The Atwood machine: measuring gravity in the living room

The Reverend George Atwood (1745–1807), fellow of Trinity College, Cambridge, invented his celebrated machine in 1784 to solve a practical and frustrating problem: free fall is too fast to be measured accurately with the stopwatches of the time. In a few tenths of a second a body falls metres, and there was no way to follow its motion instant by instant.

Atwood’s idea is elegant. Two nearly equal masses, $m_1$ and $m_2$, connected by a rope running over a pulley: the system behaves like a single body with acceleration

$a = \frac{m_2 - m_1}{m_1 + m_2}\,g$

If the two masses are nearly equal, the numerator $m_2 - m_1$ is small while the denominator $m_1 + m_2$ stays large: the acceleration becomes as small as desired. Gravity is “diluted” and the motion slowed down to the point where it can be timed by eye. Atwood was thus able to measure $g$ with remarkable accuracy for the period (Bertoloni Meli 2006, ch. 3).

The key idea

The Atwood machine does not change the physics: it changes the time scale. By adjusting the difference between the masses one chooses how slowly the experiment unfolds, turning an impossible measurement into a convenient one.

The numerical check of how it works (a cat and a giraffe connected by a rope) is worked out in The Atwood machine; the analysis of the force on the pulley axle is in The Atwood pulley: the force on the axle.

Links

Topics: Dinamica Concepts: Tensione · Seconda legge di Newton Objects: Macchina di Atwood · Puleggia

Related exercises: Inclined plane with pulley and hanging mass · The Atwood pulley: the force on the axle · Two masses on a triangular plane