# Imagine You're Shrunk Down to a Size of a Nickel

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Hi,

My professor introduced this question (attached files) to us a couple weeks ago and gave some brief explanation saying that even though our mass will reduce our muscles should be relatively stronger than our bigger form and we should be able to simply jump out of the blender.
I was wondering if anyone could give a more detailed explanation about which physical law does he base this claim.

Thank you !

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BvU
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2019 Award
Hello Daniel,

Interesting challenge. Not meant to be delegated to others, but something for you to think about. PF culture wants you to make a start and he we'll help you with questions like
"If I'm shrunk by a factor of 100, what will be my length, weight, lung capacity, etc ?" and "How does that relate to my jumping capabilities ?"

hilbert2
Gold Member
Suppose a mini human of mass $m$ is able to give itself an upward kinetic energy of $K = Cm$ by jumping. The $C$ is a constant with units Joule/kilogram. This should be reasonable as the mass of the muscles is proportional to the mass of the whole person. If you equate $K = mgh$, where $m$ is the same mass, $g$ is the gravitational acceleration and $h$ is the maximum height that can be reached by jumping, what happens to the relative magnitude of $h$ compared to the length of the mini human when mass $m$ is decreased?

sophiecentaur
Gold Member
Think about elephants and fleas then do some googling for ideas.
which physical law
Any relevant Laws need to be applied with a bit of poetic licence and open-mindedness. If you look at small animals that jump, they are not built with the same proportions that you are. What's different about them?

Dale
Mentor
Think about what determines a muscle’s strength. Hint: it is not its mass.

BvU
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sophiecentaur
Gold Member
I guess the answer wasn't presented on a plate and he's gone elsewhere.

I guess the answer wasn't presented on a plate and he's gone elsewhere.
Haha no sir. I just couldn't read the comments right away. I also didn't expect you guys to reply so fast !

Think about elephants and fleas then do some googling for ideas.

Any relevant Laws need to be applied with a bit of poetic licence and open-mindedness. If you look at small animals that jump, they are not built with the same proportions that you are. What's different about them?
Well if we look on grasshopper so obviously its legs are way longer than the rest of the body. But in the problem I posted obviously the body doesn't change its structure but only gets shrunk.

Hello Daniel,

Interesting challenge. Not meant to be delegated to others, but something for you to think about. PF culture wants you to make a start and he we'll help you with questions like
"If I'm shrunk by a factor of 100, what will be my length, weight, lung capacity, etc ?" and "How does that relate to my jumping capabilities ?"
Thank you very much !
I'm sorry I guess I missed 'PD culture' thread. I'll make sure to respect the culture from now on.

Suppose a mini human of mass $m$ is able to give itself an upward kinetic energy of $K = Cm$ by jumping. The $C$ is a constant with units Joule/kilogram. This should be reasonable as the mass of the muscles is proportional to the mass of the whole person. If you equate $K = mgh$, where $m$ is the same mass, $g$ is the gravitational acceleration and $h$ is the maximum height that can be reached by jumping, what happens to the relative magnitude of $h$ compared to the length of the mini human when mass $m$ is decreased?
If we're saying that $Cm = mgh$ and C is constant, so we get that $h = g/C$
So you're saying that the maximum height doesn't change ?

But are you sure we can assume that C is constant? isn't that as we get smaller, it's not necessarily that the relationship between the kinetic energy produced by our muscles (K) and our mass (m) would be linear ?

Think about what determines a muscle’s strength. Hint: it is not its mass.
From 9 years of doing track and field I learned that the muscle strength depends on the quality of the muscle fibers.
But I just googled and read this article that says that a muscle strength depends on the cross-sectional area. Is that what you mean ?
If that so then how does it make sense?
say that my muscle width and depth are:
$10 cm x 15 cm = 150 cm^2$,
if I get shrunk to a size of a nickel my muscles dimension will be about:
$1 mm x 1.5 mm = 1.5 mm^2$

so it's a way small decrease than a linear one.

Dale
Mentor
But I just googled and read this article that says that a muscle strength depends on the cross-sectional area. Is that what you mean ?
Yes, exactly.

say that my muscle width and depth are:
10cmx15cm=150cm210cmx15cm=150cm210 cm x 15 cm = 150 cm^2,
if I get shrunk to a size of a nickel my muscles dimension will be about:
1mmx1.5mm=1.5mm21mmx1.5mm=1.5mm21 mm x 1.5 mm = 1.5 mm^2

so it's a way small decrease than a linear one.
Yes, it is way smaller than linear, in fact it is proportional to x^2. Another way of saying it is to say that your muscle's cross sectional area (strength) scales by the square of the length.

Now, how does your weight scale? Is it proportional to x, or to x^2, or to some other power of x? Now, compare those two scaling factors. How does your strength/weight scale? If you could lift 1x your body weight now, then how many times your body weight could you lift when shrunk?

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sophiecentaur
Gold Member
Well if we look on grasshopper so obviously its legs are way longer than the rest of the body. But in the problem I posted obviously the body doesn't change its structure but only gets shrunk.
So you are getting what I mean. If your teacher is any good, he/she will appreciate plenty of talking around the topic and if you give reasons why is could / couldn't work and, if there are any marks to be earned, you need to give a full explanation of how you are thinking.
But the bottom line of all these scenarios is that 'things' don't scale in a simple manner. For a start, if you were that size, you would need to be eating vastly more just to stay at the right temperature. At school, we were asked why there are no mice near the North Pole but there are polar bears and that stimulated a good discussion.
It's the discussion rather than the answer that counts here.

hilbert2
Gold Member
If we're saying that $Cm = mgh$ and C is constant, so we get that $h = g/C$
So you're saying that the maximum height doesn't change ?

But are you sure we can assume that C is constant? isn't that as we get smaller, it's not necessarily that the relationship between the kinetic energy produced by our muscles (K) and our mass (m) would be linear ?
That's how I though about it. I would guess that the capacity of a muscle to hold adenosine triphosphate or whatever nutrients it can immediately convert to mechanical energy is quite linearly proportional to the mass. I'm not sure about it, though.

I remember a mathematical biology course back when I was an undergraduate that handled these problems of scaling when applied to living organisms. For instance, a human scaled to the size of an ant couldn't bathe because the surface tension of water would prevent getting immersed in it. For the same reason, reading a book scaled to the same relative size would be impossible because the pages would adhere together with too much force for the ant-sized human to open it.

sophiecentaur