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Re: Chain Problem.

On Fri, 10 Oct 1997, Richard W. Tarara wrote:

In the idealized problem the rope/chain falls straight down, but if it is
moving horizontally just before leaving the table, then there must be a
horizontal force that opposes that motion just as the chain leaves the
table. That force must increase with time since the horizontal momentum of
each chain segment increases with time (or as we get near the end of the
chain) and the time period in which the chain segment must change its
motion from horizontal to vertical is decreasing. This is certainly a
problem that one does not want to look at too closely--at least not until
graduate school!

Oh, it's not that bad. At the corner of the table you can have a 90 degree
bend of the chain passing through some of that good frictionless tubing.
The net force exerted on the chain passing through that tube is directed
at 45 degrees below the horizontal (proof left as exercise for reader).
Its horizontal component acts on one end of the horizontal segment of the
chain and the vertical component acts on the vertical hanging portion of
the chain. The two components have the same size. Their effect on the
motion of the chain is therefore zero, that is horizontal retarding force
on the horizontal portion is numerically equal to the vertical
accelerating force on the hanging portion. Draw the free body diagram for
the infinitesimal segment rounding the edge of the table, and you'll see
that the force on *this* portion will not contribute to acceleration or
deceleration of the chain.

Think of the chain as a total mass being subjected to an accelerating
force. The accelerating force on the entire chain is entirely due to the
weight of the hanging portion, the kink at the table edge doesn't affect
that, for our perfectly flexible ropes or chains. (Some, I suppose, might
object that to properly analyze the chain, one must know the link size and
apply quantum mechanics.)

That's what makes this an interesting problem. The *apparent difficulties*
don't affect the outcome, and can be easily bypassed in the analysis. They
give the more creative students a chance to brainstorm the problem. The
plodders who plug and chug their way through a course generally hate these

As I said earlier, these problems are a staple of *undergraduate* courses
and textbooks in engineering mechanics. I fault them for the analysis
usually given in the instructor's manuals which ignore these interesting
questions about the bent portion of the chain at the edge of the table.

If you want a better-posed problem, consider the ball-chain over a pulley,
sort of a continuous Atwood machine. Or pass the chain through an inverted
glass U-tube. Here you can do this as a lab experiment or demo, and find
the friction drag in the U-tube. Don't forget that the finite half-circle
of chain in the U-tube has angular momentum.

-- Donald

Dr. Donald E. Simanek Office: 717-893-2079
Prof. of Physics Internet:
Lock Haven University, Lock Haven, PA. 17745 CIS: 73147,2166
Home page: FAX: 717-893-2047