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Re: correct answer



The problem states that a 12 m long firehose is uncoiled by
pulling the nozzle end horizontally along a frictionless surface at the
steady speed of 2.3 m/s. The mass/length of the hose is 0.25kg/m. How
much work has been done on the hose by the applied force when the entire
hose is moving?

The solution manual applies the "Work-Energy" theorem and
calculates 0.5mv^2 for the entire hose in motion to get the answer 7.9 J.
However, I believe that this is an incorrect application of the W-E
theorem because the mass being moved by the applied force is not constant.
i.e. from Newton's second law F = d(mv)/dt = (dm/dt)v in this case, so the
total work done is the integral of Fdx, which with a little manipulation I
calculate to be rho*v^2*L where rho is the mass per unit length of the
hose and L is the total length of the hose. This gives an answer twice as
big.

This problem is somewhat complicated by the details of "uncoiling" which
are hidden from the student in the problem. So long as the uncoiling
process does not involve dissipation it should be sufficient to look at
the initial and final states of the system to calculate the work done in
this over-idealized problem. I take it to be implicit here that the
final state of the system is one in which the firehose has a kinetic
energy 1/2 mv^2, where v is given. I have been able to gedankenconstruct
an uncoiling scheme that is nondissipative, so I agree with the answer
given.

Last week a couple of students came to me with a related problem which
they were assigned in their mechanics course.

A rope* of length L is laid out perpendicular to the edge of a
frictionless tabletop. Initially the rope is at rest with a
portion of length x hanging over the edge. The rope is released
and falls to the floor. How much time t(x) is required for the
rope to leave the tabletop?

I believe the problem as stated cannot be solved. The simple
view that only the gravitational force can do work I'll grant,
but the motion of this rope is more complicated than the poser
of the problem recognized. One way to make the problem well-
formed is to specify, for example, the radius of curvature of
the edge of the table, since clearly the edge exerts a
horizontal force on the rope. having said that I'll confess
that I'm glad the radius wasn't specified, since the solution
would then perhaps have been determinate, but I was not up to
being the one to determine it.

Leigh

*inextensible, uniform, ideally limp, etc.