Re: Sauropod browsing energetics

[Mike Taylor <mike@indexdata.com>]

On 24 March 2011 22:22, Augusto Haro <augustoharo@gmail.com> wrote:
>> plus mechanically speaking
>> simply having the longer neck necessitates greater muscle exertion
>> _all the time_ (less so if held upright, but still extra work
>> regardless),
>
> At equal mass, wouldn't be necessary also greater muscular exertion to
> support an equivalent mass in other parts of the body of the similarly
> massive animal?

If you isometrically increase the size of a sauropod by a linear
factor of x, then its mass scales with x^3 and the cross-sectional
area of its muscles scale with x^2, so that mass/area scales with x.
However, the neck and tail suffer worse because they are cantilevered
out from the body.  Not only does their mass scale with x^3, but the
distance out from the fulcrum at which this mass acts scales with x,
so that the moment to be countered scales with x^4.  Does this mean
that it's hard for necks to scale isometrically with the rest of the
body?  Well, not really -- because the height above the fulcrum at
which the epaxial tension members acts is ALSO increased by a factor
of x, so that the counter-moment goes with x^3 (i.e. x^2 for the force
generated by muscles with cross-sectional area proportional to x^2,
times the moment arm x).  So the increases in the moment arms cancel
out.

So using this very simple (read: grossly oversimplied) model, it seems
that the difficulty of scaling necks and tails as animals grow
isometrically is exactly the same as that of scaling legs and other
body parts, i.e. that stresses grow linearly with x.

(Of course, larger animals are not in general isometrically similar to
smaller ones.)