Atwood machine. We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inelastic string and an ideal massless pulley the only forces we have to consider are: tension force (N), and the weight of the two masses (mg). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton's laws (if m 1 > m 2 ) we can derive a system of equations for the acceleration (a). Forces affecting m 1 : forces affecting m 2 : and adding the two previous equations we obtain, and at last,

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. Lagrangian method It can be useful to know an equation for the tension in the string. To evaluate tension we substitute the equation for acceleration in either of the 2 force equations. For example substituting into m 1 a = N − m 1 g, we get The tension cannot accurately be found in using this method due to torque of the pulley..

Atwood Example: Using an Atwood machine m 1 = 4kg m 2 = 3.5kg Find the acceleration of the setup and the tension in the wire.

Answer: a = 9.81 m/s 2 (4kg – 3.5 kg) (4kg kg) a =.654 m/s 2 N = 9.81 m/s 2 2(4kg)(3.5 kg) (4kg kg) N = N