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 Rudy Arthur & Nishit

Calling the acceleration of the masses a and the tension on the cord T, Newton's 2nd law dictates that on the left side of the pulley,

        (M+m)g - T = (M+m)a,

while on the right side,

        T - Mg = Ma.

Eliminating T gives,

        a = mg / (2M+m).


        v2 = 2ah = 2 (mg/(2M+m)) h,

or rearranging,

        g = (2M+m)v2 / 2mh.