The linear mass density, i.e., mass per unit length of the rope shown in fig. varies 0 to λ from one end to another. The acceleration of the combined system of figure will be

Consider a small mass dm of the rope at a distance x and x + dx as shown in fig.Linear mass density at x is given by  λx=λLx∴dm=λxdx=λLxdxSo, m=∫0LλLxdx=λL[x22]0L=λL2If a be the combined acceleration, thena=FM+m=F[M+(λL/2)].