You, sir, cannot be helped.
That's the impression I get, yes.
Here's a better experiment, one you can even do at home. Get a bathroom scale and put it on the ground and turn it on. Put your hand on the scale and push on the scale with 20lbs of force, and see that the bathroom scale reads 20lbs.
Next, put a 10 foot pole on the scale (or a hundred foot pole for that matter, but you might not be able get that one done at your home), with one end on the scale and make the pole perpendicular to the earth, and zero out the scale. Push on the end of the pole opposite the scale with 20lbs of force. Are you seriously contending that the bathroom scale, which is measuring force being applied at the opposite end of this rigid body, will read something less than 20 lbs?
Sir, if you place the other end of the pole against a scale that is
supported by a floor or a wall and cannot be pushed away, the scale will read 20 lbs. This is because there is not
one force on one edge applied to the scale, but
two forces: you apply 20 lbs, and the floor/wall applies 20 lbs. The sum of the local effects of these two forces, as you can calculate, is 20 lbs.
If you apply force
from one side only - if the scale is not supported by a floor/wall, and it reads zero before you start pushing - then the scale will read F = 20 * m2 / (m1+m2) lbs, where m1 is the mass of the pole and m2 is the mass of the scale.
Ask a trusted physicist of your choice to confirm this to you. Or better yet,
do the experiment and verify that this is the case.
I mean, the measurement is being taken some distance from the point where the force is being applied, right? Is the 20lb force being felt throughout the entire body or not?
No, sir. If you apply 20 lb to one side of the body (and
not also to the other side of the body), the force is
not felt throughout the entire body. Acceleration is. The force is not.
This is why your neck vertebrae are not compressed with 160 lbs of force that the ground applies to your feet. This is why, if you sit in an accelerating car, you do not feel the acceleration force exerted by the car on the ground, but only feel F2=F1*m2/(m1+m2), where F1 is the acceleration force exerted by the car on the ground, m1 is the mass of the car, and m2 is your mass.
But if you are supported by a stone wall, or in another way held still with respect to the ground as the car begins to accelerate, then you
will feel the full force exerted by the car. This is because your support will exert a second force on you.
If you cut the pole in half and put the bathroom scale in between the two sections of the pole, press down on the pole with 20 lbs of force, the bathroom scale will still read 20lbs.
No, sir, it will not. If the other end of the pole is not supported by a wall or a floor as you push, the bathroom scale will
not read 20 lbs. It will read F=20 * (0.5*m1+m2) / (m1+m2) lbs, where m1 is the mass of the pole and m2 is the mass of the scale.
Please make an effort to understand these simple facts.
Let's circle this experiment back around now. Leave the pole perpendicular with the ground, but pick up the bathroom scale and put it between your hand and the upper end of the pole. When you push down with 20lbs of force, what will the scale read? You guessed it, 20 lbs. It's the same 20lbs of force no matter which end of the pole you measure it from.
Yes. In this case, the scale will read 20 lbs, because you measure it at the same point where you apply the force.
The point here is that no matter where you measure the force being applied to the end of the pole, it will register as 20lbs of force. You're wrong.
This is not the case, and I urge you to
do the experiments that you described (without the scale being supported by a floor or a wall) and find out for yourself.
I think you're confused. You are correct that, for a building sitting on the earth, the further away you get from the earth, the less gravitational force is being placed on any differential slice of the building you examine. This has nothing to do with where the force is being applied, and everything to do with how much weight is above the building.
Sir, if the ground (or another support) does not exert force on the bottom edge of the building, the force exerted between any two parts of the building, as it falls down freely, will be zero.
Please make an effort to understand this.
It also has nothing to do with the case of two bodies impacting, which is what Bazant is trying to model with the Twin Towers. There is a gravity force accelerating on the upper floors towards the earth during the collapse, no question about it. However, the force I'm talking about is the mechanical work force (ie. the force applied through a distance) that causes the upper floors to decelerate and deform for a period of time after impact, which is not a gravitational force, and thus is not dependent on its distance from the earth.
Oh, the humanity.
Finally, consider the case of two homogeneous cube shaped bodies of equal size and mass impacting each other at one face of each cube. Assume that at impact the bodies do not deform.
If that's the case, the cubes will exert an infinite amount of force in an infinitely short amount of time as they impact.
I will assume that you mean that the bodies do decelerate over a finite amount of time, and you measure the effect of the force that the cubes apply to each other as they decelerate upon impact.
If you measure the force experienced by any differential slice of either cube that is parallel to the face being impacted, the result has to be that the forces are equal for every such differential slice. This is because F=m*a.
No, they are not.
The
acceleration in any part of the cubes is the same. If you cut through the cubes and measure the force with which the two parts of the cube are pushed together, you will find that it falls down to zero towards the outer edges of the cubes.
This is why, if two cars impact each other, experiencing impact acceleration
a, the front of the car will experience force F=a*m1, where m1 is the mass of the entire car, but the rear bumper will only experience force F=a*m2, where m2 is the mass of the rear bumper.
This is why, when two cars decelerate by impacting each other, the front bumpers will be hopelessly deformed, but the rear bumpers will not.
Please make an effort to understand that your claims of equal impact forces throughout the entire car completely contradict reality.
Every differential slice should have the same mass because the cube is symetrical and homogeneous. Every differential slice must also accelerate at the same rate because the bodies don't deform. (Indeed, every discrete point on each body must have accelerated at the same for the body not to deform.)
I am talking about the force with which the two parts of the sliced cube are pushed together at the point of the slice. This depends on the mass of the two parts of the sliced cub, and therefore depends on the point where you slice.
Solving the equation for each differential slice clearly shows that the force F experienced by each differential slice is equal because the masses and the accelerations are equal.
Sir, you misapply the equation. You would be able to see that if you actually tried to enumerate it.
Acceleration is uniform throughout the body, yes. The force that you incorrectly calculate using F=a*m, incorrectly using the
mass of the differential slice for m, is F=a*epsilon (where epsilon is the infinitesimally small mass of the differential slice), which is infinitesimally small. It should be apparent that the force with which the two parts of the cube are pushed together at the point where you measure it, cannot be infinitesimally small.
The correct value of m to use in the equation F=a*m is the mass that is being accelerated by the force, at the point where you measure the force. You measure the force with which the two parts of the cube are pushed together - i.e. the force with which the inner part of the cube (with mass m1), on whose edge the impact deceleration force is exerted, decelerates the outer part of the cube (with mass m2). This is F = a*m2. Furthermore, a = F0/(m1+m2), where F0 is the force applied to the edge of the cube. This gives F=F0*m2/(m1+m2).