The Atwood machine is a simple device which was invented by Rev. George Atwood in 1784 to illustrate the dynamics of Newton's laws. It consists of a massless, inextensible string which connects two masses, ${m}_{1}$ and ${m}_{2}$ through an ideal pulley. (An ideal pulley is one which is assumed to have negligible mass and no friction between itself and the string). A straightforward application of Newton's laws can predict the acceleration of the blocks and the time it takes for them to reach the ground.
The mass on the left, ${m}_{1}$, experiences two forces: ${m}_{1}g$ from gravity, and $T$, the tension force from the rope. Newton's second law for ${m}_{1}$ then states that:
${m}_{1}{a}_{1}\={m}_{1}g\+T\,$
where ${a}_{1}$is the acceleration of ${m}_{1}$ in the upwards direction. The second mass, ${m}_{2}$, experiences a net force of:
${m}_{2}{a}_{2}\={m}_{2}g\+T\.$
where ${a}_{2}$ is the acceleration of ${m}_{2}$. Notice that in order for the rope to maintain its total length, the accelerations must be equal and opposite, hence ${a}_{1}\={a}_{2}$. Now subtracting the second equation from the first equation gives:
${m}_{1}{a}_{1}{m}_{2}{a}_{2}\={m}_{1}g\+{m}_{2}g\.$
Finally, making the substitution ${a}_{1}\={a}_{2}$ and rearranging for ${a}_{1}$ yields:
${a}_{1}\=g\cdot \frac{{m}_{2}{m}_{1}}{{m}_{2}\+{m}_{1}}$.
Note that if ${m}_{2}\>{m}_{1}$, the first mass will accelerate upwards and ${m}_{2}$ will accelerate downwards, and if ${m}_{1}\>{m}_{2}$ the opposite will happen. In an experiment, you could adjust the masses and measure the time it takes for a mass to reach the ground. The time is given by the solution to the equation $0equals;{y}_{0}plus;a{t}^{2}$, and this provides a way to determine $a$ and ultimately find $g$. By choosing blocks with very similar masses, the acceleration is much slower and so air resistance is less important, thereby giving a more accurate method of computing $g$. (On the other hand, the assumption of a frictionfree rope becomes untenable if the masses are too similar.)
