Atwood machine

Illustration of the Atwood machine, 1905.

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.

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 m₁ accelerates downward and that m₂ accelerates upward, as would be the case if m₁ > m₂

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 $m_1>m_2$) derive a system of equations for the acceleration (a).

As a sign convention, assume that a is positive when downward for $m_1$ and 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 m₁: $$m_{1}g-T=m_{1}a$$ Forces affecting m₂: $$T-m_{2}g=m_{2}a$$ and adding the two previous equations yields $$m_{1}g-m_{2}g=m_{1}a+m_{2}a,$$ and the concluding formula for acceleration $$a=g\frac{m_1-m_2}{m_1+m_2}$$

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

See also

• Kater's pendulum – Reversible free swinging pendulum
• Spherical cow – Humorous concept in scientific models
• Swinging Atwood's machine – Variation of Atwood's machine incorporating a pendulum

External links

• A treatise on the rectilinear motion and rotation of bodies; with a description of original experiments relative to the subject by George Atwood, 1764. Drawings appear on page 450.
• Professor Greenslade's account on the Atwood Machine
• Atwood's Machine by Enrique Zeleny, The Wolfram Demonstrations Project