Illustration of Atwood machine, 1905.
The Atwood machine (or Atwood's machine) was invented in 1784 by Rev. 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 m_{1} and m_{2}, connected by an inextensible massless string over an ideal massless pulley. ^{[1]}
When m_{1} = m_{2}, the machine is in neutral equilibrium regardless of the position of the weights.
When m_{1} m_{2} both masses experience uniform acceleration.
Equation for constant acceleration
The free body diagrams of the two hanging masses of the Atwood machine. Our sign convention, depicted by the acceleration vectors is that m_{1} accelerates downward and that m_{2} accelerates upward, as would be the case if m_{1} > m_{2}
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 (W_{1} and W_{2}). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of m_1>m_2) we can derive a system of equations for the acceleration (a).
As a sign convention, we assume that a is positive when downward for m_1, and that a is positive when upward for m_2. Weight of m_1 and m_2 is simply W_1 = m_1 g and W_2 = m_2 g respectively.
Forces affecting m_{1}:
\; m_1gT=m_1a
Forces affecting m_{2}:
\; Tm_2g=m_2a
and adding the two previous equations we obtain
\; m_1gm_2g=m_1a+m_2a,
and our concluding formula for acceleration
a = g{m_1m_2 \over m_1+m_2}
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: d = {1 \over 2} at^2 .
The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion. ^{[2]}
Equation for tension
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.
a = g{m_1m_2 \over m_1 + m_2}
For example substituting into m_1 a = m_1 gT, we get
T={2 g m_1 m_2 \over m_1 + m_2}
The tension can be found in using this method.
Equations for a pulley with inertia and friction
For very small mass differences between m_{1} and m_{2}, the rotational inertia I of the pulley of radius r cannot be neglected. The angular acceleration of the pulley is given by the noslip condition:
\alpha = {a\over r},
where \alpha is the angular acceleration. The net torque is then:
\tau_{net}=\left(T_1  T_2 \right)r  \tau_{friction} = I \alpha
Combining with Newton's second law for the hanging masses, and solving for T_{1}, T_{2}, and a, we get:
Acceleration:
 a = \over {m_1 + m_2 + }}
Tension in string segment nearest m_{1}:
 T_1 = + \over {r g}})} \over {m_1 + m_2 + }}
Tension in string segment nearest m_{2}:
 T_2 = + \over {r g}})} \over {m_1 + m_2 + }}
Should bearing friction be negligible (but not the inertia of the pulley and not the traction of the string on the pulley rim), these equations simplify as the following results:
Acceleration:
 a = }}
Tension in string segment nearest m_{1}:
 T_1 = )} \over {m_1 + m_2 + }}
Tension in string segment nearest m_{2}:
 T_2 = )} \over {m_1 + m_2 + }}
Practical implementations
Atwood's original illustrations show the main pulley's axle resting on the rims of another four wheels, to minimise 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 — 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.
See also
Notes

↑ Chapter 6, example 613, page 160.

↑ Section 16, example 2, pages 2627.
ca:M quina d'Atwood de:Atwoodsche Fallmaschine es:M quina de Atwood fr:Machine d'Atwood it:Macchina di Atwood nl:Toestel van Atwood pl:Spadkownica Atwooda pt:M quina de Atwood ru: zh:
