A rope of length L lies in a heap on a table, except for a very small piece at o

Question

A rope of length L lies in a heap on a table, except for a very small piece at one end which hangs down through a hole in the table. This piece is released, and the rope unravels and slides down through the hole. What is the speed of the rope at the instant it loses contact with the table? (Assume that the rope is greased, so that it has no friction with itself. Also note that the SIMPLE solution for part (a) will not work here. Neither will F = ma. Instead, think about the more general force equation discussed in class. After writing down the equation of motion, try the solution x(t)=bgt2 and see what value of b will satisfy Newton’s Law.) How much energy is lost from the initial setup to the instant the rope loses contact with the table?

PLease include steps

Explanation / Answer

Since the acceleration is not uniform through the problem we will have to use jerk.

y=(1/6)jt3+(1/2)at2+vt

j is jerk, which is the change in acceleration per time. We can rewrite it for this situation as:

j=mg/t=Lg/t is the density of rope

Substituing back in

y=(1/6)Lgt2+(1/2)at2

t=[6L/(Lg+3g)]

Now we can find the velocity by using

v=at=(jt)t

v=Lgt

Hope that helps

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