## Thursday, March 11, 2010

### Honey, I Shrunk the Pisarenko [1]

It’s commonly said that ants can hold fifty times their own weight, but is this really as impressive as it sounds?  After all, they certainly can’t pick us up2.  If we want a fair comparison, we need to derive how much weight each species could lift if they were equal in size.  How much weight could a human pick up if he were ant-sized?

Compared to their body size, ants have tiny legs whereas humans have medium-sized legs and elephants have huge legs.  One reason for this is that the bigger a species gets the more weight it has to hold.  An animal’s weight will be proportional to its volume, whereas the force holding up that weight will be roughly proportional to the cross-sectional area of the legs.  If you double the height of an animal but keep all its limbs in proportion, then it will be 2 times as tall, but it will weigh 23 = 8 times as much because its weight grows as the length cubed.  Likewise, since the force holding up the weight is proportional to the area, it will grow as the square of the length and be 22 = 4 times as large.  If a human were to shrink down to the size of an ant, the amount of weight he could lift would shrink but his own weight would shrink even faster.

Let’s consider an 80 kg (~180 lbs) person that’s about 1.8 m (~6.0 ft) tall.  If we shrink him down to the height of an ant, he’ll be about 3 mm tall, or roughly 600 times shorter.  Using the fact that volume scales as the cube of the length, we can calculate his new mass,

new mass = (length scaling factor)3 · (old mass)
= (1/600)3 · (80 kg)
= 0.37 mg.

A fairly strong human should be able to lift about 100 kg (~220 lbs).  Using the fact that strength scales roughly as the square of the length, we can estimate how much weight his new tiny self can pick up,

new mass = (length scaling factor)2 · (old mass)
= (1/600)2 · (100 kg)
= 277 mg.

Comparing these two numbers, we see that an ant-sized human could lift an astonishing 750 times his own weight.  If you want to play around with these scaling relations, try estimating how much a human-sized ant could pick up.  Is there any chance that giant ants will take over the world?

[1] Kudos to anyone who’s both geeky enough to read this blog and well versed enough in weightlifting sub-culture to get this reference.  I admit, I had some difficulty choosing between Pisarenko and Süleymanoğlu, but I couldn’t justify shrinking an already pocket-sized Hercules.
[2] In How Many Licks?, I estimate that it would take 65,000 ants to hold up one human.