On Atwood's machine with a nonzero mass string

Let us consider a classical high school exercise concerning two weights on a pulley and a string, illustrated in Fig. 1(a). A system like this is called an Atwood's machine and was invented by George Atwood in 1784 as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Nowadays, Atwood's machine is used for didactic purposes to demonstrate uniformly accelerated motion with acceleration arbitrarily smaller than the gravitational acceleration g. The simplest case is with a massless and frictionless pulley and a massless string. With little effort one can include the mass of the pulley in calculations. The mass of a string has been incorporated previously in some considerations and experiments. These include treatments focusing on friction, justifying the assumption of a massless string, incorporating variations in Earth's gravitational field, comparing the calculated value of g based on a simple experiment, taking the mass of the string into account in such a way that the resulting acceleration is constant, or in one exception solely focusing on a heavy string, but with a slightly different approach. Here we wish to provide i) a derivation of the acceleration and position dependence on the weights' masses based purely on basic dynamical reasoning similar to the conventional version of the exercise, and ii) focus on the influence of the string's linear density, or equivalently its mass, on the outcome compared to a massless string case