Atwood Machine - Equation For Constant Acceleration

Equation For Constant Acceleration

We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inextensible string and an ideal massless pulley, the only forces we have to consider are: tension force (T), and the weight of the two masses (W₁ and W₂). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of ) we can derive a system of equations for the acceleration (a).

As a sign convention, we assume that a is positive when downward for, and that a is positive when upward for . Weight of and is simply and respectively.

Forces affecting m₁:

Forces affecting m₂:

and adding the two previous equations we obtain

,

and our concluding formula for acceleration

Conversely, the acceleration due to gravity, g, can be found by timing the movement of the weights, and calculating a value for the uniform acceleration a: .

The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.