A heavy chain with a mass per unite length `p` is pulled by the constant force `F` along a horizental 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 `mu_(k)` determine the velocity `v` of the chain when `x = L` . The force `F` is hreater than `mu_(k) ogL` in under to initate the motion ,

Explanation

`nu = (dnu)/(x) = ("Net force")/("mass") = (F - mu_(k)rho (L - x) g)/(rho L)`
`:. int_(0)^(nu) nu dnu = int_(0)^(L) (F - mu_(k) rho (L - x) g)/(rho L) dx`
`:. (nu^(2))/(2) = (F)/(rho) - mu_(k) g L + (mu_(k) g L)/(2)`
`:. nu = sqrt((2 F)/(rho) - mu_(k) g L)`