The linear mass density, i.e., mass per unit length of the rope shown in fig. varies 0 to $\mathrm{\lambda }$ 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

$\begin{array}{lll} & {\mathrm{\lambda }}_{\mathrm{x}}& =\frac{\mathrm{\lambda }}{\mathrm{L}}\mathrm{x}\\ \therefore & {\mathrm{d}}_{\mathrm{m}}& ={\mathrm{\lambda }}_{\mathrm{x}}\mathrm{dx}=\frac{\mathrm{\lambda }}{\mathrm{L}}\mathrm{xdx}\\ \mathrm{So}, & \mathrm{m}& ={\int }_0^{\mathrm{L}}\frac{\mathrm{\lambda }}{\mathrm{L}}\mathrm{xdx}=\frac{\mathrm{\lambda }}{\mathrm{L}}{\left[\frac{{\mathrm{x}}^2}{2}\right]}_0^{\mathrm{L}}=\frac{\mathrm{\lambda L}}{2}\end{array}$

If a be the combined acceleration, then

$\mathrm{a}=\frac{\mathrm{F}}{\mathrm{M}+\mathrm{m}}=\frac{\mathrm{F}}{[\mathrm{M}+(\mathrm{\lambda L}/2\left)\right]}$.