Title: Atwood machine
Content:   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 m1 and m2, connected by an inextensible massless string over an ideal massless pulley. Both masses experience uniform acceleration. When m1 = m2, the machine is in neutral equilibrium regardless of the position of the weights.
Title: Atwood machine
Content:   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 (T), and the weight of the two masses (W1 and W2). To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of {{nowrap| The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.
Title: Atwood machine
Content:  Atwood's original illustrations show the main pulley's axle resting on the rims of another four wheels, to minimize friction forces from the bearings. Many historical implementations of the machine follow this design. An elevator with a counterbalance approximates an ideal Atwood machine and thereby relieves the driving motor from the load of holding the elevator cab &mdash; it has to overcome only weight difference and inertia of the two masses. The same principle is used for funicular railways with two connected railway cars on inclined tracks, and for the elevators on the Eiffel Tower which counterbalance each other. Ski lifts are another example, where the gondolas move on a closed (continuous) pulley system up and down the mountain. The ski lift is similar to the counter-weighted elevator, but with a constraining force provided by the cable in the vertical dimension thereby achieving work in both the horizontal and vertical dimensions. Boat lifts are another type of counter-weighted elevator system approximating an Atwood machine.
Title: Atwood machine
Content:  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 2 force equations.
Title: Swinging Atwood's machine
Content:   The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions. Specifically, it comprises two masses (the pendulum, mass m and counterweight, mass M) connected by an inextensible, massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight. The conventional Atwood's machine allows only "runaway" solutions (i.e. either the pendulum or counterweight eventually collides with its pulley), except for has a large parameter space of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic, quasiperiodic or chaotic, bounded or unbounded, singular or non-singular  due to the pendulum's reactive centrifugal force counteracting the counterweight's weight. Research on the SAM started as part of a 1982 senior thesis entitled Smiles and Teardrops (referring to the shape of some trajectories of the system) by Nicholas Tufillaro at Reed College, directed by David J. Griffiths.
Title: Swinging Atwood's machine
Content:  Journal of Physics Vol.63(2), p121-126. ; Ouazzani-T.H., A. and Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine," Il Nuovo Cimento B Vol. 110 (9). ; Olivier, Pujol, JP Perez, JP Ramis, C. Simo, S. Simon, JA Weil (2010), "Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study," Physica D 239, pp. 1067–1081. ; Sears, R. (1995) "Comment on "A surprising mechanics demonstration," American Journal of Physics, Vol. 63(9), p854-855. ; Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine", Mechanics Research Communications Vol. 33 (5), p711–716. 
Title: Swinging Atwood's machine
Content:  of Physics Vol.10(10), p173. ; Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," Journal de Physique Vol.51, p1693. ; Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", International Journal of Theoretical Physics, Vol. 27(11), p1405-1410. ; Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" American Journal of Physics Vol.60(10), p951-953. ; Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the Swinging Atwood Machine", Journal of Physics II France 1, p711-715. ; Nunes, A., J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine," 
Title: Swinging Atwood's machine
Content: Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems", Brazilian Journal of Physics Vol.28 n.4 São Paulo Dec. ; Barrera, Emmanuel Jan (2003) Dynamics of a Double-Swinging Atwood's machine, B.S. Thesis, National Institute of Physics, University of the Philippines. ; Babelon, O, M. Talon, MC Peyranere (2010), "Kowalevski's analysis of a swinging Atwood's machine," Journal of Physics A: Mathematical and Theoretical Vol. 43 (8). ; Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators," Physica Scripta Vol.35(1). ; Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," European 
Title: George Atwood
Content: Analysis of a Course of Lectures on the Principles of Natural Philosophy (Cambridge, 1784). ; Treatise on the Rectilinear Motion and Rotation of Bodies (Cambridge, 1784), which gives some interesting experiments, by means of which mechanical truths can be ocularly exhibited and demonstrated, and describes the machine, since named after Atwood, for verifying experimentally the laws of simple acceleration of motion. ; Review of the Statutes and Ordinances of Assize which have been established in England from the 4th year of King John, 1202, to the 37th of his present Majesty (London, 1801), a work of some historical research. ; Dissertation on the Construction and Properties of Arches (London, 1801). 
Title: Swinging Atwood's machine
Content:  If the pulleys in the system are taken to have moment of inertia.

Question: what is a real world application of an atwood machine?
Answer: An elevator with a counterbalance.