A weird question about scale.

[Vertigo]

I have question about scaling as a kind of thought experiment.

Let's say I have a guy, let's call him average Joe: He is 1.80m tall, weights 80kg and can lift 80kg

now I somehow scale him down to 1/2 size (and let's ignore all those pesky biological things that could go wrong)

So, now I have half-size Joe: He is 90cm tall, weights 10kg, and I'm not sure how much he could lift, or how his strength would scale.

I would argue that he could lift around 20kg, because his new muscles would have 1/4 of the diameter of average Joe's one.
They would also only be half the length, but I don't know if and how that figures in somehow.
I'm not a Biologist, but maybe someone here knows about Biology, muscle scaling or small scale humans.

Thanks in advance!

[Mastikator]

Classic Square-Cube law case. https://en.wikipedia.org/wiki/Square%E2%80%93cube_law

Your reasoning is correct. This is why ants can lift many times their body weight and elephants can't. If you scaled up an ant to the size of an elephant it wouldn't be able to lift itself up. It would also explode from overheating.

[Lord Torath]

I suspect this is where the Square-Cube law rears it's ugly head. In general, smaller organisms can lift a larger percentage of their mass than larger creatures. Most ants can lift ~10 times their mass. Most humans can lift maybe half their weight. Having done none of the research or math, my guess is that being scaled down to half size will result in more than one-eighth strength.

And ninja'd.

[Lvl 2 Expert]

One of the fun things about the square cube law is that some things also work out to pretty much neutral. For instance: animals good at jumping all the way from a flee to a tiger can jump to roughly the same absolute height, raising their center of gravity around 2 meters or so from its regular position. As an animal scales up their weight goes up with a third power function, their strength only by the second power. But jump height is determined by leg length as well, as it determines how much distance and time you have over which the muscles can apply force. And a first power times a second power is a third power. Run speed is similar to that as well, though maybe not as cleanly demonstrable. But the speeds of animals with different sizes but similar builds tend not to be extremely far apart.

[SirKazum]

One of the fun things about the square cube law is that some things also work out to pretty much neutral. For instance: animals good at jumping all the way from a flee to a tiger can jump to roughly the same absolute height, raising their center of gravity around 2 meters or so from its regular position. As an animal scales up their weight goes up with a third power function, their strength only by the second power. But jump height is determined by leg length as well, as it determines how much distance and time you have over which the muscles can apply force. And a first power times a second power is a third power. Run speed is similar to that as well, though maybe not as cleanly demonstrable. But the speeds of animals with different sizes but similar builds tend not to be extremely far apart.

I've read that another one such scale-invariant thing is how much time it takes an animal to empty its bladder (at least comparing mammals, of course). I've tried figuring out why, not quite sure though.

[Rockphed]

I've read that another one such scale-invariant thing is how much time it takes an animal to empty its bladder (at least comparing mammals, of course). I've tried figuring out why, not quite sure though.

Specifically, male dogs, humans, and elephants all take about twenty seconds to empty a full bladder.

[Lvl 2 Expert]

animals good at jumping all the way from a flee to a tiger can jump to roughly the same absolute height,

To expand on this: one might figure that falling is like jumping in reverse, so animals of all sizes should be about equally good at breaking a fall, at least as long as they land on their feet. This is not true in an environment with air, because speeding up is done by mass, a third power function, while slowing down is done by air drag, a second power function. So larger animals land harder. But in a vacuum, would this work? If I found the height from which you need to drop a tiger to usually kill it, and I dropped a cat or a mouse from that height in a vacuum chamber (before they suffocated or got too panicked to land, I can tell this is going to be a tricky experiment to perform), whould they actually die?

I've read that another one such scale-invariant thing is how much time it takes an animal to empty its bladder (at least comparing mammals, of course). I've tried figuring out why, not quite sure though.

I... can't really explain that one either.

Bladder volume is 3rd degree, the size of the hole is 2nd degree (although honestly larger animals could just have a relatively larger hole, there's nothing really stopping the anatomy from changing here), so assuming the relative same hole size and stuff we're missing a first degree in favor or larger animals somewhere. Maybe larger animals build more pressure? The thickness of the bladder wall is a frist degree function? It shouldn't be the extra gravity right?

[tiornys]

It shouldn't be the extra gravity right?

Apparently, it more or less is gravity. More specifically, larger bladders have higher flow rates in part thanks to the extra pressure contributed by the increased weight of the urine (and in part because of the increased cross section of the urethra, of course). Source: this video

[HouseRules]

Specifically, male dogs, humans, and elephants all take about twenty seconds to empty a full bladder.

Yet I take 1 minute to empty my bladder; so I'm an exception.

[Khedrac]

I have question about scaling as a kind of thought experiment.

Let's say I have a guy, let's call him average Joe: He is 1.80m tall, weights 80kg and can lift 80kg

now I somehow scale him down to 1/2 size (and let's ignore all those pesky biological things that could go wrong)

So, now I have half-size Joe: He is 90cm tall, weights 10kg, and I'm not sure how much he could lift, or how his strength would scale.

I would argue that he could lift around 20kg, because his new muscles would have 1/4 of the diameter of average Joe's one.
They would also only be half the length, but I don't know if and how that figures in somehow.
I'm not a Biologist, but maybe someone here knows about Biology, muscle scaling or small scale humans.

Thanks in advance!

One of the problems with your question is actually the bolded part, despite how necessary it is! It's those pesky biological things which could go wrong that determine things like how his strength would scale.

Consider, humans and chimpanzees are very very closely related from a biological viewpoint, but an adult chimp is far stronger in terms of limb muscle strength than a human is because it has a very difference balance of muscle types.
(My memory suggests that human muscles seem to be optimised for long duration low intensity use, making us very unusual among primates.)
This means you best guide here probably is the square-cube law with quite a few "buts" thrown in.
First up - comparison with animals of the same size is a lot less useful unless you know they have the same type of muscles (fibre types, density etc.).
Next, are you shrinking the existing muscle fibres or reducing the volume so the individual fibres are the same size?
The tensile strength of the materials involved has a bigger effect the smaller you go - one of the limiting factors in lifting is not tearing our muscles. If the numbers are smaller, the tensile strength limit is proportionally larger allowing small creatures to life larger amounts proportionally than we can, if you shrink the materials then the tensile strength probably also shrinks proportionally keeping the ceiling on lifting the same relative to bodyweight.
And so forth.

If you want to ignore all the biological details totally then work out what proportion of their mass the person can lift at full size, then apply the same mass:lift ratio at their reduced size (mass will probably use the square-cube rule without other factors, so at half size they are 1/8 volume therefore 1/8 mass and lift 1/8 amount.
That or allow them to lift their usual limit because *magic*.

As for the time to empty the bladder, this probably depends more on evolution than on size - if most mammals take 20 seconds that is probably the same limit for lightening the load before running from predators.

[Radar]

To expand on this: one might figure that falling is like jumping in reverse, so animals of all sizes should be about equally good at breaking a fall, at least as long as they land on their feet. This is not true in an environment with air, because speeding up is done by mass, a third power function, while slowing down is done by air drag, a second power function. So larger animals land harder. But in a vacuum, would this work? If I found the height from which you need to drop a tiger to usually kill it, and I dropped a cat or a mouse from that height in a vacuum chamber (before they suffocated or got too panicked to land, I can tell this is going to be a tricky experiment to perform), whould they actually die?

Not quite as this is much more dependent on structural integrity than air slowing down the fall. Terminal velocity in air for a cat is about 100 km/h and for a human about 200 km/h assuming a spread-out position. A 15 m fall results in a final velocity of about 60 km/h before applying air drag, which will not change the result significantly. Such a fall could be fatal for a human and is bound to result in severe injuries unless the landing is properly cushioned while cats most often will be completely unharmed unless they cannot position themselves correctly for the landing.

Basically, the larger the animal, the closer the skeleton is to its stress limit even under normal use (there is a reason elephants do not jump and their runing looks more like intense marching). Therefore it is less resistant to sudden shocks. Additionally, the pressures involved in hard landing scale linearly with size.

[halfeye]

Not quite as this is much more dependent on structural integrity than air slowing down the fall. Terminal velocity in air for a cat is about 100 km/h and for a human about 200 km/h assuming a spread-out position. A 15 m fall results in a final velocity of about 60 km/h before applying air drag, which will not change the result significantly. Such a fall could be fatal for a human and is bound to result in severe injuries unless the landing is properly cushioned while cats most often will be completely unharmed unless they cannot position themselves correctly for the landing.

Air resistance kicks in long before terminal velocity, that cat would be a lot faster in a vacuum. OTOH, I'm by no means convinced that a (35mph?) impact is always fatal for a human.

Basically, the larger the animal, the closer the skeleton is to its stress limit even under normal use (there is a reason elephants do not jump and their runing looks more like intense marching). Therefore it is less resistant to sudden shocks. Additionally, the pressures involved in hard landing scale linearly with size.

The "in a vacuum" difference is a very big one. A feather would fall as fast as a stone in a vacuum, it's a big difference, mice may fall slow enough to survive from all heights, but that's only in air, in a vacuum they'd die like humans at the height that humans would (yeah, they'd probably suffocate first).

[Radar]

Air resistance kicks in long before terminal velocity, that cat would be a lot faster in a vacuum. OTOH, I'm by no means convinced that a (35mph?) impact is always fatal for a human.

I did not say it is always fatal - only that it could be and it would surely result in severe injuries. While air drag does kick in, it has little influence on the final velocity of a 15 m fall. Without air, it would be 38 mph while with air it would be 31 mph for a cat and 34.5 for a human. Noticeable, but hardly important for the outcome.

The "in a vacuum" difference is a very big one. A feather would fall as fast as a stone in a vacuum, it's a big difference, mice may fall slow enough to survive from all heights, but that's only in air, in a vacuum they'd die like humans at the height that humans would (yeah, they'd probably suffocate first).

For a creature size of a mouse it does make a bigger difference, but still it is not correct to assume it would be as fragile as a human. An impact at the same speed results in very different pressures lasting different time depending on the mass of the falling object. To make things simpler and far less gruesome: consider concrete blocks of different sizes and think about the height you can drop them from before they start cracking. Size matters a lot.

[halfeye]

I did not say it is always fatal - only that it could be and it would surely result in severe injuries. While air drag does kick in, it has little influence on the final velocity of a 15 m fall. Without air, it would be 38 mph while with air it would be 31 mph for a cat and 34.5 for a human. Noticeable, but hardly important for the outcome.

Where did you get the figures? they look plausible and impressive, but I certainly couldn't get them myself. If they are right however, I suspect they might be significant, it's the last straw that proverbially broke the camels back.

For a creature size of a mouse it does make a bigger difference, but still it is not correct to assume it would be as fragile as a human. An impact at the same speed results in very different pressures lasting different time depending on the mass of the falling object. To make things simpler and far less gruesome: consider concrete blocks of different sizes and think about the height you can drop them from before they start cracking. Size matters a lot.

Concrete blocks are also affected by air resistance, I don't know whether that test has been performed in a vacuum.

What the objects are landing on also matters, ski jumpers land without dying from ridiculous heights, stacked mattresses can break a fall, solid concrete or granite is much less forgiving.

[Lord Torath]

Where did you get the figures? they look plausible and impressive, but I certainly couldn't get them myself. If they are right however, I suspect they might be significant, it's the last straw that proverbially broke the camels back.

Aerodynamics, which covers air drag, and physics, which covers acceleration and buoyant forces.

What the objects are landing on also matters, ski jumpers land without dying from ridiculous heights, stacked mattresses can break a fall, solid concrete or granite is much less forgiving.

What's important for ski jumpers is that they land on a 'slippery' slope. When they hit the ground, the vertical component of their velocity doesn't instantly go to zero, but is directed down the hill. When they train in the summer, they land on the same hills, but instead of snow/ice, they land on plastic bristles that let them keep sliding down the hill as they land. They can then reduce their velocity (both horizontal and vertical components) more slowly. If they overshot the landing hill and landed on flat ground, even on skis in the winter, they'd be seriously injured if not dead.

It's not the fall that kills you; it's the sudden stop at the end. Landing on a pile of mattresses brings you slowly to a stop, as the mattresses compress under the impact. All your kinetic energy needs to go somewhere, and the more you can spread that out over time, the better your chances of survival. Ski-jumpers spread it out over the 5-10 seconds between landing and coming to a stop. The stack of mattresses spreads it out over 1-2 seconds. The concrete compresses it all into about a 1/10 of a second. Guess which one is hardest on your skeleton.

As far as big vs small, mass goes as the cube of length, so someone who is half as tall, wide, and deep has 1/8th the mass. Surface area (which turns force into stress) goes by square of length, so the bigger person has 8 times the mass and 4 times the surface area of the smaller person. If they both hit the ground at the same speed, the smaller person's head exerts half as much stress on his skeleton than the larger person's head. For the cat vs the human, that's going to be much more significant than the difference between (31 m/s)² and (34.3 m/s)².

[Radar]

Where did you get the figures? they look plausible and impressive, but I certainly couldn't get them myself. If they are right however, I suspect they might be significant, it's the last straw that proverbially broke the camels back.

Newtonian mechanics plus simplistic assumption of linear air drag, which quite possibly results in an overestimate of how important air drag is. I got the air drag coefficient estimate from terminal velocity (as in that case drag cancels out gravity force). After that this is just a linear second order differential equation to solve.

Concrete blocks are also affected by air resistance, I don't know whether that test has been performed in a vacuum.

It matters even less than for living beings. At any rate, the difference in critical impact speed is the most important. Why there is a difference, was explained by Lord Torath.

[Lvl 2 Expert]

As far as big vs small, mass goes as the cube of length, so someone who is half as tall, wide, and deep has 1/8th the mass. Surface area (which turns force into stress) goes by square of length, so the bigger person has 8 times the mass and 4 times the surface area of the smaller person. If they both hit the ground at the same speed, the smaller person's head exerts half as much stress on his skeleton than the larger person's head. For the cat vs the human, that's going to be much more significant than the difference between (31 m/s)² and (34.3 m/s)².

But that's assuming they both absorb the impact over the same depth/length/whatever.

But say they land on their legs, and bend their legs as they do. The larger person will have a two times longer distance over which to absorb the impact, as their legs are twice as long.

Even if they don't land on their legs, the larger person should have a layer of muscle and fat twice as thick as the smaller person that should have the same relative effect.

Now, from personal experience I do feel like I actually did get worse at landing from higher places since I was maybe 12 to 14 years old or so. I am not sure that's due to overall size increase though, it could easily be a function of length to weight ratio, or other biological changes related to either puberty, getting older or eating too much. (Or me misremembering things and being a wimp now?)

[Rockphed]

(Or me misremembering things and being a wimp now?)

Given how my children react to falls that knock the wind out of them, I am going to go with this explanation. Well, that it is a psychological effect of being older and valuing lack-of-pain more.