA1988 thank you for that, I think that is what I am looking for. But rather than having an exponential growth shown on a chart like that, I want to use like an animation of some kind, such as a circle growing rapidly and then slowing down as it gets larger
  22. Mar 4, 2014 #21
  23. Mar 4, 2014 #22
    I owe you one! Thank you so much
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