# Rope Falling Off Table

by torq123
Tags: chain, differential, falling, force, rope, table
 P: 8 1. The problem statement, all variables and given/known data A (smooth) rope of length L and mass m is placed above a hole in a table. One end of the rope falls through the hole, pulling steadily on the remainder of the rope. Find the velocity of the rope as a function of the distance to the end of the rope, x. Ignore friction of the rope as it unwinds. Then find the acceleration of the falling rope and the mechanical energy lost from the rope as the end of the rope leaves the table. Note that the rope length is less than the height of the table. 2. Relevant equations F = mA= (mg/L)x 3. The attempt at a solution My thoughts are that since $a(x)=x\frac{g}{l}$, $x(t) = e^{t\sqrt{g/l}}$, so v(t) is just the derivative of that, and $v(x) =x\sqrt{g/l}$ I cannot figure out what I have done wrong up to this point. The problem is that, as you can see, that equation simply leads to KE(gained) = PE(lost). The correct solution starts with: $mg = m\frac{dv}{dt} + vm\frac{dm}{dt}$ which yields $v^{2} = m\frac{2gx}{3}$. which corresponds to a loss $\frac{mgL}{6}$ of mechanical energy. I just don't get it...