Atwood's machine

  Atwood's machine gives a unique insight into forces acting on a body. One can find a description of it's use in Comstock's System of Natural Philosophy. The machine consists of two weights hanging from a pulley. One only needs one variable, s, to represent the position of the apparatus.

The value s is the distance along the perimeter of the pulley corresponding to its rotation through an angle θ. From the formulas for the positions we can express the acceleration of the masses and ignoring the inertia of the pulley we can look at the balance of forces acting on the masses due to gravity, the tension in the string and the "inertial force" associated with resistance of a mass to a change in its state of motion. After solving for the acceleration of the pulley we can obtain the tension in the string.

The balance forces acting on the pulley gives us the tension in the line holding it in position. The net force acting on the apparatus is the difference in gravitational force and an inertial factor equal to the sum of the masses and more generally including the moment of inertia of the pulley. The tension in the string depends on the distribution of the masses and is maximum when both masses are equal and the system is in equilibrium.

The units of the vertical scale are T/g and the horizontal scale is the ratio of one mass to their sum.

A difference in tension would be needed to accelerate the pulley so in general one cannot say that the tension in the string on both sides of the pulley would be the same.