Re: Aircraft Scaling Problem / Question



On Jan 30, 8:19 am, "Jon Slaughter" <Jon_Slaugh...@xxxxxxxxxxx> wrote:
<mensana...@xxxxxxx> wrote in messagenews:1170140810.050598.299410@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

On Jan 30, 12:34?am, "Jon Slaughter" <Jon_Slaugh...@xxxxxxxxxxx>
wrote:





<mensana...@xxxxxxx> wrote in
messagenews:1170137502.006052.240730@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

On Jan 29, 8:22?pm, "David T. Ashley" <d...@xxxxxxxx> wrote:

<mensana...@xxxxxxx> wrote in
messagenews:1170122241.139905.250880@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

On Jan 29, 7:38 pm, "David T. Ashley" <d...@xxxxxxxx> wrote:

<snip>

This is similar to how forces work on your joints and explains why we cannot
have huge creatures because there surface area only increases with the
square while there mass is with the cube. Apply the same analysis above and
scale your body to be 10 types larger and then find out your mass. it will
be 1000 times larger. That means you would way 100 tons or so. We could
even go as far to say that you would probably die since you would not be
getting enough oxygen. The body absorbs oxygen in relation to surface area
and your multiplying the amount you need by 1000 times yet your only getting
100 more oxygen. So now you have 10 times less. This can also be used to
say that smaller people are able to extract more oxygen per unit mass from
the air. (ofcourse though we are talking about exact copies and my analysis
doesn't take that into account).

As a child, I wondered why small creatures could often jump very high
in relation to their own size. The most extreme cases are insects but
they differ from us in many ways so there could be many explanations.
However, the effect is also demonstrated by more closely related
animals such as cats. Domestic cats can often make very impressive
jumps, many times their own height. If a tiger could do an equivalent
jump, he could clear a tree. So, I tried to develop a simple model.
Of course, all cats are perfect scale models of each other.

The mass of the cat is m which will be proportional to the cube of the
cat's size.

I assumed that the force F that his muscles could produce would be
proportional to the cross-sectional area A, F = kA. This will be
proportional to the square of the cat's size.

I assumed that the muscle could exert this force until it reached its
maximum extension x. I assumed that the centre of mass of the cat is
moved by a distance l during this extension. The relation between x
and l will depend on the leverage effect of the bones and muscle
configuration but I would expect that the length l should be
proportional to x and the size of the cat.

So while the muscle is exerting this force, the cat accelerates at a =
F / m. So, the force should be exerted for a period t = sqrt(2l/a).

The cat will reach a velocity of v = at = sqrt(2la) = sqrt(2klA/m)

Now l should be proportional to the size of the cat. A should be
proportional to the square of the cat's size. m should be
proportional to the cube of the cat's size. The proportionality
factors cancel and the model suggests that all cats can jump to the
same absolute height. A tiger can only clear a fence of the same
height as a domestic moggy. This seems plausible from my limited
observation of jumping tigers.

A different way to view the problem is that the force of the muscle is
proportional to its area. It moves the cat by a distance proportional
to the size of the cat. So the work done by the muscle is
proportional to the cube of the size of the cat. So is its mass.

<snip>

--
Seán O'Leathlóbhair

.

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