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Re: A problem: conserving L during deceleration



At 08:06 PM 8/1/01 +0530, D.V.N.Sarma wrote:

A uniform cube with a edge 'a' rests on a horizontal plane whose
friction coefficient equals 'k'. The cube is set in motion with an
initial velocity, travels some distance over the plane and comes to
rest. Explain the disappearance of the angular momentum of the
cube relative to an axis lying in the plane at right angles to the
cube's direction of motion.

At 09:53 PM 8/1/01 +0500, B.Surendranath Reddy wrote:
Force of normal reactioon due to the surface would not act through the
center of the bottom surface. It would act closer towards the front edge.

That's true, and it is an important step in the correct answer.

But let me make explicit the rest of the answer:
a) The effect of gravity acting on the cube remains the same. The total
effect of gravity can be summarized by a downward vector acting on the
cube's center of mass.
b) When the cube is at rest, the upward force due to contact is colinear
with this weight vector, so there is no torque. But during the
deceleration, the weight and the contact force are not colinear, resulting
in a torque. This torque carries away "just enough" angular momentum to
ensure that the angular momentum of the block is proportional to its linear
momentum, as it should.

The underlying physics can be more easily seen by considering a wheeled
cart rather than a sliding block. In particular imagine a six-wheeled
cart, shown here in profile:
_____________
|___________|
_____*____O____*_________

At rest, all the weight rests on the middle pair of wheels; the others are
just idle outriggers. But during strong deceleration, the cart transfers a
goodly amount of weight to the front pair of wheels. In extreme cases, it
might even pop a wheelie.

This weight-transfer creates the required torque, because the weight vector
and the contact-force are not colinear.

If the cart has a floppy suspension, the tilting will be easy to see.
In contrast, in the case of the stiff cube, only an infinitesimal amount of
tilt is needed for shifting the point of contact by a goodly amount, so it
is harder to notice what's going on.

(If the torque weren't "just enough", the cube would tilt a little more and
move the contact-point further forward.)

In the numerical analysis business, the word "stiff" is used to describe
any system where an infinitesimal change in one variable has a macroscopic
effect. It is a pejorative term. It is a warning that the system is
pathological or nearly so, and will be hard to analyze.