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An answer of balance



Okay, I've been puzzling on this question of how a big meal would have
affected T. rex balance for a while now, and I think I've finally got it
worked out.  The biggest problem I had was figuring out which variables are
extraneous--which, as it turned out, most of the ones I was looking at were.

Here is my current thinking on the mechanics of load compensation in the
rex.  I've omitted a few trivial considerations:

The first place I ran off track was in trying to figure out the relative
mass of the torso versus the tail.  This doesn't matter.  All we need to
know to start with is that rex balances (presumably on empty).  Next we need
to know where the stomach was in relation to the hips.  I estimated this at
about five feet forward (which should be far enough forward to keep out of
the way of the legs).  Next we dump a meal in.  Lets figure the meal weight
at the remarkable two ton figure GSPaul presented in _PDW_ (a figure which I
was totally unable to accept on faith).  At this point, I found it handy to
think in terms of torque.  A two ton meal that is five feet from the hips
exerts ten foot-tons torque at the hips.  To support this, the rex would
have had to impart a cancelling ten foot-tons torque to its pelvis.  To
accomplish this, the rex rotates the legs forward, which has the effect of
moving the body back in relation to its feet.  To know exactly how much it
has to do this, all we need to know is the desired torque, and the total
weight of the rex.  (We don't even need to know leg length, or whether the
knees were bent--which came as a surprise to me.)  Assuming an eight ton rex
ate a two ton meal, that would be ten tons total, which means that to
develop ten foot-tons lifting torque at the hips, the rex has to move it's
body back by just one foot.  Assuming the legs were about eight feet tall,
that would mean the hips move back about seven degrees (on average) from the
point of contact with the ground.

At this point, I had to conclude that even a rex with GSPaul's amazing
two-ton meal might not have been seriously impaired in the locomotion
department.

I did find it interesting to note what this says about the rex's leg
strength.  If the muscle which pulls the femur rearward attaches at about a
foot behind the hip joint, that muscle would be carrying an average of five
tons tension over and above it's normal load--just to stand still.  That's
ten tons load when the rex shifts all it's weight onto that leg.  This
tension naturally generates a comparable load in the hip joint, so an eight
ton rex with a two ton meal would have twenty tons (40,000 lbs.) load on the
hip of the loadbearing leg while walking, and that's BEFORE you add in
impact and deceleration loads.  (Just to put this in perspective, we have a
dump truck which when fully loaded to capacity with gravel has a gross
vehicle weight of just over 22,000 lbs. so you could stack two of those on
one rex hip joint.)

At this point, the sharp and awake among you have already figured out that
this way of figuring balance isn't just for meal loads.  Since all you have
to know is the weight and distance from the hips, it should work for any
kind of load.  For example, in a recent thread, it was suggested that rex
tooth marks in a ceratops frill could have been from the rex relocating the
corpse.  Someone inquired why it would have done this, but I think we only
need to look at *whether* it could have done this.  Assuming the nose of the
rex (with head pointed down) is about 15 feet forward of the hips, every
pound lifted requires a compensating 15 foot pounds torque at the pelvis.
To lift a four ton ceratops would have required 60 foot-tons torque at the
pelvis (120,000 foot-pounds) or a five-foot rearward displacement of the
body (adding in the ceratops weight).  Even dragging the ceratops would have
involved lifting a considerable portion of its weight, plus overcoming
ground friction.  I know the rex is an amazing animal, but I'm doubting it
could have been this amazing.

I have to say, I feel a little silly posting an answer to a question I
originally posted, but I figured I should eat humble pie for having doubted
GSPaul, and I thought if any of you were actually spending any time on my
original question, I should let you move on to productive things.  So,
unless someone sees where my reasoning has gone horribly wrong, I think that
wraps up this particular thread.

Nicholas (Wren)