At 3:23 PM -0400 11/5/10, Josh Gates wrote:
>_____
>
>From: Chuck Britton [mailto:cvbritton@embarqmail.com]
>
>
> It seems to me (naively) that we better come up with exACTLY the same
> total force when we integrate force around all SIX sides of this
> mathematical volume as we did the first time.
> That only follows if the behavior of the cube is the same as
>before - that it floats or sinks or stays with the same accel. as
>before. If it stays on the bottom now instead of accelerating
>upward, then I don't agree.
It's a mathematical construct. It doesn't beHAVE in any way except as
the axioms and postulates of whatever maths you are using calls for.
OK... If we're saying that:
(integral of the pressure dA) - mg = ma_1 in the first scenario, and that:
(integral of the pressure dA) - mg + Normal force of bottom = ma_2, then:
N = first integral - second integral only follows if a_1 = a_2.
I'm calling the the acceleration behavior of the cube, and it'll do what it pleases, regardless of my mathematical construct. If my construct doesn't agree with its behavior, then that's a problem with my construct.
So...
your argument depends on the behavior of the cube in the two circumstances. Same behavior -> your normal force equality. Different behavior -> different size of N.