# Physics of scaling up an animal: the neck

Consider an animal like a horse. Now scale its neck longer and longer.

How can a giraffe, or even worse a huge dinosaur, raise its neck without the tendons snapping? The dinosaur case in particular seems ridiculous. Is there a "physics trick" the animals use to make this more manageable? Or does the tendon tension not scale as badly as my intuitiion is claiming?

Let's scale up an animal.

If length is $L$, and we don't change the proportions, then the mass of the head+neck will scale as $L^3$. If the neck & head are being held horizontally, the lever arm scales as $L$, so the torque at the base of the neck scales as $L^4$.

The width of the neck scales as $L$, so the force on the tendons/muscles scales as $L^3$. The yield strength of a tendon/muscle scales with its cross-sectional area: $L^2$.

The force is going as $L^3$, but the yield strength is going as $L^2$. My math agrees with your intuition: as you make an animal bigger, eventually the neck won't be able to handle the stress.

The reason why the giraffe can get away with long necks is twofold, I'd guess:

1) They have a proportionally thicker neck. Look at a picture of giraffe. Note that the neck gets extremely thick toward the base along the front-to-back axis (which is the the axis along which they lower their head), while it's slender side-to-side. Of course, this you can only change proportions so far: eventually the animal will be all neck.

2) We've got a lot of overhead with our short human necks. Ferinstance, I can support weights much heavier than my head with my arms, despite my arms being much longer AND a bit thinner than my neck.

Why not? Even though, as Anonymous Coward pointed out very well, the force scales as a cube while strength goes as a square, this only implies that infinitely long necks aren't possible. Obviously, lots of coefficients are being dropped in this rough sketch - stuff that pertains to the placement of tendons and arrangement of muscles.

Giraffes do exist; this guarantees that necks can be at least 7 or 8 feet once the optimal neck-construction is done. Beyond that, it could be that the optimal neck-construction lets necks get to 10 or 20 or 40 feet. Impossible to say. All your intuition really tells you - assuming you don't know a whole lot about the architecture of giraffe necks - is that the neck can't be infinitely long.