A heavy chain with a mass per unit length ? is pulled by the constant force F 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 rough surface is µ , determine the velocity of the chain when x=L . I am applying work energy theorem . Work done by constant Force will be Force × displacement of centre of mass i.e FL but not able to find work done by friction .The friction force at an instant when chain length x lies on the rough surface should be µ?xg.This force is continuously decreasing .i feel calculus is involved here but i am unable to apply it.Please help me .thanks.

A heavy chain with a mass per unit length ? is pulled by the constant force F 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 rough surface is µ , determine the velocity of the chain when x=L .

I am applying work energy theorem . Work done by constant Force will be Force × displacement of centre of mass i.e FL but not able to find work done by friction .The friction force at an instant when chain length x lies on the rough surface should be µ?xg.This force is continuously decreasing .i feel calculus is involved here but i am unable to apply it.Please help me .thanks.