Atwood machine

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.¹

Both masses experience uniform acceleration. When m₁ = m₂, the machine is in neutral equilibrium regardless of the position of the weights.

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 ${\displaystyle m_{1}>m_{2}}$) derive a system of equations for the acceleration (a).

As a sign convention, assume that a is positive when downward for ${\displaystyle m_{1}}$ and upward for ${\displaystyle m_{2}}$. Weight of ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ is simply ${\displaystyle W_{1}=m_{1}g}$ and ${\displaystyle W_{2}=m_{2}g}$ respectively.

Forces affecting m₁: $${\displaystyle m_{1}g-T=m_{1}a}$$ Forces affecting m₂: $${\displaystyle T-m_{2}g=m_{2}a}$$ and adding the two previous equations yields $${\displaystyle m_{1}g-m_{2}g=m_{1}a+m_{2}a,}$$ and the concluding formula for acceleration $${\displaystyle a=g{\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}}$$

The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.²