Poggendorff in 1853 designed an apparatus to demonstrate the changes in the tensions in Atwood's Machine that result from the acceleration of the masses. Mach gives a brief description in his Science of Mechanics. A compound Atwood's Machine is easier to analyze since the suspension points from the upper pulley don't change while the ends of the beam in Poggendorff's apparatus do. Only two parameters are required to specify the configuration of the compound machine, the rotation angles of the two pulleys or the arcs s ant t.
The positions of the masses are specified by the vertical coordinates, xi. From the balance of forces for each mass we obtain expressions relating the tensions and torques to the two accelerations, s̈ and ẗ.
One can then find the formulas for the individual tensions and by using the difference in tensions acting on the two pulleys we can derive a pair of linear equations for the unknown accelerations.
If we have cylindrical pulleys we can relate their moment of inertia to their masses as follows.
We are now in a position to compute the accelerations and tensions for the compound pulley by solving the system of linear equations.
All the tensions are reduced except for that of the first mass when the masses are accelerating.