We are going to use the mechanism (figure below) of a disk with radius L rotating on a fixed axis through its center and a massless link of length L connected to a massless piston. The problem we will look at is that for a constant moment (torque), M₀, applied to the disk, find the differential equation of motion for theta and the expression for x as a function of theta. No damping or friction exists between the piston and the walls in this problem.   However, there is gas sealed in the cylinder which compresses from an original volume of 4LA where A is the cross-sectional area of the cylinder.  To find the pressure in the cylinder and therefore the force on the piston, we will use the ideal gas law, PV=nRT, where P is the pressure in the cylinder, V is the volume of the cylinder, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin which we will assume remains constant.  Recall that P=F/A when setting up the equations.

Extra Problem: derive the equations of motion for this system by drawing free body diagrams for each of the three bodies in this problem.  Assume the disk has mass, m, and the cross-sectional area of the cylinder is A (this will not appear in your final equation of motion).  Also, so there is no confusion, assume that the cylinder is not in the same plane as the disk, i.e. the piston is free to travel the entire length of the cylinder.   The results here will be used later in a computer simulation.