- #1

z0r

- 11

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A rope rests in a bundle on a table with a small hole in it. One end of the rope slides through the hole and gravity steadily pulls on it until the total length of the rope slides through the hole. The length of the rope hanging out of the hole can be [tex] x [/tex]. What is the velocity of the rope when it has all slid through the hole? Ignore all friction.

Let's say the rope has mass [tex] M [/tex] and length [tex] L [/tex]. Then we say the linear density is [tex] \mu = \frac{M}{L} [/tex]. The force on the rope is then [tex] F = \mu x g [/tex], where [tex] g [/tex] is just gravity. Then I need to figure out the momentum, take the time derivative, and set it equal to the force. Then we should have a simple differential equation to solve and integrate from zero to the length [tex] L [/tex] to find the final velocity. Simple enough, but I don't know how the momentum is defined here...

Both the mass and velocity of the length of rope hanging out of the hole is changing, so then would we not say that momentum [tex] P = \dot{m}x + m\dot{x} [/tex], with [tex] m = \mu x [/tex]? That yields something that doesn't look correct. Nor does just [tex] P = \mu x \dot{x} [/tex].

Thanks in advance for any help.