A heavy chain with a mass ρ per unit length is pulled by the constant force P along a ho...

A heavy chain with a mass ρ per unit length is pulled by the constant force P along a horizontal surface consisting of a smooth section and a rough section. The chain is initially at rest on the rough surface with x = 0. If the coefficient of kinetic friction between the chain and the rough surface is μ_k, determine the velocity v of the chain when x = L. The force P is greater than μ_kρgL in order to initiate motion.