Rope hanging from teh ceiling

stunner5000pt

ASsume teh z axis is the vertical axis

A homogenous rope of mass M and length L hangs vertically from a point of hte ceiling.
Find the magnitude of the force F(z) by which the top part of the rope acts on teh lower segment of the rop at the distance z from the free end of the rope.

Not quite sure here Would i have to use something like mass density of each segment like how you do for a line of charge calculation??
so then define [tex] \lambda = \frac{M}{L} [/tex]

so then [tex] F(z) = m_{eff} g [/tex]
[tex] F(z) = \lambda z g = \frac{Mg}{L} z [/tex]
good so far??

Now the rope is spun in a horizontal circle with constant angular velocity omega. Ignore gravity.
Find the magnitude of the force F(r) by which the inner part of the rope acts on the outer segment of the rope at the distance r from the center of the circular motion.
so use the same lambda as before
The question above has been copied from teh book verbatim. I'm wondering if the rope is being spun about its center or from one end?
[tex] F(r) = m_[eff} \omega^2 r [/tex]
[tex] F(r} = \lambda r \omega^2 r = \frac{M}{L} \omega^2 r^2 [/tex]

now i know this is wrong becuase th book gives the answer to be [tex] F(r) = \frac{M \omega^2}{2L} (L^2 - r^2) [/tex]
where di i go wrong??

ramollari

Why did you take r as the radius? It is like there's a point mass of mass [tex]m_{eff}[/tex] moving in a circle of radius r. Certainly this is not correct. Divide the rope into tiny imaginary points of mass dm, find the force applied on each individual rope segment as [tex]dF = \omega ^2r_{pt} dm[/tex]. Try to express r_{pt} in terms of the mass that remains from the extreme, total mass, and length L. Then integrate from 0 (mass supported at the extreme) to m that is the mass supported at the point where r_{pt} = r.