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# Re: are normal reaction and tension conservative ?

• From: "John S. Denker" <jsd@MONMOUTH.COM>
• Date: Sat, 30 Jun 2001 12:58:42 -0400

At 08:26 PM 6/30/01 +0530, Chetan wrote:

Consider the case of a plank (with a block on it ) at rest on the floor.
the work done by normal reaction is zero.

Now imagine a man pushes the plank upward with an acceleration.
Now the work done by normal reaction is nonzero.

can normal reaction in the second case be considered non conservative.

The question is still a bit ambiguous. The label "normal reaction" is not
sufficiently specific. In problems such as this, there are lots of things
that are normal to other things. If you are not careful, it is easy to be
suckered into false generalizations: something that is true in one example
may not be true in general.

True statement: We can define the notion of a _force of constraint_. By
definition, such a force is normal TO THE MOTION. As an example, a
roller-coaster is constrained to stay on its track, but to a good
approximation the track applies no forces along the direction of motion. A
simple application of the F dot dX formula proves that any force of constraint
-- is conservative, and
-- indeed does no work whatsoever,
-- all provided the constraint itself doesn't move.

Now imagine a man pushes the plank upward with an acceleration.
Now the work done by normal reaction is nonzero.

If "the normal reaction" here refers to a certain component of the force
between the plank and the block, namely the component normal to the plank
(and hence normal to the motion) then the question embodies a false
premise. The work done by !this! component is always zero (assuming the
plank itself doesn't move).

Meanwhile, the normal component is not the whole story for ordinary
sliding. If you want to emphasize the normal component and neglect the
non-normal component, try putting a heavy block on light, efficient wheels.

=================

Here's an example on the other side of the coin, again illustrating that it
is crucial to specify what is normal to what: Suppose I pull a paddle
broadside through the water in a stream. The force is normal TO THE FACE
OF THE PADDLE. (That's what I mean by broadside.) This force does work
and is highly nonconservative.

A similar situation is the case of a body at the lower end of a rope being
pulled vertically upward by a man moving the upper end. could we label
tension as non conservative?

Well, now, that's an interesting question. If we took a vote, I'll bet
most people would assume that yes, the tension is conservative. Some of
them would even "prove" that it is conservative. But once again, it is bad
luck to prove things that aren't true.

For homework, come up with an example along the following lines:
-- We have an object at the lower end of a long rope.
-- I move the upper end of the rope.
-- The result is spectacularly non-conservative.

Notes:
== You can assume the rope is non-stretchy if you like.
== You can constrain me to moving the upper end purely vertically if you
like.
== You can choose the initial conditions over a wide range, but I can
veto certain initial conditions (a set of measure zero).

Extra credit:
What is such a system called?

Hint:
I have mentioned the concept by name in a fairly recent posting.