# How can ants carry items much heavier than themselves?

This morning I saw an ant and suddenly a question came to my mind: how do ants actually carry items much heavier than themselves?

What's the difference (in physics) between us and them?

• But i mentioned ,in physics... I dont want to know about biological details..i want to know about the physics. – Paul Dec 16 '14 at 2:36
• Michal basically got it right. The cool thing about this is, that Galileo's last book "Discourses and Mathematical Demonstrations Relating to Two New Sciences" ("Discorsi e dimostrazioni matematiche, intorno à due nuove scienze"), published in 1638 already contained a very good treatment on the idea of mechanical scaling laws. – CuriousOne Dec 16 '14 at 3:13
• This question appears to be off-topic because it is about entomology and not physics. – Kyle Kanos Dec 16 '14 at 3:22
• This is absolutely a physics question. The only biology you need to know is "ants are small". After that, it's all physics. Keep this open. – Floris Dec 16 '14 at 3:29
• en.wikipedia.org/wiki/Square-cube_law#Biomechanics – Tim S. Dec 16 '14 at 14:49

Strength is proportional to surface area divided by volume, but since volume is directly proportional with mass and I can't get an accurate density (I am guessing approximately both for mass and size.), I will use mass instead.

According to Wolfram Alpha, the average mass of human body is 70 kilograms. The surface area of a person weighing 70 kg with a height of 170 cm is 1.818 square meters. This gives us a weight/surface area ratio of about $0.0385 \frac{kg}{m^2}$.

So now, how much does an ant weigh?

This article provides a variety of different numbers, varying from 1 mg to 60 mg. Since the biggest ants will be soldiers, I assume that the approximation will be slightly smaller than 30 mg. Say 25 mg or 0.000025 kilogram.

Now comes the interesting part. Not Wolfram, not even uncle Google knows the surface area of an ant.

This Britannica page says that ants range from 2 mm to 25 mm. Let's eliminate the soldiers since they are huge. (A big worker would be as long as 8 mm.) That gives an approximation of 5 mm.

I gave animation industry a shot and tried to measure the surface area of this free ant model. The length of the ant is now 0.005 - let's call it meter.

This gives us a surface area of about $4.87\cdot 10^{-5}$, or $0.0000487$ square meters. So an ant that weighs 0.000025 kg with a length of 5 millimeters has a surface are a of about $0.0000487 m^2$. This gives a weight/surface area rate of about $0.5335 \frac{kg}{m^2}$.

So, the uniform strength of an ant is about thirteen times more than a human's.

How much can a human carry while walking a long distance, maybe even climbing? Maximum 20 kilograms for most people. That is slightly more than a quarter of our weight (about 0.28).

How much can an ant carry? About 1 gram - the weight of a leaf, or 40x the weight of an average ant.

4 divided by .28 = 14. So ants are about 14 times stronger than we are. (Carrying capacitiy according to body mass.)

• We presume that the person weighs 70 kg. 1/3 of it makes 20,3 kg.. For mountain climbing, this weight seems quite average to me. – altac bori Dec 16 '14 at 5:20
• I wouldn't consider loads that cause consistent injuries in some of the most well trained and strong individuals of the human species to be representative of "how much can a human carry while walking a long distance" – March Ho Dec 16 '14 at 9:31
• He's ignoring ant soldiers, so he should ignore human soldiers too. – RemcoGerlich Dec 16 '14 at 11:03
• Four times 25mg is 100mg = 0.1g, not 1g. – immibis Dec 17 '14 at 7:55
• Can you provide a reference for your "strength proportional to surface area divided by mass"? That would imply a $strength\propto \frac{1}{r}$ relationship when buckling strength in fact goes as $\frac{1}{r^2}$ - see details in my answer. Can you tell us where your $1/r$ relationship comes from? – Floris Dec 18 '14 at 2:24

I think the answer has less to do with their construction and more to do with their smaller size

Basicly the mass of a object scales as it's size cubed so a ant 10 times the size will be 1000 times heavier. But the strength of an organism depends on the cross sectional area of muscle (I've heard this somewhere, not sure about the details), and hence scales as the size squared. So an ant 10 times the size will only be 100 times stronger.

Putting those two facts together the strength to weight ratio of an organism varies inversely with it's size. Hence smaller organisms even with the same construction will be able to lift more in relation to its mass.

Note: When I say size I'm referring to the linear size of a body as measured with eg a ruler

• Assuming the same muscle material it is pretty obvious that strength is proportional to the cross-sectional area of the muscle. Just imaging putting to muscles next to each other (in parallel). You've clearly doubled both the strength and the cross-sectional area. Good answer btw. – Timmmm Dec 18 '14 at 11:03
• @Timmmm: Just imagine putting muscles next to each other (in parallel). You've clearly doubled the strength, cross-sectional area, mass, and volume. So it's pretty obvious that strength is proportional to, uh, . . . one of those. (As it happens, it is in fact true that strength is proportional to the cross-sectional area. But you should be wary of confirmation bias, of deciding that things are "obvious" because you already know them. The history of science is full of people continuing to accept falsehoods that seemed "obvious" to those who "knew" them.) – ruakh Dec 22 '14 at 7:25
• @ruakh It's obvious that stength is a function of cross-sectional area. I was just using the doubling thing to show that the function is linear. In this case it really is obvious. I'm pretty sure sailers from centuries ago knew if you put two ropes together they could carry twice the load because they're twice as thick (in terms of cross sectional area). – Timmmm Dec 22 '14 at 10:15

Strength / weight is a funny thing. The stress on a long thin rod (like an ant's leg) is limited by the Buckling strength which is given (for rod that can freely rotate at each end) by

$$F = \frac{\pi^2EI}{L^2}$$

where $I$ is the second moment of area which scales with $r^4$ - so

$$F \propto \frac{r^4}{L^2}$$

So when you make an object 2x smaller, the mass is 8x smaller but the strength is only 4x smaller. This means that smaller objects are stronger for their weight.

AFTERTHOUGHTS

Ants have an exoskeleton meaning that their legs derive most of their strength from the outermost part of their body (think "skin as tough as bone"). This makes the "second moment of area" of the support structure much larger than you would expect - see that $r^4$ term above... This is one reason why the skinny legs of the ant are quite so strong - all their strength is on the outside.

Having established that the (exo)skeleton of the ant has greater structural strength, weight for weight, than that of larger species, we still need to address the question of muscle strength. Here we need to look at the surface-to-volume ratio. Doing work with a muscle requires oxygen - which is obtained by exchange of oxygen with the atmosphere. Now if we assume that the volume of muscle scales with the volume of the animal, and thus with $r^3$, and the surface area of the lungs, or spiracles in the case of ants (tubes from the skin to the muscles) scales as $r^2$, then you can see that the "lung to muscle ratio" (LMR) is

$$LMR \propto \frac{1}{r}$$

so the smaller you are, the less likely you are to run out of breath. Even if the lung is a fractal surface with a fractional dimensionality greater than 2, it will be less than 3 and the LMR is still larger for smaller animals. Diffusion of oxygen - same story, because it has much less far to go.

In short- by dint of their size, the structure of an ant is more resistant to buckling; and their metabolism (ability to burn oxygen) is better which means their muscles can work harder.

Clever little things, really.

## protected by Qmechanic♦Dec 18 '14 at 8:41

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