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From: MJackson.Wbst@Xerox.ARPA
Newsgroups: net.physics
Subject: Re: Monkey Query
Message-ID: <711@sri-arpa.ARPA>
Date: Wed, 30-Oct-85 13:31:01 EST
Article-I.D.: sri-arpa.711
Posted: Wed Oct 30 13:31:01 1985
Date-Received: Sun, 3-Nov-85 11:52:26 EST
Lines: 40

Simple answers to these questions quickly get confounded in the
ambiguity of "The Monkey. . .starts climbing the Rope."  It is simplest
if we take this to mean that the Monkey exerts a force Fm on the Rope
which exceeds MG (where M is n Kg and G is the acceleration of gravity).
Then by considering tension in the Rope it is clear that:

If the Rope is massless and the Pulley is frictionless and massless then
the Monkey and the Weight both accelerate upward at the same rate A =
Fm/M - G.

If the Pulley has friction (let us assume speed-independent friction Fp)
then the Monkey accelerates upward faster than the Weight, which may not
move at all.  The Monkey's acceleration is as before and the Weight's
acceleration is (Fm - Fp)/M - G [if Fm > MG + Fp], or zero [if Fm<= MG +
Fp].

If the Rope has mass (let us assume mass per unit length R and total
length L [Monkey and Weight are at opposite ends]) then the Monkey's
acceleration is as before and the Weight's acceleration Aw is initially
(Fm - MG)/(M + RL).  However, as rope passes over the Pulley this
increases the weight on the left (Monkey) side, which raises the rate of
acceleration of Rope and Weight.  If we denote the length of Rope on the
left by X we have:

		(Fm - MG - LRG) + 2XRG = (M + RL)Aw

and since Aw is also d(dX/dt)/dt the Rope acceleration increases
exponentially until the Weight hits the Pulley.

Note well the first paragraph.  If one takes "The Monkey. . .starts
climbing the Rope" to mean the Monkey begins moving with a fixed upward
velocity with respect to the Rope then the answers may look somewhat
different, and further specification of the "start-up conditions" (how
did he accelerate to this velocity?) may be required.  In particular, if
the Rope has mass and is long enough the Monkey will eventually find
himself *descending* under this interpretation.

Loved your diagram, by the way.

Mark