Re: T. rex mechanix

[ngear@gvtc.com]

Jim Cunningham wrote:
>Try bending the tail inboard when only one foot is on the ground. 
>Straighten it when both feet are on the ground to create a larger
>resisting moment,

Yes, as I noted in the original post, T. rex could have exerted torque
against the ground more effectively when both feet were on the ground, but
this would have required substantial weight on both feet to work, and that
would have been a very brief period when the rex was striding (and not even
that if the rex could run).  Also, any torque generated during this instant
would have been entirely attributable to work done by the legs.  Bending the
tail would have contributed nothing to the net torque delivered to the
combined mass of body and tail.

>When you drop a cat upside down, how does it flip over prior to ground
>contact?   This isn't an old wives' tale.  I've owned a number of cats,
>and they do.  My point being that ground contact isn't required.

To understand how a cat does this, let's go back and make use of Dr.
Baeker's excellent office chair and weights again.  (Sorry I missed your
title last time, Doc.)  When seated in this chair take your weight, hold it
over your head, and sweep it around in a horizontal circle.  You will find,
due to what Dr. Baeker correctly identified as conservation of angular
momentum, that if you swing the weight clockwise, you will rotate
counterclockwise.  Since your hand is attached to you, you will notice that
it too rotates counterclockwise even though you are sweeping it around in a
clockwise motion.  This is what a cat makes use of, except the axis of
rotation is horizontal (it arches its back and sweeps its head and pelvis
around).  Another way of looking at this is inertial torque.  The torque
required to change the velocity at which something rotates about an axis has
to be transmitted to something, and the result is a counter-torque, but the
important thing to note here is that the axis of counter-torque is always
parallel to the axis of the initial torque.

One other consideration for making use of inertial torque.  It does not
alter the net momentum of the body (hence the term 'conservation').  If you
had a truly frictionless office chair, for example, you would find that you
would continue rotating only so long as you kept the weight rotating, and as
soon as it stopped, you would stop.

So, in order for T. rex to have been able to capitalize on inertial torque
to rotate its body in a turn, it would have had to have been able to rotate
some significant portion of it's mass around a vertical axis, and do so for
the entire duration of the turn.  Frankly, I was unable to imagine how it
could have done that, which is why I considered inertial torque irrelevant
to the rex rotation problem.

***

Apparently some people are still having difficulty seeing that banking over
in a turn of itself does not contribute to rotating the body in the
direction of the turn, without there being some way to impart more "lift"
(i.e. centripetal force) to the front than the back.  In the case of the
rex, I do think there would have been a slight rotation effect just because
the rex's feet were located away from its center of gravity, and when angled
over in a corner the mere act keeping the feet "under" the center of gravity
(in line with the lean angle) would have had the effect of providing more
lift in front.  How much would this effect have contributed to rotating the
rex?  I don't know, but I know how we can find out.  However, I'll need some
help from a volunteer.  If someone out there has a decent scale model of a
T. rex (preferably solid and unbreakable--not one of the old ones with the
upright posture) could you try this and report what you find?:

    Tie a thread to a foot, or a bridge between the feet, and let the rex
hang (yes, upside down) until there is no twist left in the thread.  The
length of the thread doesn't matter, but I would guess two or three feet
would be the most convenient for this.  Without touching the rex, swing it
around in a circle so that the thread traces out a cone.  The angle of the
cone is your lean angle around a turn.  I'm expecting the rex to rotate some
as it swings around, but I'm also expecting the amount of rotation to be
small.  The effect will be more pronounced with a steeper lean angle, but I
doubt the rex could have leaned over very far.  (Thirty degrees off vertical
would probably be more than generous.)  What we want to know is how much the
rex rotates after one full swing around at something less than a 30 degree
lean angle.  We can then take fractions of this amount for turns of less
than 360 degrees. (Well, not exactly, but close enough for us.  This will
actually overstate the strength of this effect because it is cumulative, so
the model will rotate more in the last quarter of the swing than it did in
the first.)

I would do this myself but, I'm ashamed to admit, I don't have a good model
rex. 
I hope someone can help on this.  I'm a bit curious whether I've correctly
anticipated what will happen.

-------------
Nicholas Wren


P.S. Someone will probably point out that the rex is swinging around
upside-down, but the physics should be symmetrical.  All we are looking for
here are the right proportions.  So long as the line from the foot through
the center of gravity is in line with the bank angle, the foot will impart
the same rotational torque to the body about the rex's center of gravity
whether it does so from above or below.  If we put the rex on a zero-mass
zero-friction turntable and had the turntable bank around in a circle, we'd
get the same results.  I just thought the thread approach would be easier
and cheaper.