# Why are smaller animals stronger than larger ones, when considered relative to their body weight?

I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat.

It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects/lifeforms to support several times their own weight, but this is harder as objects/animals become larger?

• I remember Bill Nye doing this one! I believe it has to do with a constant tensile strength leading to member strength being proportional to $l^2$ while $m g$ is proportional to $l^3$, and apparently that holds for muscle strength as well. Jun 5, 2011 at 14:35
• You have to google for the appropriate expressions! "scaling of animals" will produce a lot of links. Jun 5, 2011 at 14:57
• A related question is why animals with exosceleton (chitin) are smaller (about some decimeters maximal) whereas animals with endoskeleton (bones) reach(ed) size of elephants or brontos Jun 6, 2011 at 12:59

Scaling up a ant to human size means volume (weight) increasing by length proportional $l^{3}$, but the force of muscles is determined by cross section (not muscle weight), so muscle force goes proportioal to $l^{2}$.

Smaller factors are likely:

• stiffness (or strentgh of the skeleton)
• balance point (center of mass)
• leverage (human skeleton is "sub-optimal" for this, we are afaik best optimized by evolution for long runs, more than any other animal)

i did some quick further search on "robot insects" on this interesting topic. This article is quite worth reading and relating biological to technological limits as well as current state of the art in nanobionics:

Interestingly, the force generated from a wide variety of actuator materials and devices has been found to be surprisingly invariant when compared with the actuator mass. A few years back, a comparison of the force-to-weight ratio of various organisms and machines found a striking similarity, with the force scaling linearly with mass over 20 orders of magnitude – from individual protein molecules to rocket engines ("Molecules, muscles, and machines: Universal performance characteristics of motors"). Remarkably, this finding indicates that most of the motors used by humans and animals for transportation have a common upper limit of mass-specific net force output that is independent of materials and mechanisms. Therefore any actuating device produces the same force per mass regardless of the material from which it is constructed and the mechanism by which it operates. This study also makes clear that biological systems dominate at the small mass, small force, range. In contrast, human-made machines dominate at the large mass range.

short example as Sonny asked for in comment:

ant with 10 mm length & 10 mg mass

$\Rightarrow$ lets scale up to human size (2m) $\Rightarrow$ means a factor of 200. So the mass scales with 200x200x200=8000000 (Volume $\propto$ $l^{3}$ ) $\Rightarrow$ human sized ant=80 kg. But muscle forces scales only by factor 200x200=40000. The small ant can carry 100x10mg of her own mass=1g, the human sized ant should be able to carry 1g x 40000=40 kg.

Conclusion: pretty comparable to a avg. 80 kg human man able to carry 40 kg!

• Oh oh, surface area of lungs scales as $l^2$ and thus limits aerobic activity by that much! Well... I thought that and then realized that I'm treating this too naively. A large animal will not raise and drop their legs as much. It would seem that inclusion of time scaling would be useful, but that opens up a can of worms that I didn't feel like getting into. To crack that can a little... If you decreased the pull of gravity for an elephant you could possibly get a mechanically equivalent system to an ant (with a different time scaling). Jun 6, 2011 at 1:30
• You can also see the effect of scaling observationally. Look at the relative thickness of the legs of large mammals; elephants, hippopotamus etc. versus smaller ones, say dogs, cats, deer. The latter's legs diameter is a small fraction of body length, while the former groups legs are quite fat. This is an indication that it is really tough to scale these animals up much further. Jun 6, 2011 at 2:48
• lungs are fractals, so I'm not sure the surface area goes as $l^2$. Jun 6, 2011 at 3:16
• @Werner Schmitt ""(or strentgh of the skeleton, human mainly consist of H2O)"" This about H2O is somewhat strange. What does that mean in this context? DTO "balance point" and "leverage" Jun 6, 2011 at 4:30
• @Zassounotsukushi afaik lungs have a fractal dimensionalty, so $l^{2.x}$ Muscle force is mainly defined by short time anaerobic energy supplies. Biggest problem imo is the the heavy H2O in all our extremities. Probably one could look up on youtube some bigger robotic insects made out of stiff and light carbon fibre, how good this scales up and what the current state of the art is :) @Georg H20 portion may be just a indication, that our overall stiffness is lower than that of a ant(?) The scaling of extremities weight seems crucial to me too, insects cant stretch/bow their ones Jun 6, 2011 at 11:17

To answer this question you just need to know that the force scale like the transection surface of a muscle. In other words the bigger the muscles the stronger. Therefore you have

$$F ∝ σ_s S$$,

where $$σ_s$$ is the maximum developed muscle stress. It so happens that on earth the muscles work pretty much the same in all the animals (from the ants all the way up to the elephant) so it is pretty universal (it can still vary from one species to another but not by one order of magnitude). For mammals it is around 10 N/cm$$^2$$.

Then the general allometric law (or spherical cow argument) gives :

$$l∝ M^{1/3}$$,

$$S ∝ l^2 ∝ M^{2/3}$$,

where M is the animal weight.

Ok now let's consider the problem of lifting up a weight of mass $$M_0$$. To do so you have to lift up your own weight plus this mass using your muscles :

$$F \sim (M+M_0)g$$

$$\dfrac{σ_s}{ρ^{2/3}}M^{2/3} \sim (M+M_0)g$$

$$\dfrac{M_0}{M} \sim a M^{-1/3} -1$$

You thus see that the mass that an animal can lift up compare to its weight scale like $$M^{-1/3}$$. This leads to the impressive strength of ants.

Interestingly enough, you can also use this in a single species to compare the stronger and the heaviest :

$$M_0 \sim a_h M^{2/3}(1-\dfrac{M{1/3}}{a_h})$$

This function as a maximum ($$\dfrac{dM_0}{dM}(M_{strongest}) = 0$$) and 2 values for which M_0 = 0 (M=0, boring ; and $$M_{heaviest} = a_h^3$$. You will easily check that :

$$M_{strongest} =\dfrac{8}{27}M_{heaviest}$$.

Ok so what? well if you use it for human, we can estimate the value of $$a_h$$ using the weight of the heaviest man (\$M_{heaviest} = 635 kg, Jon Brower Minnoch). Therefore the strongest man should be around 190kg. Well this is exactly the weight of the "strongest man" the Iranian Hossein Reza Zadeh...

Best,

Rémi

To lift anything, a life form on this planet needs muscles. If you want to lift heavier things you need

• more muscles and
• a stronger body/legs to support that additional weight.

Stronger muscles and bones need to be supplied with more oxygen, nutrion and so on which leads to needing a stronger heart, a better digestive system and so on. Or simpler: if you want to be stronger, you will get heavier and you need to support your own weight too. And the additional weight increases faster then the additional force you can get out of that.

• Why would capabilities of the support systems not scale at the same rate as the support needs? Your final sentence articulated that it would but evidence as to why is missing. Jun 5, 2011 at 20:22