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Rex Towing Capacity (Aaargh!)

At 10:24 PM 2/24/98 -0600, Jack Conrad wrote:
>...A very useful post sent
>by Nicholas (Wren) (thanks Nicholas) suggests that GSPaul's estimation of
>a two-ton powerlunch was not impossible for a _T. rex_ (something I had
>never bothered to do the math on). Nicholas' analysis of the ability to
>drag a four ton ceratopsid was very logical, though I'm not sure that his
>conclusion was accurate...

To quote Homer Simpson...  DOH!  It seems I can't get one foot extricated
from my mouth without inserting the other one in the process.  In my last
post I showed how a rex would indeed have been able to load-compensate
GSPaul's Big Happy Meal (which I had been so publicly skeptical of) and then
mentioned how this process could be applied to calculate the rex lifting
capacity of any sort of load.  So far so good.  Then I applied this argument
to the thread debating whether the tooth marks in the ceratops frill were
made pre or post mortem (the post mortem argument being that the rex was
relocating the corpse).  I showed the rex could not have had sufficient lift
capacity at its head, did a bit of hand-waving, and congratulated myself
that I had dispatched the relocation scenario.  More fool me.  Jack is quite
right, of course.  My argument was both logical and completely irrelevant.
I figured rex lift capacity correctly (I think) but then utterly failed to
calculate rex towing capacity--which would have been the most likely way a
rex would relocate a corpse (ie. drag it).  I basically proved the
improbable impossible.  (And for my next magical trick, I will show
conclusively that the rex definitely could not have ridden a bicycle.)

Okay, so here's the question.  Could a rex have generated enough towing
force to drag a ceratops?  Right off the bat, we have a problem in that we
don't know the coefficient of drag for a ceratops, but I don't think this
will matter much.  If we can show the rex could have gotten it moving, we
can be pretty sure it could have dragged it, so let's just figure what the
rex would have been capable of first.  To know how much pulling force a rex
exerts, we need to know three things.  We need to know the weight of the
rex, we need to know the angle from the foot to the hip (I'm fudging a
little here, this assumes the hip was normally over the foot, which it
wasn't quite, but close enough for us), and we need to know the angle from
the bite up to the hip, and we can make pretty good guesses about all of these.

Now, we can figure this formulaically or graphically.  Since we are just
estimating (and I'm already fudging details anyway) the latter approach is
plenty precise enough, it's lots easier, and it's more intuitive.  We just
draw a load (vector) diagram and measure.  First we represent rex's weight.
This is a down force of (say) eight tons, so we make a vertical line eight
units long (any units, just remember that one unit equals one ton).
Cancelling the vertical weight line are two support components.  One will be
the legs and the other will be any side force (ceratops drag resistance in
this case).  The leg component starts at the foot of your vertical line and
goes up at the angle from foot to hip.  (With zero side force, the leg
component is also vertical and eight units long.)  The side force starts at
the top of your vertical line and goes up at the angle of bite to hip until
it intersects the leg component.  Then all you have to do is measure the
length of the side force vector and you have your towing load.

Let's try it with an extreme case of rex's legs angled back at 45 degrees,
and a purely horizontal towing load (bite at same height as hips, in other
words).  Here we have our classic 45,45,90 triangle, and you can see the
horizontal component would be the same length as the vertical--meaning eight
tons pulling force.  However, if you angle the side-force vector down in
front (bite lower than hips) that raises the point of intersection with the
leg component, and results in more pulling force.  With a ceratops on its
side (the most likely tow position, I would think) and the rex biting the
frill two feet off the ground, that gives a bite to hip angle of 13 degrees,
raising the tow force to 10 tons.  Short of getting wedged between rocks or
trees, this would almost certainly have been enough to get a four or five
ton ceratops moving.  Even at the more modest figure of 60 degrees leg
angle, 20 degrees bite to hip angle (this angle increased because the hips
went up but the bite height stayed the same) that still gives you about 6.3
tons tow force.

If you want to measure how much towing loads up the legs, just measure the
length of the leg component.  In the first example, (45 deg. foot to hip, 13
degrees bite to hip) that would have been almost 15 tons, and in the second
example (60/20) that would have been nearly 12 tons.  If you will remember,
in my last post, I showed that the rex hip would have had to have been
overrated by more than 100% to carry a Big Happy Meal, so these leg loads
would probably have been well within spec.  (Yes, that's right.  Dragging
with ten tons force would have put LESS load on the hips than eating two tons.)

Other considerations.  The highest drag is in getting a body moving.  Once
moving, drag drops.  Wet vegetation would help reduce drag.  Blood would
really help reduce drag.  Also, if you were going to tow a ceratops, the top
of the frill would probably have been the place to clamp on.  (Assuming a
rex could turn it's head sideways like that.)  I heard it suggested that the
rex could have been moving the corpse to conceal it, which doesn't strike me
as very likely.  I think I missed a couple of episodes of this thread,
though.  Did anyone think to suggest this rex could have been trying to drag
this body back to a den/nest/whatever to feed little rexlings?

I'm still skeptical of any post mortem scenario, just because I think when
you combine a rex and a dead ceratops, you get shredded ceratops, and I
gather that wasn't the case here.  But thanks to Jack pointing out my woolly
thinking, I see now that I can't invoke my previous argument against the
relocation scenario.  If anyone sees where I've put another foot in it while
trying to extricate this one, don't just stand there, grab a shoehorn and do


PS  Something I was wondering about.  How precisely do rex teeth mate up?
Close enough to remove meat purely by cutting, or would the rex have had to
tear loose the sinews that pass between the teeth?  I used to think the rex
would bite most of the way through and then lift to tear hunks of meat out,
but after my last post, I realized it would not have had enough lift
capacity.  If the teeth don't mate well enough for a clean cut, would
shaking the head have been sufficient to tear the meat out?  If not, I was
thinking the rex might have been able to use it's prodigious towing ability
to pull hunks of meat loose, since it apparently had several tons capacity
to play with.

Just a thought.

PPS  If any of you math masochists out there want the formulaic version for
figuring tow loads, I'll post that.