I have seen an interesting problem as follows:
Both ends of a rope with the mass of $\rho$ kg/m and length of $L$ are attached to a horizontal surface (see photo attached). At $t=0$, one end of the rope detaches from the surface and starts falling vertically. When this end falls the height $X$, what is the tension on the other end of the rope?
I also have seen two different solutions from a professor which result in two different answers!
Approach 1 is based on falling part of the rope is freely falling (acceleration $= g$).
Approach 2 is based on energy conservation $KE + PE = $ Constant (no loss).
Now I would like to ask
1- Is the falling part of the rope really freely falling?
2- Is energy conserved when the rope is falling (for $X \lt L$)?