Chalkdust, Issue 13

Have you ‘herd’? The world’s largest cow is over six feet tall and weighs more than 1.3 tonnes. Is a bigger cow possi-bull? Will the future contain infinitely large cows? The steaks have never been higher!

To answer this question, let’s take a look at the cow’s legs. If the main (meaty) bit of the cow has a volume 𝑉 and density 𝜌 then its weight is 𝜌𝑉 𝑔. So each leg supports a load of about 𝜌𝑉 𝑔 𝑁= . 𝟦 In pursuit of glory, let’s now make the length, height and width of the cow bigger by a factor 𝑎. The cow’s new volume is 𝑎𝟥 𝑉 and so the load on each leg is 𝑎𝟥 𝑁: it grows cubically as 𝑎 increases.

Can the legs cope? If we model the legs as cylinders (since they already ‘lactose’...), we can use a 1757 result from the famous cow enthusiast Euler: if a cylinder has height 𝐿 and radius 𝑟 , the maximum load it can support standing upright is 𝐸π𝟥 𝑟 𝟦 . 𝟦𝐿𝟤 𝐸 here is just a property of the material: its stiffness, or Young’s moo-dulus. 𝑁max =

With our scaling, 𝐿 and 𝑟 are now 𝑎 times bigger. Our new maximum load is 𝐸π𝟥 𝑎𝟦 𝑟 𝟦 = 𝑎𝟤 𝑁max . 𝟦𝑎𝟤 𝐿𝟤

Uh oh... this only scales as 𝑎𝟤 : quadratically.

So even though 𝑁max starts above 𝑁 (it has to, given that these cows exist!), there will come a maximum possible 𝑎, after which there will beef-ar too much cow and its legs will give way... an udder disaster.

This analysis tells us something really important about biology—that there is a natural maximum size for land mammals. But have we reached it for cows? Brody & Lardy’s 1000-page tome Bioenergetics and Growth from 1946 has all the de-tail you need. We’ll leave you to ruminate on the cow-culations.

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spring 2021