The Atwood machine (or Atwood's machine) was invented in 1784 by the English mathematician George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Atwood's machine is a common classroom demonstration used to illustrate principles of classical mechanics. The ideal Atwood machine consists of two objects of mass and , connected by an inextensible massless string over an ideal massless pulley. Both masses experience uniform acceleration. When , the machine is in neutral equilibrium regardless of the position of the weights. Equation for constant acceleration An equation for the acceleration can be derived by analyzing forces. Assuming a massless, inextensible string and an ideal massless pulley, the only forces to consider are: tension force (), and the weight of the two masses ( and ). To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of derive a system of equations for the acceleration (). As a sign convention, assume that a is positive when downward for and upward for . Weight of and is simply and respectively. Forces affecting m1: Forces affecting m2: and adding the two previous equations yields and the concluding formula for acceleration The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion. Equation for tension It can be useful to know an equation for the tension in the string. To evaluate tension, substitute the equation for acceleration in either of the two force equations. For example, substituting into , results in where is the harmonic mean of the two masses. The numerical value of is closer to the smaller of the two masses. Equations for a pulley with inertia and friction For very small mass differences between and , the rotational inertia of the pulley of radius cannot be neglected. The angular acceleration of the pulley is given by the no-slip condition: where is the angular acceleration. The net torque is then: Combining with Newton's second law for the hanging masses, and solving for , , and , we get: Acceleration: Tension in string segment nearest : Tension in string segment nearest : Should bearing friction be negligible (but not the inertia of the pulley nor the traction of the string on the pulley rim), these equations simplify as the following results: Acceleration: Tension in string segment nearest : Tension in string segment nearest