Speed of a rope sliding off a peg?

A flexible rope of mass m and length L = L1 + L2 hangs over a frictionless peg. What is the speed of the rope when it just slides off the peg? (it is reasonable to assume that the rope is uniform and that the masses of each section 1 and 2 are concentrated at the midpoint)

In the picture L1 is shorter than L2. I set the change in kinetic energy equal to the negative change in gravitational potential energy, with the final height (of the center of mass of the rope) being L/2 and the initial height being L2/2 + L1:

(1/2)mv^2 = mg(L/2 – L2/2 – L1)
The masses cancel and I got that v = √(gL1)

I’m just asking about this because setting the heights right in terms of L, L1 and L2 is slightly confusing, and the question seems to ask for an actual speed but I think you can only find it in terms of the length.


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