Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Rigid bodies, or hard bodies?





On Thu, 13 Mar 1997, Al Bachman wrote:

1. Most of the difficulties with W-E come from pushing the rigid body
approximation too far. NO REAL OBJECT IS COMPLETELY RIGID. If it were
then it would have no internal degrees of freedom and could not have
an internal energy (no heat transfer).

The term "rigid" is is entrenched in physics and engineering textbooks,
but is misleading to students. Perhaps we should simply speak of "hard"
bodies. At least we should point out that the rigid bodies talked about
*are* deformable.

An interesting analysis of this considers collisions between two 'rigid'
bodies. I'll try to describe it without diagrams. This isn't original with
me, but not often seen in textbooks.

Consider the collision of two perfectly elastic equal mass balls, moving
toward each other with speed v. "We all know" that the balls rebound with
equal and opposite velocities. This is required by conservation of energy
and momentum.

But the time of contact cannot be infinitesimally small, for the finite
impulse Ft would require an infinite force if t goes to zero. Since
infinite forces are hard to come by in the universe, we conclude that t is
small, but finite. During this time, both balls are moving forward for
half that time, then moving away the other half of the time. This forces
us to conclude that either they co-penetrate, or they deform. In either
case, they can't be rigid.

It boils down to this: If bodies were perfectly rigid, then in collisions,
infinite forces must occur between them. If infinite forces can't occur,
then bodies can't be rigid.

"Can we tear out that chapter titled 'Rigid Bodies' from our mechanics
textbooks now?" I hear students ask. No, just rename it 'pesudo-rigid
bodies'. Now what about pseudo-forces acting on pseudo-rigid bodies...?

You can get yourself in all sorts of pseudo-paradoxes with 'simple'
mechanics when you idealize things overmuch. Consider the usual
discussions of ideal-gases, considering point particles bouncing about in
a box, hitting the walls, but never hitting each other. Typically, the
students swallow the assertion that the particle bouncing off the massive
wall rebounds with the same speed, just altered direction. One concludes
that the kinetic energy of the particle isn't changed, but its momentum is
changed by amount 2 x (normal component of velocity). This process gives
the wall an impulse Ft, which force added up for all particles accounts
for the pressure on the walls.

But how can the particle rebound with unaltered speed, suffering no change
in kinetic energy, but a large change in momentum? Do students wonder
about this? Not usually. It becomes resolved if you consider collision
between a moving particle and a stationary mass of larger size. In the
limit as the mass ratio goes to infinity, the desired result falls out
naturally. But textbooks seldom do this, or justify *in any* way the
*obvious* statement about rebounding without speed change. When this is
pointed out to students, they see the problem, but if asked to resolve it
as a homework assignment, the results are usually disastrous.

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: http://www.lhup.edu/~dsimanek FAX: 717-893-2047
......................................................................