Atwood's machine

An early arrangement for measuring the acceleration of gravity, called Atwood's Machine, is shown in the figure. The pulley P and cord C have negligible mass and friction. The system is balanced with equal masses M on each side as shown (solid line), and then a small rider m is added to one side. The combined masses accelerate through a certain distance h, the rider is caught on a ring and the two equal masses then move on with constant speed, v. Find the value of g that corresponds to the measured values of m, M, h, and v.

Solution by Michael A. Gottlieb

Gravity can do no (net) work on the balanced masses, so the only work done is on the rider, equal to mgh. This work must equal the kinetic energy of the system just before the rider is caught, equal to (2M+m)v²/2 . Thus

        mgh = (2M+m)v²/2 ,

or rearranging,

        g = (2M+m)v² / 2mh .