# 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?

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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.

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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.

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