## Tuesday, March 2, 2010

### The supposedly herculean feats of ants

Most of us have heard that ants can carry weights hundred times of their own. Just recently, BBC News ran a story related to this ability.

Image Source: Wikimedia Commons.

In most people, this conjures up images of SUV-sized ants effortlessly lugging around buildings, reminiscent those in classic Sci-Fi film THEM!; more still, of mimicking the ants and creating super-strong materials and actuators that are vastly stronger than anything we have access to today.

This is however a pipe dream. This apparent strength is a consequence of being small, and not of some specific capability of ants. The villain is scaling laws -- the rules that describe how physical properties change with size.

Here is how it works:

The mass of an object that has characteristic length L (that is, is roughly of size L in height, width and depth) is described by m = ρL3, where ρ is the density of the object.

The weight a muscle can lift is given by Mmax = cL2, where c is some material constant that is independent of size. You can see why this is by realizing that this is very much like the strength of a string -- and making a string longer does not make it stronger, but making it thicker does.

What we are interested in is the ratio of these quantities (i.e. maximum carry weight over mass):
Mmax / m = c/(ρL)

Ignoring c and ρ as they are independent of size, a pattern emerges. If we put in a very small value of L--a very small characteristic length--this ratio becomes very large. If we put in a large L, it becomes small.

So, what does this mean? Well, the smaller the muscle, the more it can carry relative to it's own weight.

In conclusion:

An ant that is 1,000 times larger than it is weighs a billion times more, but is only a million times stronger, so it's strength to mass ratio decreases by a factor 1000.

A human that is 1,000 times smaller than it is weighs a billion times less, but is only a million times weaker, so her strength to mass ratio increases by a factor 1000.

Returning to the SUV-sized ants, such ants would not be able to support their own weight, let alone carry buildings around, and an ant-sized human would be stronger than anything it's own size.

this value is a bit larger than SUV-size, but correct in an order of magnitude ballpark sort of way.